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-A Comprehensiue Introduction to . '

DIFFERENTIAL GEOMETRY

VOLUME ONE Third Edition

MICHAEL SPIVAK

PUBLISH OR PERISH, INC.

Houston, Texas 1999

ACKNOWLEDGEMENTS 1 am greatly indebted to

without his encouragement these volumes would have remained a short set of mimeographed notes and

without his TE/X program they would never have become typeset books

PREFACE

T

he preface to the first edition, reprinted on the succeeding pages, excused this book's deficiencies on grounds that can hardly be justified now that these "notes" tmily have become a book. At one time I had optimisticaliy planned to completely revise ali this material for the momentous occasion, but I soon realized the futility of such an undertaking. As I examined these five volumes, written so many years ago, I could scarcely believe that I had once had the energy to learn so much material, or even recall how I had unearthed some of it. So I have contented myself with the correction of errors brought to my attention by diligent readers, together with a few expository ameliorations; among these is the inclusion of a translation of Gauss' paper in Volume 2. Aside from that, this third and final edition diffiersfrom the previous ones only in being typeset, and with figures redrawn. I have merely endeavored to typeset these books in a manner befitting a subject of such importante and beauty As a final note, it should be pointed out that since the first volumes of this series made their appearance in 1970, references in the text to "recent" results should be placed in context.

HOW THESE NOTES CAME TO BE

m d how they d i d not come t o be a book For many years 1 have wanted t o w r i t e t h e Great American D i f f e r e n t i a l Geometry book. Today a d i l e m a c o n f r o n t s any one i n t e n t on p e n e t r a t í n g t h e mysteries of d i f f e r e n t i a l geometry. On t h e one hand, one can consult numerous c l a s s i c a l t r e a t m e n t s of t h e s u b j e c t i n an attempt t o f orm some i d e a how t h e concepts w i t h i n it developed. Unf o r t u n a t e l y , a modern mathemat i c a l education t e n d s t o make c l a s s i c a l mathematical works i n a c c e s s i b l e , p a r t i c u l a r l y t h o s e i n d i f f e r e n t i a l g e o m e t r y . Onthe o t h e r hand, one can now f i n d t e x t s a s modern i n s p i r i t , and a s c l e a n i n e x p o s i t i o n , a s Bourbaki's Algebra.

But a thorough study of t h e s e books

u s u a l l y l e a v e s one unprepared t o c o n s u l t c l a s s i c a l works, and e n t i r e l y ignorant of t h e r e l a t i o n s h i p between e l e g a n t modern c o n s t r u c t i o n s and t h e i r c l a s s i c a l c o u n t e r p a r t s . Most s t u d e n t s e v e n t u a l l y f i n d t h a t t h i s ignorance of t h e r o o t s of t h e s u b j e c t has i t s p r i c e

-- no one d e n i e s t h a t

modern d e f i n i t i o n s a r e c l e a r , e l e g a n t , a n d p r e c i s e ; i t ' s j u s t t h a t i t ' s impossible t o comprehend how any one e v e r thought of them, And even a f t e r one does master a modern treatment of d i f f e r e n t i a l geometry , o t h e r modern t r e a t m e n t s o f t e n appear simply t o b e a b o u t t o t a l l y d i f f e r e n t s u b j e c t s . Of course, t h e s e remarks merely mean t h a t no matter how w e l l some of t h e present day t e x t s achieve t h e i r o b j e c t i v e , 1n e v e r t h e l e s s f e e l t h a t an i n t r o d u c t i o n t o d i f f e r e n t i a l g e o m e t r y ought t o have q u i t e d i f f e r e n t aims. There a r e two main premises on which t h e s e n o t e s a r e based. The f i r s t premise i s t h a t i t i s a b s u r d l y i n e f f i c i e n t t o eschewthe modern language of manifolds, bundles, forms, e t c . , whichwas d e v e l o p e d p r e c i s e l y i n order t o r i g o r i z e t h e concepts of classicaldifferentialgeometry. Rephrasing every t h i n g i n more element a r y t e m s involves i n c r e d i b l e

c o n t o r t i o n s which a r e not only unnecessary , but misleading. The work of Gauss , f o r example, which uses i n f i n i t e s i m a l s throughout , i s most n a t u r a l l y rephrased i n terms of d i f f e r e n t i a l s , even i f it is p o s s i b l e t o r e w r i t e i t i n terms of d e r i v a t i v e s . For t h i s reason, t h e e n t i r e f i r s t volume of t h e s e notes is devoted t o t h e theory of d i f f e r e n t i a b l e manifolds, t h e basic language of modern d i f f e r e n t i a l geometry. This language i s compared whenever possible with t h e c l a s s i c a l l a n p a g e , so t h a t c l a s s i c a l works can then be read. The second premise f o r t h e s e notes i s t h a t i n order f o r a n i n t r o d u c t i o n t o d i f f e r e n t i a l geometry t o expose t h e geometric aspect of t h e s u b j e c t , an h i s t o r i c a l approach is necessary; t h e r e i s no point i n introducing t h e curvature t e n s o r without explaining how i t was invented and what it has t o do with curvature. 1 personally f e l t t h a t 1 could never acquire a s a t i s f a c t o r y understanding of d i f f e r e n t i a b l e geometry u n t i l 1 read t h e o r i g i n a l works. The second volume of t h e s e notes g i v e s a d e t a i l e d exposition of t h e fundamental papers of Gauss and Riemann. Gauss' work i s now a v a i l a b l e i n E n g l i s h (General I n v e s t i g a t i o n s of Curved Surfaces;

Raven P r e s s ) . There a r e a l s o two E n g l i s h t r a n s l a t i o n s of Riemann's work, but 1 have provided a (very f r e e ) t r a n s l a t i o n i n t h e second volume . Of course, 1 do not t h i n k t h a t one should f ollow a l 1 t h e i n t r i c a c i e s of t h e h i s t o r i c a l process, with its i n e v i t a b l e d u p l i c a t i o n s and f a l c e l e a d s What i s intended, r a t h e r , is a p r e s e n t a t i o n of t h e s u b j e c t along t h e l i n e s which i t s development might have followed; a s Bernard Morin s a i d t o me, t h e r e is no reason, i n mathematics any more than i n biology, why ontogeny must r e c a p i t u l a t e phylogeny. When modern terminology f i n a l l y i s introduced, i t should be a s an outgrowth of t h i s (mythical) h i s t o r i c a l

development, And a l 1 t h e major approaches have t o be presented, f o r they were a l 1 r e l a t e d t o each o t h e r , and a l 1 s t i l l play an important r o l e .

A t t h i s point 1 am reminded of a paper described i n Littlewood'

S

Mathematician' S M i s c e l l a n y . The paper began "The aim of t h i s paper i s t o prove

. .." and

it t r a n s p i r e d only much l a t e r t h a t t h i s aim was not

achieved (the author h a d n ' t claimed t h a t it was)

.

What 1 have outlined

above i s t h e content of a book t h e r e a l i z a t i o n of whose b a s i c p l a n and t h e incorporation of whose d e t a i l s would perhaps be impossible ; what 1 have w r i t t e n i s a second or t h i r d d r a f t of a preliminary version of t h i s book, 1 have had t o r e s t r i c t myself t o what 1 could w r i t e and l e a r n about within

t h e present academic year, and a l l r e v i s i o n s and c o r r e c t i o n s have h a d t o be made within t h i s same period of time. Although 1 may some day be a b l e t o devote t o i t s completionthe time which s u c h a n u n d e r t a k i n g deserves, a t p r s s e n t 1 have no plans f or t h i s . Consequent l y , 1 would l i k e t o make t h e s e n o t e s a v a i l a b l e now, d e s p i t e t h e i r d e f i c i e n c i e s , a n d w i t h a l 1 t h e compromises 1 learned t o make i n t h e e a r l y hours of t h e morning. These notes were w r i t t e n while 1 was teaching a year course i n d i f f e r e n t i a l g e o m e t r y a t Brandeis University, during t h e academic year 1969-70. The course was taken by s i x juniors and s e n i o r s , and audited by a few graduate s t u d e n t s . Most of them were f a m i l i a r with t h e m a t e r i a l i n Calculus on Manifolds, which i s e s s e n t i a l l y regarded a s a p r e r e q u i s i t e . More p r e c i s e l y , t h e complete p r e r e q u i s i t e s a r e advanced c a l c u l u s using l i n e a r algebra and a b a s i c knowledge of metric spaces. An acquaintance with t o p o l o g i c a l spaces i s even b e t t e r , s i n c e it allows one t o avoid t h e t e c h n i c a l t r o u b l e s which a r e sometimes r e l e g a t e d t o t h e Problems, but 1 t r i e d hard t o make everything work without i t . The m a t e r i a l i n t h e p r e s e n t v o l u m e was covered i n t h e f i r s t t e r m , except f o r Chapter 10, which occupied t h e f irst couple of weeks of t h e second term, and Chapter 1 1 , which was not covered i n c l a s s a t a l l . We f ound it necessary t o t ake r e s t cures of n e a r l y a week af t e r completing Caapters 2 , 3, and 7 . The same m a t e r i a l could e a s i l y be expanded t o a f u11 year course

i n manifold theory with a pace t h a t few would describe a s excessively leisurely

.

1 am gratef u1 t o t h e c l a s s f ú r keeping up w i t h my accelerated

pace, f o r otherwise the second half of these notes would not have been w r i t t e n . 1 am a l s o extremely g r a t e f u l t o Bichard P a l a i s , whose expert knowledge saved me innumerable hours of labor.

TABLE OF CONTENTS Although the chapters are not divided into sections. the listing for each chapter @ves some indication which topics are treated. and on what pages.

CHAPTER 1 . MANIFOLDS Elementary properties of manifolds . . . . . . . . . . . . . . . . 1 Exarnples of manifolds . . . . . . . . . . . . . . . . . . . . . 6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 20 CHAPTER 2 . DIFFERENTI AL STRUCTURES

Cm structures . . . Cm functions . . . Partial derivatives . Critica1 points . . . Immersion theorems Partitions of unity . Problems . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27 31 35 40 42 . . . . . . . . . . . . . . . . . . . . . 50 . . . . . . . . . . . . . . . . . . . . . 53

CHAPTER 3. T H E TANGENT BUNDLE The tangent space of IRn . . . . . . . . . Tlle tangent space of an imbedded manifold . Vector bundles . . . . . . . . . . . . . . The tangent bundle of a manifold . . . . . Equivalence classes of curves. and derivations Vector fields . . . . . . . . . . . . . . . Orieil tation . . . . . . . . . . . . . . . Addendum . Equivalence of Tangent Bundles Problems . . . . . . . . . . . . . . . .

xiu CHAPTER 4. TENSORS The dual bundle . . . . . . . . . The difkrential of a function . . . Classical versus modern terminology Mul tilinear functions . . . . . . . Covariant and contravariant tensors Mixed tensors. and contraction . . Problems . . . . . . . . . . . .

. . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107 109 111 115 . . . . . . . . . . . . . . 117 . . . . . . . . . . . . . . 121 . . . . . . . . . . . . . . 127

CHAPTER 5. VECTOR FIELDS AND DIFFERENTIAL EQUATIONS Integral curves . . . . . . . . . . . . . . . . . . . . . . . . Existence and uniqueness theorems . . . . . . . . . . . . . . . The local flow . . . . . . . . . . . . . . . . . . . . . . . . One-parameter groups of difkomorphisms . . . . . . . . . . . Lie derivatives . . . . . . . . . . . . . . . . . . . . . . . . Brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . Addendum 1. Dserential Equations . . . . . . . . . . . . . . Addendum 2. Parameter Curves in Two Dimensions . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . .

135 139 143 148 150 153 164 167 169

CHAPTER 6. INTEGRAL MANIFOLDS Prologue; classical integrability theorems . . Local Theory; Frobenius integrability theorem Global Theory . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . .

. . . . . . . . . . 179 . . . . . . . . . . 190 . . . . . . . . . . 194

. . . . . . . . . . 198

CHAPTER 7 . DIFFERENTIAL FORMS Alternating functions . . . . . . . . . . . . . . . . . . . . . 201 The wedge product . . . . . . . . . . . . . . . . . . . . . . 203 Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . .207 DSerential of a form . . . . . . . . . . . . . . . . . . . . . 210 Frobenius integrability theorem (secondversion) . . . . . . . . . 215 Closed and exact forms . . . . . . . . . . . . . . . . . . . . 218 Tlle Poincaré Lemma . . . . . . . . . . . . . . . . . . . . . 225 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 227

CHAPTER 8. INTEGRATION Classical line and surface integrals Integrals over singular k-cubes . . The boundary of a chain . . . . Stokes' Theorem . . . . . . . . Integrals over manifolds . . . . . Volume elements . . . . . . . . Stokes' Theorem . . . . . . . . de Rliarn cohomology . . . . . Problems . . . . . . . . . . . CHAPTER 9. RIEMANNIAN METRICS Inner products . . . . . . . . . . . . . . . . . . . . . . . . 301 Riemannian metrics . . . . . . . . . . . . . . . . . . . . . 308 LRngth of curves . . . . . . . . . . . . . . . . . . . . . . . 312 The calculus of variations . . . . . . . . . . . . . . . . . . . 316 The First Variation Formula and geodesics . . . . . . . . . . . 323 The exponential map . . . . . . . . . . . . . . . . . . . . . 334 Geodesic com ple teness . . . . . . . . . . . . . . . . . . . . 341 Addendum . Tubular Neighborhoods . . . . . . . . . . . . . . 344 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 348 CHAPTER 1 O . LIE GROUPS Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . 371 L f t invariant vector fields . . . . . . . . . . . . . . . . . . . 374 Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . 376 Subgroups and su balgebras . . . . . . . . . . . . . . . . . . 379 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . 380 One-parame ter subgroups . . . . . . . . . . . . . . . . . . . 382 The exponential map . . . . . . . . . . . . . . . . . . . . . 384 Closed subgroups . . . . . . . . . . . . . . . . . . . . . . 391 LRft invariant forms . . . . . . . . . . . . . . . . . . . . . 394 Bi-invariant metrics . . . . . . . . . . . . . . . . . . . . . . 400 The equations of structure . . . . . . . . . . . . . . . . . . . 402 Probleins . . . . . . . . . . . . . . . . . . . . . . . . . 406

Conlents

xuz

CHAPTER 1 1. EXCURSION IN T H E REALM OF ALGEBRAIC TOPOLOGY Complexes and exact sequences . . . . . . . . . . . . . . . . 419 The Mayer-Vietoris sequence . . . . . . . . . . . . . . . . . 424 Triangulations . . . . . . . . . . . . . . . . . . . . . . . . 426 The Euler characteristic . . . . . . . . . . . . . . . . . . . .428 Mayer-Vietoris sequence for compact supports . . . . . . . . . . 430 The exact sequence of a pair . . . . . . . . . . . . . . . . . 432 Poincaré Duality . . . . . . . . . . . . . . . . . . . . . . . 439 The Thom class . . . . . . . . . . . . . . . . . . . . . . . 442 Index oí' a \lector field . . . . . . . . . . . . . . . . . . . . .446 Poincaré-Hopf Theorem . . . . . . . . . . . . . . . . . . . 450 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 453 APPENDIX A To Chapter I . Problems . . . To Chapter 2 . Problems . . . To Chapter 6 . To Chapters 7.9. Problem . . . .

. . . . . . . . . . . . . . . . . . . . . . . 459 . . . . . . . . . . . . . . . . . . . . . . . 467 . . . . . . . . . . . . . . . . . . . . . . . 471 . . . . . . . . . . . . . . . . . . . . . . -472 . . . . . . . . . . . . . . . . . . . . . . . 473 10 . . . . . . . . . . . . . . . . . . . . . 474 . . . . . . . . . . . . . . . . . . . . . . -475

NOTATlON INDEX . . . . . . . . . . . . . . . . . . . . . . . 477 INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . 481

A Comprehemive Introduction to

DIFFERENTIAL GEOMETRY

VOLUME ONE

CHAPTER 1 MANIFOLDS

T

he nices t example of a metric space is Euclidean n-space R", consisting of al1 n-tuples x = ( x l , . . . ,Y) with each x' E IR, where R is the set of real numbers. Whenever we speak of R" as a metric space, we s h d assume that it has the "usual metric"

unless anotlier metric is explicitly suggested. For n = O we will interpret R0 as the single point O E R. A manifold is supposed to be "locally" like one of these exemplary metric spaces R". To be precise, a manifold is a metric space M with the following proper ty: If x E M, then there is some neighborhood U of x and some integer n 2 O such that U is homeomorphic to R". The simplest example of a manifold is, of course, just Rn itself; for each x E R" we can take U to be ali of R". Clearly, R" supplied with an equivalent metric (one which makes it homeomorphic to R" with the usual metric), is also a manifold. Indeed, a hasty recollection of the definition shows that anything l-iomeomorphic to a manifold is also a manifold-the specific metric with which M is endowed plays almost no role, and we s h d alrnost never mention it. [If you know anything about topological spaces, you can replace "metric space7'by "topological space" in our definition; this new definition allows some pathological creatures which are not metrizable and which fad to have other properties one might carelessly assume must be possessed by spaces which are locally so nice. Appendix A contains remarks, supplementing various chapters, which should be consulted if one allows a manifold to be non-metrizable.] The second simplest example of a manifold is an open bali in R"; in this case we can take U to be the entire open ball since an open ball in R" is homeomorphic to R". This example immediately suggests the next: any open

subset V of R" is a manifold-for each x E V we can choose U to be some open ball with x E U c V. Exercising a mathematician's penchant for generalization,

we immediately announce a proposition whose proof is left to the reader: An open subset of a manifold is also a manifold (calIed, quite naturally, an open submanifold of the original manifold). The open subsets of R" already provide many dXerent examples of manifolds Cjusi 11ow maiiy is the subject of Problem 24), though by no means all. Before proceeding to examine other examples, which constitute most of this chapter, some preliminary remarks need to be made. If x is a point of a manifold M, and U is a neighborhood of x (U contains some open set V with x E V) which is homeomorphic to RR by a homeomorphism #: U + R",then #(V) c R" is an open set containing #(x). Conse-

quently, there is an open ball W with #(x) E W c $(V). Thus x E #-'(w) c V c U. Since # : V + R" is continuous, the set #-'(W) is open in V, and thus open in M; it is, of course, homeomorphic to W, and thus to R". This complicated little argumentjust shows tliat we can always choose the neighborhood U in our definition to be an open neighborhood.

Witli a little tliought, it begns to appear that, ii-i fact, U mtwf be open. But to prove this, we need the following theorem, staied I-iere without proof.*

c Rn is open and f : U + Rn is one-one and continuous, then f (U) c IR" is open. (It foUows that f (V) is open for any open V c U, 1. THEOREM. If U

so f

-'

is continuous, and f is a homeomorphism.)

Theorem 1 is called "Invariance of Domain", for it implies that the property of being a "domain" (a connected open set) is invariant under one-one continuous maps into R". The p o o f that the neighborhood U in our definition must be open is a simple deduction from Invariance of Domain, left to the reader as ail easy exercise (it is also easy to see that if Theorem 1 were false, then there would be an example where the U in our definition was not open). We next turn our attention to the integer n appearing in our definition. Notice that n may depend on tlie point x. For example, if M c IR3 is

ihen we can clioose n = 2 for points in MI and

ri

= 1 for points in M2. Tliis

example, by the way, is an unnecessarily complicated device for ~roducingone manifold from two. In general, given Mi and Ms, with metrics di and d2, we can firsi replace each di witli an equivalent metric di such that d i ( x , y ) .E 1 for al1 X, y E Mi; for example, we can define

* Al1 proofs require some amount of macliiner)' The quickest routes arc probably provided by Vick, Hont o1o.g TJleory and Massey, Singular Hont ology Theoy. An old-fashioned , but pleasanily geometric, treátment may be found in N e w a n , %pologp ofPlenr Se&.

Then we can define a metric d on M = Mi U M2 by d(x,v) =

di(x, y)

if therc is some i such that x , y o therwise

E

Mi

(we assume that Mi and M2 are disjoint; if not, they can be replaced by new sets which are). In tlie new space M, both Mi and M2 are open sets. If MI and M2 are manifolds, M is clearly a manifold also. This coiistruction can be applied to any number of spaces-even uncountably many; the resulting metric space is called the disjoint union of the metric spaces Mi. A disjoint union of manifolds is a manifold. In particular, since a space with one point is a manifold, so is any discrete space M, defined by the metric

Although ddfeient 11's may be required at ddferent points of a manifold M, it would seem tliat only one n can work at a giveii point x E M. For the proof of this ii-ituitivelyobvious assertion we llave recourse once again to Invariance of Domain. As a first step, we note that R" is not liomeomorphic to R m wlien 11 # 111, for if 11 > 111, t1-iei-i iliei-e is a one-oiie continuous map from R m into a iloii-open subset of R". The furtlier deduction, tliai the n of our definition is uilique at each s E M, is lefi to the reader. This unique n is called the dimension of M at x. A manifold has dimension 11 01- is n-dimensional or is an a-manifold if it lias dimension i~ at each point. It is convenient to refer to tlie inaiiifold M as M" wlieii we want to indicaie tliai M has dimensioii n. Consider once more a discrete space, whicli is a O-dimensional maiiifold. The oiily compact subsets of such a space are finite subsets. Consequently, an ui-icountablediscrete space is not o-compact (it cannot be written as a countable uiiioi~of com11act sul~sets).Tlie same plieiiomenon occurs with higher-dime~isional manifolds, as we see by taking a disjoint uilion of uncountably many manifolds liomeomorpl~icto R". In tliese examples, however, h e manifold is not coi~i-iected.Mie \vil1 often ileed to know that tliis is the only way in wliicl-i o-compactness can fail to hold.

2. THEOREM. If X is a connecied, Iocally coinpaci metric space, tl-ien X is o -compact. PR001;. Foi- each ball

1- E

X cconsider tl-iose numbers r > O such that the closed

(y

E

X : d(x-,y) 5 i ' )

is a compact set (there is at least one such r > O, since X is locally compact). The set of al1 such I. > O is an interval. If, for soine A-, tliis set includes all r > 0 , then X is o-compact, since

If not, then for each x E X define r ( x ) to be one-half the least upper bound of al1 su& r. The triangle inequality implies that

so that

which implies that

Jnterchanging xi and

x2

gives

so tlle function r : X + IR is continuous. This has tlle following important consequence. Suppose A c X is compact. Let A' be the union of all closed balls of radius r (y) aiid cei-iter y, for all 11E A . Then A' is also compact. The poof is as follows. Let z1,22,23,. . . be a sequence in A'. For eacli i there is a yi E A such that zi is in the ball of radius r(yi)with center yi. Since A is compact, some subsequence of the y i , which we might as well assume is the sequence itself, converges to some point y E A . Now the closed ball B of radius and

center y is compact. Since yi eventually the closed b d s

y and since the function r is continuous,

: i

are contained in B. So the sequence ri is eventually in the compact set B, and consequently some subsequence converges. Moreover, the limit point is actually in the closed ball of radius r(y) and center y (Problem 10). Thus A' is compact. Now let AO E X and consider the compact sets

Their union A is clearly open. It is also closed. To see this, suppose that x is a point in the dosure of A . Tlicn there is sorne y E A with d(x, y) c i r ( x ) .

This shows tliat jf y E A,, tlien x E A,', so x E A . Siiice X is coi-ii-iected,and A # 0 is operi a ~ i dclosed, it must be tliat X = A , wliich is o-compact. 6 After this liassle with point-set topology?we present the long-promised exarnples of manifolds. Tlie oiily connected i-manifolds are tlie line IR and the circle, or I -dimensional sphere, S', defined by

The function f : (0,2ir) + S' defined by f (O) = (cos O, sin O) is a homeomorphism; it is even continuous, though not one-one, on [0,2n]. We will orten denote the ~ o i n (cos t O, sin O) E S' simply by O E [O, 2x1. (Of course, it is always necessary to check that use of this notation is valid.) The function g : (-s, s ) + S', defined by the same formula, is also a h o m e o m ~ r ~ h i s m ; together with f it shows that S' is indeed a manifold. There is another way to prove this, better suited to generalization. The projection P from the point (O, 1) onto the line IR x (-1) c R x R, illustrated in

the above diagram, is a homeomorphism of S' - ((0, 1)) onto R x (- 1): this is poved most simply by calculating P : S' - ((O,])) + R x (- 1 ) explicitly. The point (O, 1) may be taken care of similarly, by projecting onto R x (I), or it suffices to note that S' is "homogeneous"-there is a homeomorphism taking any point into any other (namely, an appropriate rotation of IR2). Considerations similar to these now show that the n-sphere

is an n-maiiifold. The 2-spliere S2,commonly known as "the sphere", is our first example of a compact 2-manifold or surface. From these few manifolds we can already construct many others by noting tl-iat if M iare manifolds of dimension ir j (i = 1,2),then M I x M2 is an (ni +n2)manifold. In particular

n times

is called the n-torus, while S' x S' is commonly called "the torus". 1t is obviously liomeomorphic to a subset of IR4, and it is also liomeomorphic to a certain subset of IR3 which is what most people have in mind when they speak of

"the torus": This subset may be obtained by revolving the circle {(O,y,a)

E

IR3 : (JJ - 1)2 + z2 = 1/41

around the z-axis. The same construction may be applied to any 1 -manifold

coiitaiiied ili {(O,jp,r) E IR3 : 1):> O). Tlie resul tiiig surface, called a surface of revolution, lias coinpoiients lioineomorpl~iceither to tlie toi-us or to the cyliiideiSi x R, tlie latter of which is also homeomorphic to the annulus, the region of tlie plai-ie contail-ied be tweeii two concentric circles.

l'hc i-iestsimplest compact 2-maiiifold is tlie 2-lioled torus. To provide a more

explicit description of tlie 2-lioled torus, it is easiest to begin witli a "l-iandle", a space liomeomorphic to a torus witli a hole cut out; more precisely, we tlirow

away d the points on one side of a certain circle, wbich remains in our llandle, and which will be referred to as the boundary of the handle. The 2-holed

torus may be obtaiiied by piecing two of tliese together; it is also described as

the disjoint uiiion of two handles witli corresponding points on the boundaries "identified". Tlie >I-holedtorus may be obtained by re~eatedapplications of this procedure. It is homeomorpliic to iIie space obtained by starting with the disjoint

union of i 7 liandlcs and a spliere with n holes, and then identif9ng points on the boundary of the ithIiandle with corresponding points on the ithboundary piece of the sphere.

There is one 2-manifold of which most budding mathematicians make the acquaintance wheli tliey still know more aboui paper and paste than about

metric spaces-ihe famous Mobim S@, which you "make" by giving a strip of paper a half twist before pasting its ends together. This can be described

analytically as the image in R3 of the function f : [O, 2x1 x (- 1,l) + R3 defined by e c o s B ,2sinB+icosIsinB, e f(B,i)= (2cosB+icos~

If we define: f on [O, 2x1 x [- I,1] instead, we obtain the Mobius strip with a boundary; as iiivestigation of the paper model will show, this boundary is homeomorphic to a circle, not to two disjoint circles. With our recently introduced terminology, tlie Mobius strip caii also be described as [O, I] x (- 1,l) with (O, i) and ( 1, -1) "identified".

We have not yet liad to make precise this notion of ccidentification'y,but our next example will force tlie issue. We xvisli to identify eacli point x E s2witli

its aiitipodal poiiit -x E S2.The space whicli results, the projecúve plane, IP2, is a lot harder io visualize than previous examples; indeed, there is no subset of IR3 which represents it adequately.

The precise definition of IP2 uses the same trick that mathematicians always use when they want two things wliich are not equal to be equal. The p0in.hof P 2 are defilied to be the sets ( p , - p ) for p E s2.We wilI denote this set by [ p ] E IP2, so tliat [ - p ] = [ p ] . We thus have a map f : S2 + IP2 @ven by f ( p ) = [ p ] , for whicl-i f ( p ) = f ( q ) implies p = fq . We will postpone for a while the problem of defining the metric giving the distance between two points [ p ] and [ q ] , but we can easily say wliat the open sets will turn out to be (and this is all you need to know iil order to check that P 2 is a surface). A subset U c p2 will be open if aiid only if f ( U ) c S2is open. This just means that the open sets of P 2 are of tlie form / ( V ) where V c S2 is an open set with the addiiional important property that if it contains p it also contains - p .

-'

In exacily ihe same wa): we could have defined ihe poinis of the Mohius sirip M io be al1 points (s,f) E (O, 1) x (-1,I) iogeiher wiih allseis {(O,r),(l,-111,

denotedby[(O,f)Ior[(I,-111.

(S, f ) if s # O, 1 [(s,f)] i f s = O or 1,

-'

and U c M is open if and only if f (U) c [O, 11 x (- 1 , l ) is open, so tliat tl-ie open seis of M are of the form f ( V ) ~rliereV is open and coniains (S, -1) wlienevcr it coniains (S, f ) for s = O or 1.

To gei an idea of wliat p 2looks Iike, we can make tliings easier for ounelves by first ilirowiiig away all poiiiis of s2helow tlie (x, y)-plane, siiice iliey are ideniified with poinis ahmre ihe (x, y)-plane anyway. This leaves the upper I-icniisplier-e(including tlie bounding ci~-clc), wliicli is Iiomeoinor~~li ic io ilie disc

aiid wc must idcntify eacli p E Si wiili - p E S'. Squaring tliiiigs o K a bit, tliis is ilie same as identifying points on ilie sides of a square according to tlie sclienie shown below (poinis on sides wiil-i ihe same lahel are ideniified in sucli a way iliat ille heads of tlie arrows are ideniified wiih each otlier). Tlie dotied liiies iii ihis piciui-e are h e key io undersiaiiding IP2. If we disiori ilie

region beiween tl-iem a bit we see tliat the froni pari of B followed by t l ~ e I~ackpart of A, ai the upper left, is to be ideniified wiih the carne thing ai the 10wer righi, in reverse direction; in other wods, we obtain a Mobius sirip with

a boundary (narnely, ihe doited line, which is a single circle). If this h4obius strip is removed, \ve are lefi wiih two pieces \vI~ichcan be rearranged to form something homeomorphic to a disc. The projectiw plane is thus oh 1. (h) l'rove directly from the Generalized Jordan Cunre Theorem thai IRm is no1 Iiorneomorphic to IR" for nr # n . 10. In ihe proof of TI~eoirin2, sliow that the limii of a convergent subsequence oi'the zi is aciudly i i i the closed ball of radius r(y) and cenier y. 11. Every connecied maiiifold (which is a metr-ic space) has a couniahle 11asc li~i-its iopology, and a countahle dense subset.

P

12. (a) Compute tlie composiiion f = S' - ((0, 1)) + Ri x (- 1 ) + Ri i:spIiciily for ihe map P on page 7, and show that it is a homeomorphism. (h) Do the same for f : S"-' - ((O,. . .,O, 1)) + IRn-'. 13. (a) Tlie text describes tlie open subsets of iP2 as seis of ihe form f(V); \\*liei-eV c S' is open aiid coiitaiiis - p wlienever it contains p. Show tliat this last condition is aciually unnecessary. (II) The andogous condiUon Zr necessary for the Mobius sirip, which is discussed iinmediately afterwards. Explain how ihe two cases differ.

14. (a) Check tliat tlie inetric defined for IP2 gises tlie open sets described in ihe texi. (II) Check that P* is a surface.

15, (a) S h o ~that l P' is homeomorphic to S ' . (b) Since we can consider S"-' c S " , and since aniipodal points in S"-' are siill antipodal when considered as points in S", we can consider P"-' C P" in an obvious way. Show that P" - iPn-' is Iiomeomorphic to interior D" = (x E R" : d(x,O) -= 1).

16. A classical tlieorem of topology states that every compact surface otl~er ~liaiiS 2 is obiained by gluiiig iogether a certain number of ton and projective spaces, and that all compact surfaces-~4th-boundaryare ohtained from these by cutting out a finite number of discs. To whích of these "standard" surfaces are the following homeomorphic?

a hole in a hole in a hole

17. Lei C c R c IR2 be the Cantor sei. Show thai W 2 - C is homeomorpliir to tlie surface shown at the top of the next page.

illis uii.clc is tiol in tlic sur fa c.^

these circles are

18. A locdly compact (but non-compact) space X "has one end" if for every voinpact C c X there is a compact K such that C c K c X and X - K is c-onnected. (a) R" Iias one end if n > 1, but not if n = 1. (13)

Rn - (O) does not "have one end" so Rn - ( O ) is not homeomorphic to Rm.

l 9. This proaem is a seque1 to the previous one; it will be used in Problem 24. Aii end of X is a function E which assigns io each compaci subset C c X a iion-empty component E(C) of X - C, in sudi a way that Cl c C2 hplies ~ ( C ZC) E(C1). (a) If C

c R is compact, ihen R - C has exactly 2 unbounded components, the

"left" component coiltaining dl iiumbers < some N, the "right" one containing al1 ilumbers > some N. If E is an end of R, show thai E(C) is either dways tlie "lefi" componeni of IR - C, o r dways the "rhiglli" one. Thus R has 2 ends. (11) Show that R" has only one end E for n > 1. More generdly, X has exactly oiie end E if and o~llyif X "has one end" in tlie sense of Problem 18. (c) This part requires some knowledge of topological spaces. Let E(X ) be the set of d ends of a coi-ii-iected, locally connected, locdly compact Hausdofl sl~aceX. Define a topology on X U E(X) by choosing as neighborhoods N c (EO) of an end EO die sets

for- dl compact C. Sl-iow ihat X U G(X) is a compact HausdorK space. Whai is R U E(R), and R" U E(Rn) for n > l ?

20. Coilsider ihe folloiving- tliree surfaces. (A) The infiniie-lioled torus:

(B) The doubly infiniie-holed torus: (C) The infinite jail cell window:

_ k l l-~ : ~-~Ji!+ lli ~J\t -

e

*

*

-

-

-

e

*

.

(a) Surfaces (A) and (C) liavc oiie end, wllile surface (B) does not. (b) Surfaces (A) and (C) are homeomorphic! Hi?zt: Tl-ie region cut out by ihe lines in tlie piciure below is a cylinder, wliicl-i occurs ai tlle left of (A). Now draw in two more liiies enclosirlg more holes, aild consider ihe region between the iwo pairs.

21. (a) The three open subsets of RZ shown below are Iiomeomorphic.

(b) The points kside the three surfaces of Problem 20 are homeomorphic.

22, (a) Every open subset of ES is homeomorphic to the disjoint union ofintervals.

(b) Ther-e are only countably many non-homeomorphic open subsets of R.

23. For the purposes of this problem we will use a consequence of the Urysohn Metrization Theorem, that for any connected manifold M , there is a homeomorphism f from M to a subset of the countable product R x R x - . (a) If M is a connected non-compact manifold, then there is a continuous

function f :M + R such that f "goes to ca at m", i-e., if (x,)is a sequence which is eventually in the complement of every compact set, then f (x,) + m. (Compare with Problem 2-30.) (b) Given a homeomorphism f : M + R x R x - - - and a g : M + R which goes to ca at m, define f : M + R x ( R x R x - - - )by f ( x ) = ( g ( x ) ,f ( x ) ) . Show that f ( ~ is)closed. (c) Thei-e are ai most c non-liomeomorphic coniiected manifolds (where c = ZN" is the cardinality of ES).

24. (a) It is posible for IRZ - A and IR2 - B to be homeomorphic even though A and B are non-liomeomorphic closed subsets. (b) If A c IRz is closed arid totally disconnected (the only components of A are points), then E(RZ- A) is liomeomorphic to A. Hence IR2 - A and IRZ - B are non- homeomorph ic if A and B are non-homeomorphic closed to tally disconnected sets. (c) The derived set A' of A is the set of al1 non-isolated points. We define A(") inductively by A ( ' ) = A' and A ( ~ + ' = ) (A("))'.For each n there is a subset A, of R such that A,(") consists of one poini.

*(d) There are c non-homeomorphic closed totally disconnected subsets of IRZ. Hint: Let C be the Cantor set, and ci C: cz C: c3 C: . -. a sequence of points in C. For each sequence nl r m r - - - , one can add a set A,, such that its n jth derived set is (ci). (e) Tliere are c non-homeomorphic connected open subsets of IRZ.

25. (a) A manifold-with-boundary could be defined as a metric space M with the property tliat for each x E M there is a neighborhood U of x and an inieger n 2 O such thai U is homeomorphic to an open subset of W". (h) If M is a manifold-with-boundary, then aM is a closed subset of M and aM and M - aM are manifolds. (c) If Cj, i E 2 are the components of aM, and Ir c I , then M - UiEl, Ci is a manifold-with-boundary.

26. If M c Rn is a closed sei and an 11-dimensional manifold-with-boundary, ilien the iopological boundary of M , as a subsei of IR", is aM. This is not iiecessarily true if M is not a closed subset. 27. (a) Every poini (a, b, c ) oii Steiner's surface satisfies bZc2+ ozcZ+ 02b2 = abc. @) If (a, b,c) saiisfies tliis equation and O # D = JbZcZ + oZcZ + oZbZ,then (a, b, c) is on Sieiner's suríiace. Hint: LRt x = bclD, etc. (c) The sei {(a, b, e) E IR3 : bZcZ+ ozcz + azbz = abc) is the union of tlie Steiner surface and of the portions (-m, - 1 12) and (1 12, m) of each axis.

CHAPTER 2 DIFFERENTIABLE STRUCTURES

e are now ready to apply analysis to the study of manifolds. The necessary tools of "adva~lcedcalculus", which the reader should bring along fieshly sharpened, are contained in Chapters 2 and 3 of Cizku1u.r on A4an8Ms. We will use freely the notation and results of these chapters, including some 11roblems, notably 2-9, 2-15, 2-25, 2-26, 2-29, 3-32, and 3-35; however, we wül denote the identity map from W" to Wn by 1, rather than by n (which will be used oRen enough iii other contexts), so that l i ( x ) = xi. Ori a general manifold M the notion of a continuous function f : M + W inakes sense, but tlie notiori of a differentiable function f : M + W does not. This is the case despite the fact that M is locally like IR", where differentiability of funclions can be defined. If U c M is ari open set and we choose a homeomorpliism #: U + IRn, it would seem reasonable to define f to be differentiable on U if f o #-' : R" + W is differeiitiable. Unfortunately, if q : V + W" is anotl~erhomeomorphism, and U n V # 0, then it is not necessarily true that f o SI-': W" + W is also dfirentiable. Indeed, since

we can expect f o VI-' to be differentiable for al1 f wliich make f o#-' diñereiitiable only if # o q-' : Wn + W" is differeiitiable. This is certainly not always

tlie case; for example, one need merely choose # to be h o@,where h : R" + R" is a liomeomorphism that is not differentiable. If we insist on defining differentiable functions on any manifold, there is no way out of this impasse. It is necessary to adorn our manifolds with a little additional structure, the precise nature of which is suggested by the previous discussion. Among all possible homeomorphisms from U c M onto Rn,we wish to select a certain collection witli tlie properry that # o @ - ' is differentiable whenever #, @ are in the collection. This is precisely whai we shall do, but a few refinements will be introduced along the way. First of all, we M i l 1 be interested almost exclusively in functions f : R" + R" wliich are C m (that is, each component function f i possesses continuous partial derivatives of all orders); sometimes we will use the words "dflerentiable" or c'smoothy'to mean Cm. Moreover, insiead of considering h o m e o m ~ r ~ h i s mfrom s open subsets U of M orlto R" ,it will sufice to consider Iiomeomorphisms x : U + x ( U ) c R" onto open subsets of R". The use of tlie letters x, 11,etc., for tliese liomeoniorphisms, henceforth adIiered to almost religiously is meant to encourage the casual confusion of a point p E M with x ( p ) E R n : wliich has "coordinates" x' ( p ) , .. . ,x n ( p ) . The only time tliis notation will be confusing (and it will be) is when we are referring to ihe maiiifold R", whcre ii is hard not to lapse back into the practice of denoting points by x and y. IVe will often mention tlie pair (x, U), instead of x aloile, just to provide a coii\rei~ieniname for the domain of x . If U and V are opeii siibsets of M, two homeomorphisms x : U + x ( U ) c R" arid 11:V + j c ( V ) c R" are called Cm-related if the maps

are Cm. This make seiise, since x ( U nV ) and y(U nV ) are open subsets of R". Also, it makes sense, aild is automatically true, if U n V = 0. A family of mutually Cm-related homeomorpliisms whose domains cwer M is called an atlas for M. A particular member ( x , U ) of an atlas 36 is called a chart (for tlie atlas 36):. or a coordinate system on U, for the obvious reason iliai it provides a way of assigiiing "coordinaies" to points on U , namely, tlic coordinates x' ( p ) , . .. ,x n ( p ) to the point p E U . I4Je can even imagine a mesh of coordinate lines on U , by considerirlg the

l)2jintz/iable Structure~ iiiverse images uiider x of lines in Rn parallel to one of the axes. . . . . ......... . . . . ..........

.. .. .. .. .. .. .. .. .. . . . . -....... . . -..... ............ .. .. .. .. ,........ . . . . ............. . . . . . .. .. ,.. ... ..m..

L.......

......s......

.................... . . . . . . . . . . .. .. .. .. .. .. .. .. .. d...

.. .. .. .,........... . . . . . ......... .. . . ., -.,

.. .. .. .. .-........ .. .. . . . . ............ . . . . . . . .-...

The simplest example of a manifold together ~ r i t han atlas consists of Rn with ari atlas A of only one map, the identity 1: R" + Rn. We can easily make the atlas bigger; if U a ~ i dV are llomeomorphic operi subsets of R", we can adjoin any homeomorphism s: U + V with the property that x and x-l. are Cm. Indeed, we can adjoiii as many such x's as we like-it is easy to check that they are al1 Cm-relaied io each other. Tlie advantage of this bigger atlas U is ihai the single word "cliart", wlieri applied to this atlas, denotes something which must be described in cumbersome lariguage if one can refer only to 36. Aside from this, U differ-s only superficially from 36; one can easily construct U from 36 (and one would be foolish not to do so once and for all). What has just been said for ihe atlas (1)applies to any atlas: 1. LEMMA. If 36 is an atlas of Cm-related cliarts on M, then 36 is contained in a unique maximal atlas 36' for M .

PR001;. Let A' be il-ic set of al1 cliarts y wliicll are Cm-related to al1 charts x E 36. It is easy to check that al1 charts in 36' are Cm-related, so 36' is an atlas, and it is clearly tlie uriique maximal atlas coiitaining 36. *:* Mle now dcfine a C m manifold (or differentiable manifold, or smooth manifold) to be a pair (M, A): wliere A is a niam'rna/ atlas for M. Thus, about the simplest example of a Cm manifold is (R", U), wliere U (the "usual Cm-structure for Rn") is tlie maxi~naladas containing (1).Arioiher example is (R, V) where V contains the l~~rneornor~liism x H x3, wliosc inverse is no/ C m , together with al1 charts Cm-relaied io it. Although (R, U) and (R, V) are not tlie same, tliere is a one-one onto function f : R + R such thai A-

E

U ifand only if x o f

E

V,

namely, the obvious map f( x ) = x Thus (R, U) aiid (8, V) are the sort of structures one would want to cal1 "isomorphic". Tlie term actually used is

"diffe~mor~hic":two C* manifolds (M, A) and (N, a) are diffeomorphic if there is a one-one onto function f:M + N such that x

E

J? if and only if

A-

o

f E A.

-'

Tl-ie map f is called a diffeomorphism, and f is clearly a difeomorphism also. If we had not required our atlases to be maximal, the definition of diffeomor.pliism would have had to be more complicated. Normally, of course, we will suppress mention of tlie atlas for a dfireritiable mariifold, and speak elliptically of "the difirentiable manifold M"; the atlas for M is sometimes referred io as the dzjierentiabk structure for M. Ii wdl always be undersiood that IR" refers to the pair (IR", U). Ii is easy to see that a difkomor-phism must be coritinuous. Consequently, its inverse musi also be coniinuous, so ihat a difeomorpliism is automatically a homeomorpl~ism. This r-aises the natural question whetlier, conversely, two liomeomorpliic maiiifolds are necessarily diffeomor-phic. Later (Pi.oblem 9-24} we will be able to prove easily that IR with any ailas is difkomorphic to (R, U). A proof of tlie corresporiding assertion for is much harder, the proof for IR3 would ccrtainly be ioo difficult for- iriclusion llere, aiid the proof of the essential uniqueness of C* structures on IR" for n 2 5 requires very difficuli techniques from topology, In ihe case of spl-ieres,tlie projections Pl and P2 from the points (0,. . .,O, 1) and (0, . .. ,O, - 1) of S"-' are easily seen to be Cm-related. They iherefore clctei-minean ailz5-tl-ie "usual C* structure for S"-'". T1-iis atlas may also bc described in terms of the 2n homeomorpliisms

defined by J(s)= gi(x) = ( x r , . . ., s i - ' , x i + ' , .. . ,xn),wbicli are Cm-related to P1and P2.Tlrei-e are, up io diffeoniorpl-iism,unique differentiable struciures on S" for 17 5 6. But tl-iere are 28 difeomorpl-iism classes of differeniiable structures on S7. aiid over- 16 million on S3'. However, we shall not come close to proving iliesc asser-tioi~s,whicli are pari of tlie field called "differential topology", rather then diferential geometry. (hrhaps most astonishing of al1 is ilie quite receiit discovery that IR4 lias a difeirritiable structure that is not cliffromorphic io tlie usual diffei-entiablesiructure!) Otl-ier examples of differentiable manifolds will be @ven soon, but we cari already describe a diffcrentiable structure A' 011 any open submarlifold N of

a differentiable manifold ( M , A); tlie atlas A' consists of all (x, U) in 36 witli U c N. Just as dfiomor-phisms are analogues for C* manifolds of homeomorphisms, there are analogues of continuous maps. A function f : M + N is called differentiable if for every coordinate system (x, U) for M and (y, V) for N, the map y o f o x-' : IRn + IRm is differentiable. More particularly, f

is called differentiable at p E M if y o f o x-' is differentiable at x(p) for coordinate systems (x, U) and (y, V) with p E U and f (p) E V. If this is true for one pair of coordinate systems, it is easily seen to be true for any otlier pair. We can thus define differentiability of f on any open subset M' C M ; as one would suspect, tliis coincides with differentiabilit~of the restricted map f 1 M ' : M' + N. Clearly, a differentiable map is continuous. A differeniiable function f : M + IR refers, of course, to the usual d f i r e n tiable structure oii IR, and lience .f is differentiable if and only if f o x-' is diflerentiable foi- cach chart x. It is easy to see that (1) a function f : IR" + IR" is diflerentiable as a map between C* manifolds if and only if it is difler-entiable in tlie usual sense;

(2) a fuiiction f : M + IRm is differentiable ifand only if each f i : M + W m is diferen tiable;

(3) a coordinate system (x, U) is a diffeomorphism from U to x(U); (4) a fuiiction f : M + N is diffei-eiitiable if and only if each y' difirentiable for each coordinate system y of N;

o

f is

(5) a difler-eritiable function f : M + N is a dfleomorphism if and only if f is one-o~ieonto and f U 1 : N + M is differentiable.

Tlie differe~itiablesiructui-es on inaiiy manifolds are designed to make ccrtain fuilciions differ-eniiable. Consider first the product MI x M2 of two dfleren-

iial~lemanifolds Mi, arid the two "projections" ni : M I x M2 + Mi defined by ni(pl, p2) == pi. It is easy to define a differentiable structure on MI x M2 which makes eac1-i ni difirentiable. For each pair (xi, Ui) of coordinate systems on Mi, we constmct the homeomorphism

defined by

Tlien we cxiend tliis atlas to a maximal one. Similarly, ihere is a difirentiable structure on Pn which makes the map . f : Sn + Pn (defiried by f (p) = [p] = (p,-p)) difirentiable. Consider any coordinate system (x, U) for S",where U does not contain - p if it contair~sp, so tllat f 1 U is o~i,e-one.Tlie map x o (f 1 U)-" is a homeomorp1~ism on J'(U) c Pn,and any two such are Cm-related. The collectiori, of tliese lioineoi~iorpliismscan then be extended to a maximal atlas. To obtain differentiable structures on other surfaces, we first note that a C m manifold-witli-bour~darycan be defined in a n obvious way. It is only necessary to know wlien a map f : W n + Rn is to be considered differentiable; we cal1 f differentiable d e n it can be extended to a dflerentiable function on a n open i-ieiglibor-hoodof W".A "Iiandle" is then a Cm manifold-with-boundary.

A diffei.eriiiable structure on tlie 2-holed torus can be obtained by "matcliiiig" tRe dflereiitiable structure on two l-iandles. Tl-ie details involved in tliis process are reserved for Pro blem 14. To deal witl-i C m fuilctions effectively,one needs to know that there are lots of ilicm. Tlie esistcrice of COQfurictions oii a manifold depends oii tlie existence of C m funciions on R" ~ I i i c lare i O ouiside of a compact set. We briefly recall liere ilie necessat-y facis aboui such Cm fuiictions (c.f. Cakulus on MaliifoMs, pg. 29).

D@ben/iabL Slructu res (1) The function h : R + R defined by

is Cm, and h(")(0) = O for al1 n.

(2) The function j: IR + R defined by e-(x-r)-2 e-(x+~)-2 x E (-1, 1) =

{

o

x

(-41)

-1

I

I

is Cm. Similarly, there is a Cm function k : W + R which is positive on (0,S) and O elsewhere.

I

S

(3) The fuiiction I : R + R defined by

is Cm; ii is O for x 5 O, increasing on (O, S), and I for x 2 S. (4)Tlie function g : R" + R defined by

is Cm; ii is positive on (-E,

E)

x . - - x (-E,

E)

and O elsewliere.

O n a Cm manifold M we can now produce many non-constant Cm functions. The closure (x : f(m) # O) is called tlie support of f,and denoted simply by support f (or sometimes supp f). 2. LEMMA. Let C c U c M witli C conipact aild U open. Then there is a Cm function f : M + [O, 11 such tliat f = 1 on C and support f c U. (Compare h e 2 of the proof of Theorem 15,)

PROOF. For each p E C, choose a coordinate system (x, V) with c U and x(p) = O. Then x(V) 3 (-E,&) x - - - x (-E,&) for some E 3 O. T h e function g o x (wliere g is defined in (4))is CDgon V. Clearly it remains Cm if we extend

it to be O outside of V. IRt

Sp be the extended function. The function fp can be

coiisir~uctedfor each p, aiid is positive on a neighborhood of p whose closure is coniained iii U. Since C is compact, finitely many such neighborhoods cover C, aiid the sum, fp, + - - + -fp,, of the correcponding functions has support c U. 0 1 1C it is positive, so on C it is 2 S for some 8 3 O. Let f = Io (fp, + - . -+fpt,), where I is defined in (3).e:* By tlie way, we could have defined C r rnanifolds for each 1. 1, not just for i- = m". (A function f : IRn + IR is C r if it Iias continuous partid derivatives up to order i-). A "COfunction" is just a continuous function, so a CO manifold is jusi a maiiifold in tlie sense of Chapter l. We caii also define analytic manifolds (a function f : W n + W is analytic at u E IRn if f can be expressed as a power series in the ( x i - í i i ) which converges in some neighborhood of U ) . Tlie symbol C Wstaiids for analytic, and it is conveiiient to agree that r .:w c: o for each iiiteger i . > O. If a < B, tlien tlie cliarts of a maximal CB atlas are al1 Ca-related, bui ihis atlas can always be extended to a btggu atlas of Ca-related cliaris, as in Lemma 1. Tlius, a CB structure on M can always be extended io a C" structure i i ~a uiiique way; tlie smaller siructui-e is tlie "stronger" one, tlie CO structure (coiisistiiig of al1 homeomoi~pliismsx: U + W " ) beiiig tlie lüigest. The coiiverse of this trivial remark is a hard tlieorem: For a 2 1, ewry C" structurc roiitains a CB structurc for eacli /3 3 a;it is not unique, of c,ourse, but it is ui~iqueup to diffeomorpliism. Tliis will not be proved liere.* In fact, C" maiiifolds for a # w will hardly ever be mentioned again. One irinark is iii ordei- 1 3 0 ~ tlie ~ ; proof of h m m a 2 produces an appropriate C" fiinciion f oii a C" inaiiifold, for O 5 a 5 w. Of course, for a = o the proof 6c

* hi.a p1-00fsee h4unkrcs, EIentefztay Ditpereniial %/~olog)l.

fails completely (aiid ihe result is fahe-an analytic function which is O on a n open set is O everywhere). With difirentiable functions now at our disposal, it is fitting that we begin differentiating them. What we S hall define are the partid derivatives of a differentiable function f : M + R, with respeci to a coordinate system (x, U). At this point classical notation for partid derivatives is systematically introduced, so it is wortli recalling a logicai notation for the partial derivatives of a function f : R" + R . We denote by Di f (u) the number lim

f ( u ' y * . * y u i + h , - . - , u "-) f(u) I?

h-O

The Chaiii Rule states that if g : Rm + R" and f : R" + R, then n

Dj (f

0

Di f (g(u)) -

g) (u) =

~j

gi (u).

i=21

Now, for a function f : M + R and a coordinate system (x, U ) we define

a f = D i(f o x-') o r, as an equation behveen functions). If we (or simply axl define tlie curve ci : (-E, E) + M by

then this partial derivative is just

so it measures the rate change of f along tlie curve c;;in fact it is just (f oci)'(0). Notice tliat ax3 1 ifi= j -(p) = 6; -

a A-

J

1f x liappens to be the identity map of Rn, then Dt f (p) = 8f/axi(p), whidi is the classical synibol for this partid derivative.

Anot her classical instance of this notation, often not completely clarified, is thc use of the syrnbols a/ar and a/a9 in connection with "polar coordinates". On the subset A of JRZ defined by

by' we can iii troduce a "coordinate system" P : A + IW

where r ( x , y) =

m

and 9(x, y) is the unique numbcr in (O, 2rr) with

Tliis really is a coordinate system on A in our sense, with its image being the set (r : r > O) x (0, k).(Of course, the polar coordinate system is often

not restricted to tlie set A . One can delete any ray other than L if 9(x, 1)) is restricted to lie in the appropriate interval (60, 90 + 2 ~ ) many ; results are essentially independent of which line is deleted, and this sometimes justifies the slopl~inessinvolved in the definition of the polar coordinate system.) IVe liavc 1-eallydefined P as an inverse function, whose inverse P-' is defined simply by P-'(r,9) = (r cos9,r sino).

From this formula we can compute f i a r explicitly: (f

af

-(x? ar

o

Y) = Di (f

P-')(r, O) = f (r cos O, i- sin O),

0

p-'1 (P(x9 Y))

= DI f (P-' ( ~ ( xY)) , Di [P-'11 (P(x, Y)) + ~ z (p-'(p(x, f Y)) - Di [ P - ' I ~ ( P ( XY)) ~ by the Chain Rule = DI f (x, y) tos O(x, y) Dzf (x, Y) sino(x, Y).

+

Tliis formula just gives tlie value of the directional derivative of f at (x, JI), along a uiiit \lector v = (cos O (x, y), sin O (x, y)) poin ting outwards from the origin to (x, y). Tliis is to be expected, because cl, the inverse image under P

sin B(x,y)

of a culve along t l ~ e"1.-axis", is just a line in tliis direction. A similar computation gives

to v , and thus tlle The lector w = (- siii O(x, y), cos 0(x, y)) is direction, at tlie point (A-, y), of the curve cz ivliicli is tlie inverse image under P of a cunfe along tlle "O-axis". Tlie factoi- i. (x, y) appears because this cuive

goes around a circle of that radius as 8 goes from O to 2n, so it is going i-(x, y) times as fast as it should go in order to be used to compute the directional derivative of f in the direction w , Note that af/aO is independent of which line is deleted from the plai~ein order to define the function 8 unarnbiguously. Using the notation af/ax for Di f, etc., and suppressing the argument (x, 11) evenutliere (thus writing an equation about functions), we can write the above equations as

af

af

- c = -

ay

ax

8.f = ?f ae ax -(-)e

a f sin 8 cos 8 + ay sin 8) + -ra.f cos 8. ay

111 particular, these formulas also te11 us what ax/ai- etc., are, where (x, y) rlrnotes ilie identit~coordinate system of RZ.We have ax/ar = cose, etc., so oui- formulas can be put in the form

af = -afax +-al-

a~-al-

aya~

In classical notation, the Cl~ainRule would always be written in this way. It is a pleasui-e to repoi-t thai Iienceforih this may always be done:

3 . PROPOSITION. If ( S , U ) and (y, V) are coordinate systems on M, and f : M + R is differentiable, then on U n V we have

PROOF. It's tlie Chaii~Rule, of course, if you just keep your cool:

At this point we could introduce the "Einstein summation convention". Notice that the summation in this formula occurs for the index j, which appears both "above" fin a x j / a y i ) and "below" (in a f / a x j ) . There are scads of formulas in which this happens, often with hoards of indices being summed over, and the convention is to omit the C sign completely-double indices (wliicli by luck, tlie nature of t hings, and felicitous choice of notation, almost always occur above and below) being summed over. 1 won't use this notation because wheiiever 1 do, 1 sooii forget I'm supposed to be summing, and because by doing things "riglit", one can avoid what Élie Cartan lias called the "debauch of indices". We will often write formula (1) in the form

here alay' is considered as aii operator taking the function f to a f / a y i . The operator taking f to af /ayi ( p ) is denoted by

For later use we record a property of f = a/axil,: it is a "point-derivation". 4. PROPOSITION. For any diflerentiable f , g : M + IR,and any coordinate system (A-,U ) with p E U, the operator l = a/axilPsatisfies K f g ) = f (p)f(g)+f(f )g(p).

PROOF. Lft to the reader. If

(A-,U)

and (x', U ' ) are two coordinate systems on M, the n x n matrix

is just the Jacobian matrix of x1 o inverse is clearly

A--'

at x(p). It is non-singular; in fact, its

Now if f : M" + N m is Cm and (y, V ) is a coordinate system around f(p), t he rank of the m x 11 matrix

clearly does not depeiid on the coordinate system (x, U ) or (y, V ) . It is called tlie rank of f at p. T h e point p is called a critical point of f if the rank of f at p is 4 i?7 (the dimension of the image N); if p is not a critical point of f, it is called a regular point of f. If p is a critical point of f,the value f (p) is called a cñtical value of f. Other points in N are regular values; thus y E N is a regular value if and only if p is a regular point of f for ewry p E f (y). Tliis is true, in particular, if q $ f (M)-a non-value of f is still a "regular value". If f : W + W, tlien x is a critica] point of f if and only if f t ( x ) = O. It is ~ossiblefor al1 points of the intenral [u, b] to be critical points, although this can happen only if f is constant on [u, b]. If f : WZ + IR has al1 points as critical values, tlien Dl f = Dzf = O everywhere, so f is again constant. O n tlie other hand, a function f : WZ + IRZ may have al1 points as critical points without being constant, for example, f ( x , y) = A . In this case, Iiowever, the iinage f (WZ)= W x { O ) c WZ is still a "small" subset of IRZ. The most importan1 theorem about critical points generalizes this fact. To state it, we will need some terminology. Recall tliat a set A c W" has "measure zero" if for every E > 0 tliere is a sequeilce Bi, Bz, B 3 , . . . of (closed or open) rectangles with

-'

and

wliere u(&) is the volume of B,l. We want to define the same concept for a subset of a manifold. To do this we need a lemma, wliich in turn depends on a Icmma from Caku/us on Adanfolds, which we merely state.

5. LEMMA. Let A C IR" be a rectangle and let f : A + Rn be a function such that ~ ~ ~5 fK ' onl A for i, j = 1 ,..., n. Then

for al1 x , y

E

A.

6. LEh4MA. If f : IR" + IR" is C' and A measure O.

c IR"

has measure O, then f ( A ) has

PROOF. We can assume that A is contained in a compact set C (since IR" is a countable union of compact sets}. Lemma 5 implies that there is some K such t hat I f (x) - f (r)l 5 n2Klx - yl for al1 x , y E C. Tllus f takes rectangles of diameter d into sets of diameter - >I' Kd. This clearly implies that f ( A ) has measure O if A does. +3 A subset A of a Cm 71-manifold M Iias measure zero if there is a sequence of charts (xi, U i ) , with A c Ui Ui, S U C ~that each set x i ( A n U i ) c IR" has measure O. Using L m m a 6, it is easy to see tliat if A c M has measure O, then x(A n U ) c IR" Iias measure O for any coordinate system ( x , U). Conversely, if this condition is satisfied and M is connected, or has only countably many compoiieiits, then it follows easily from Theorem 1-2 that A has measure O. (But if M is the disjoint union of uncountably many copies of IR, and A consists of one poirit from each compoiient, then A does not have measure O). Lemma 6 thus implies another result:

7. COROLLARY. If f : M + N is a C' fu~ictionbetween two n-manifolds and A c M has measure O, then f ( A ) c N has measure O.

PR001;. Tliere is a sequelice of charts (xi, U i ) with A c Ui U; and each set

xi(A n U i ) of measure O. If ( y , V ) is a cliart on N, then f ( A ) n V = Ui f (A n U i ) n V . Each set

Iias measure O, by Lemma 6. Thus y( f ( A )nV ) has nieasure O. Since f (UiUi) is contained in tlie uiiion of at most countably many components of N, it follows that f ( A ) has measure O.

e+

8. THEOREM (SARD'S THEOREM). If f : M + N is a C i map between n-manifolds, and M has at most countably many components, then the critical values of f form a set of measure O in N. PROOI;. It clearly suffices to consider the case where M and N are Rn. But this case is just Theorem 3.14 of Calculw on M a n f o b . Tlie stronger version of Sard's Theorem, which we will never use (except once, in Problem 8-24), states* tliat the critical values of a C' map f : M" + N"' are a set of measure O if k 2 1 + max(n - in,O). Theorem 8 is the easy case, and tlie case nz > n is the trivial case (Problem 20). Although Theorem 8 will be very important later on, for the present we are more interested in knowing wliat tlie image of f : M + N looks like locally, in terms of the rank k of f at p E M . More exact information can be @ven wlien f actually has rank k i n a neighborhood of p. It should be noted that f must have rank 2 k in some iieiglihor-hood of p, because some k x k submatrix of (a(yi o f)/axi) has non-zero determinant at p , and hence in a neighborhood of p.

9. THEOREM. (1) If f : M" + N m has rank k at p, then tliere is some coordinate system (x, U) around p and some coordinate system (y, V) around f (p) with y o f ox-' in the form

Moreover, given any coordinate system J?, the appropriate coordinar e syst em ori N can be obtained merely by pei-muting the component functions of y. (2) If f has rank k in a rieiglil~orlioodof p, theii tliere are coordinate systems (A-, U) and (y, V ) such that

Ren~ark:Tlie special case M = Rn, N = Rm is equivalent to the ge~ieraltlieorem, wliicli @\res only local results. If y is tlie identity of R", part (1) says thai ly first pei-formiiig a difleoniorpliism on Rn, arid tlieri permuting the coordiiiates in Rm,we can irisure that f keeps tlie first k components of a poiiit fixed. Tliesc difTcornoiphisms on Rn and Rm are clearly necessary, since f may not everi be oiie-oiic on Rk x { O ) c RW aad itr image could, for example, contain oiily ~ o i n t swith first coordinate O.

* Foi- a proof, scc hqiliroi; Dflirenliol Geonzelv.

To/)ologilh

t n

tlic Ditp~~'nl&inblc I4cuy)oi)ir or Sternberg, LEcllrres ott

In part (2) we must clearly allow more leeway in the choice of y , since f (IRn) rnay not be contained in any k-dimensional subspace of IRn.

I'ROOF. (1) Choose some coordinate system u around p . By a permutation of ilie coordinate functions u' and y' we can arrange that

Define

Condition (1) implies that

This shows thai A- = (x o u - ' ) o u is a coordinate system in some neighborI~oodof p, since (2) and the Inverse Function Theorem show that x o u-' is a dfleomorphism in a ncighborhood of u ( p ) . Now

q = x - ' ( a ' , . . . , a n ) means hence

1

x ( q ) = (a ,. .. , a n ) , x'(~= ) ai ,

SO

Y ~ f o x - l ( a l , . . . , a J=1y) o f ( q ) = ( a 1 ,..., a k ,

for q = x-l (a 1 ,. . ., a n ) ).

(2) Choose coordinate systems x and v so that v o f o x-' has the form in (1). Since rank f = k in a neigliborhood of p, the lower square in the matrix

must vanish in a neighborhood of p. Thus we can wnte

Define ya = va y ' = vr - $ r o ( ~ ' , . . . , V k ). Since

has non-zero determinant, so y is a coordinate system in a neighborhood of f (p). Moreover,

Theorem 9 acquires a special form when the rank of f is n or nz:

10. THEOREM. (1) If m 5 ,I and f : M" -t Nm lias rank m at p , then for any coordinate sysiem (y, V ) around f (p), there is some coordinate system (x, U) around p with y o f ox-'(a',.. .,an) = (a',. . . , a m ) .

(2) If 11 5 m and $: M n + N"' has rank ir ai 11, tlien for any coordinate system (x, U) around p , there is a coordinate system (y, V ) around $ ( p ) with y o f ~ x - ~ i(, a. . . , a n ) = (a 1 ,..., a n , O ,...,0).

PROOF. (1) This is practicdly a specid case of (1) in Theorem 9; it is only rlecessary to observe that when k = rn, it is clearly unnecessary, in the proof of this case, to permute ihe yi in order to arrange that a,p = 1,. . .,m;

) 0 det ( a ( y ~ u ~ f ) ( p ) f

only the u' need be permuted. (2) Since the rank of f at any point must be 5 n, the rank of $ equals n R" in some neighborhood of p. It is convenient to think of the case M and N = RJn and produce the coordinate system y for Rm when we are Siven ihe identity coordinate system for R". Par-t (2) of Theorem 9 yields coordinate systems # for Rn and S, for Rm such that 1 1 @ o f o#-'(a ,...,a") = (a ,..-,an,O,..., O),

-

Even if we clo rioi pei-form #-'. firsi, tl-ie map f still takes EXn into the subset

(R") wliich S, takes to R" x (O) c Rm-the points of Rn just get moved to ihe wrong place in Rn x (O). This can be corr-ected by another map on R". Define h Ily h(bi ,. . . ,bJn)= (4-' (bl,. . . ,b"), b"+', . . . ,bm). Then h o S , o $ ( a i , ...,a n ) = h o S , o $ o # - ' ( b 1 ,..., b") for (b', .. . ,bn) = #(a)

= h(bl,. . .,ón,0,. . .,O) = ($-'(bl, .. ., b"),O,. . .,O)

..

= (a i , . , a R ,O,

. ,O),

so h o S, is tl-ie desired If we are Siven a coordinate system x on R" other than the identify, we just define h(bl, . . .,b"') = ( ~ ( 4 - ' ( b i ,- . . ,b')), b"+I,. . . ,bm); )J.

it is easily cliecked tliat y = h o S, is rlow the desired y. +3

Aliliough p is a regular point of f in case (1) of Theorem 10 and a critica1 point in case (2) (if n < m), it is case (2) \vhich most interests us. A dflerentiable funciion f :M n + N m is called an immersion if the rank of f is n, the dimension of the domain M, at al1 points of M. Of course, it is necessary that In > n, and i i is clear from Theorem 10(2) that an immersion is locally one-one (so it is a iopological immersion, as defiiied in Chapter 1). On the other hand, a differentiable map f need not be an immersion ewn if it is globally one-one. The simplest example is ihe function f :IR + IR defined by f ( x ) = x3, with f (O) = O. Another example is

A more illun~inatingexample is the function h : IR + IR2 defined by

alihough its image is tlie graph of a non-dflerentiable function, the curve itself manages to be differenkiable by slowing down to velocity O at the point (0,O). One can easily define a similar curve whose image looks like the picture below.

Tliree immersions of IR in IR2 are shown below. Althougll the second and iliii-d immei-sions Bi and B2 are one-one, iheir iinages are not homeomorphic

to R . Of course, even if tl-ie one-one immersion f : P + M is not a homeoniorpliism onto iis image, tliere is certainly some metric and some dflerei~tiable sti-ucture on f ( P ) which makes the inclusion map i : f ( P ) + M an inlmersion. In general, a subset MI c M, with a dflerentiable structure (iiot neccssarily compatible with the metric Mi inherits as a subset of M), is called an immersed submanifoId of M if the inclusion map i : MI + M is an immersion. l'he follo\ving picture, indicating the image of an immersion B p : R + S' x S ] ,

rS3

first time around second time around / sl-iows that Mi may even be a dense subset of M. Despite these complications, if M i is a k-dimensional immersed submanifold of M n and U iis a neighborhood in M' of a point p E M I , tllen there is a coordinate sysiem (y, V) of M around p , such that

tliis is an immediaie consequerice of Tlieorcm 10(2), with f = i. Tlius, if g ; Mi + N is Cm (co~isideredas a function oii tlie manifold M i ) in a neigliborhood of a point p E M I , ihen ihere is a Cm furiction g on a rieigliboi-liood V c M of p sucli that g = g O i on V n M i -we can define

S (4) = g (qi),

where

~ ' ~ ( q=' )~ ' " ( 4 ) a = 1, ...,k r =k + l,.s.,iz.

On the oilier hand, even if g is Cm on al1 of M I we may not be able to define g on M. For example, this cannot be done if g is one of the functions B;' j j ( M ) + R. One other complicatioi-i arises with immersed submanifolds. If M I c M is an immersed submailifold, aild f : P + M is a Cm function with f ( P ) c Mi, ii is not necessarily true that $ is Cm when considered as a map into MI, with its Cm structure. The following figure shows that f might not even be continuous

as a map jnto M I . Actually, tliis is the only thing that can go wrong:

11. PROPOSITION. If MI c M is an immersed manifold, f : P + M is a Cm function witli f (P)c M I , and f is coniinuous considered as a map irlto M I , then f is also Cm coilsidered as a map into MI.

PROOF. Let i : MI + M be tlie inclusioil n-iap. We want to show that iP1o j' is Cm if it is continuous. Given p E P, choose a coordiilate system (y, V) for M

around f (p) such that

is a iieigliboi-hood of f (p) in MI and (yi1 U], . .. ,y k 1 U]) is a coordiliate system of MI on U].

By assumption, i-'

of

f

is continuous, so

-' o i(open set)

is an open set.

-'

Since Ui is open in M I , this means that f (Ul) c P is open. Thus f takes some neighborhood of p E P into U,. Since al1 y j o f are Cm, and y ' , . ..,y k are a coordinate system on U1, the function f is Cm considered as a map into MI. 43 Most of these difficulties disappear wlien we consider one-one immersions f : P + M which are liomeomorphisms onto their image. Such an immersion is called an imbedding ("embedding" for the English). An immersed submanifold Mi c M is called siinply a (Cm) submanifold of M if the inclusion map i ; MI + M is an imbeddirlg; it is called a closed submanifold of M if MI is also a closed subset of M.

Tliere is one way of getiiilg submanifolds wl-iich is very important, aild @ves the sphei-e S"-' C W" - {O) c JR", defined as (x : 1x12 = 11, as a special case. 12. PROPOSITION. If f : M" + N lias constani railk k on a neighborhood of f (J!), then f (y) is a closed submanifold of M of dimension n - k (or is empty). Iii particular, if y is a regular value of f : M" + Nm, then f (y) is an (11 - 111)-dimensionalsubmanifold of M (or is empty).

-'

-'

-'

PROOF. Left to the reader. 43 It is to be hoped that however abstract the noiion of Cm manifolds may appear, sul~manifoldsof JRN will scem like faii-ly concrete objects. Now it turns out tliat e u q r (connected) Cm manifold can be imbedded in some J R N , so that manifolds caii be pictured as subsets of Euclidean space (though this picture is iiot always die nlost uscful one). We will prove tliis fact oilly for compaci mailifolds, but we first develop some of the macl~iileiywhich would be used in

the general case, since we xrtil1 need it later on anyway. Unfortunately, there are many definitions and theorems involved. If O is a cover of a space M, a cover O' of M is a refinement of O (or "refines O ") if for every U in O' there is some V in O with U c V (the sets of O' are "smaller" than those of O)-a subcover is a very special case of a refining cover. A cover O is called locally finite if every p E M has a neighborhood W which intersects only finitely many sets in O. 13. THEOREM. If O is an open cover of a manifold M, then there is an open cover O' of M whicli is locally finite and which refines O. Moreover, we can choose al1 members of O' to be open sets dfleomorphic to R".

PROOF. \Ve can obviously assume that M is connected. By Theorem 1.2, there are compact seis Cl,C2,C3,. . - with M = CI U C2 U C3 U - - - - Clearly Ci has an open neighborliood U] with compact closure. Then U Ci has an open neighborhood U2 ~ 4 t hcompact closure. Continuing in this way, we obtain c Ui+i, whose union contains aH C;, open sets Ui, witli compact and and lience is M. Let U-] = U. = 0.

-

Now M is ilic union for i > 1 of the "aniiular" regons A i = Ui - Ui-]. Since eacli A i is compact, we can obviously cover Ai by a finite number of open sets, eacli coiiiaiiied in some membcr of O, and eacli coiitained in Vi = Ui+, - Ui-2. \Ve can also clioose these open sets to be diffeomoi.phic to RR.In this way we obiain a cover O' wliich refines O aild wliicli is locally finite, since a point in Ui is noi in 6 for j 3 2 + i. /f,

Notice that if O is an open locally finite cover of a space M and C C M is compact, then C intersects o~ilyfinitely many members of O. This shows that aii open IocaHy finite cover of a connected manifold must be countable (like the cover constructed in the proof of Theorem 13). 14. THEOREM (THE SHRINKING LEMMA). Let O be an open locally firiiie co\rer of a manifold M. Then it is possible to choose, for each U in O, an open set U' wiui c U in such a way that the collection of al1 U' is also an open cover of M.

PROOF. 1442 can clearly assume that M is connected. Let O = (Ui, U2, U3,. . .). Then c1= u, - (U2UU3U . - - ) is a closed set co~itainedin U], and M = Ci U U2 U U3 U. . . L t U{ be an open set witli Ci c U; c c U1 Now

is a closed set contained in U2, and M = U; U C2 U U3 U - . Let U; be an c U2. Continue in this way. open set wiili C2 c U; c For aliy p E M tliere is a largest n witli p E U,, because O is locally finite. Now p E u; UU; U U u; U (u.+, U UN+2U . - . ) ; -

S

it follows that

P

E

U/UU;Ua.*,

since replacing U,+í by Un+i cannot possibly eliminate p. 34

15. THEOREL4. Let O be an open IocaHy firiite cover of a manifold M. Then ihere is a collection of Cm functioiis # u : M + [O,11, one for each U in O, such tllat (1) support $U C U for each U, (2)

$u(/)) = 1 for al1 p

E

U

iieigliborliood of p, by (1)).

M (this sum is really a finite sum in some

PROOF. Cme 1. Each U in O Rar ~omjia~t closure. Choose the U' as in Theorem 14. Apply Lemma 2 to c U c M to obtain a C m function Jru : M + [O, 11 wliicli is 1 on aiid has support c U. Since the U' cover M ,clearly

Define

2. Gerie~alc a e . Tliis case can be proved in tlie same way, provided diat Lemma 2 is irue for C c U c M with C closed (but not necessarily compact) and U open. But this is a consequence of Case 1: For eacli y E C choose an open set Up c U with compact closure. C w e r M - C with open sets V. haviiig compact closure and contained in M - C. The open cover ( U p ,V..) lias an open locally fiilite refinement O to which Case I applies. Let Cme

Tliis sum js Cm,silice it is a fi~iitesum iii a iieigliborliood of each point. Since #u ( p ) = 1 for al1 p , arid #u ( p ) = O when U c V., clearly J ( p ) = 1 fol. al1 p E C. Usirig tlic faci tliat O is locally finite, it is easy to see tliat support f c U.+:+

xu

16. COROLLARY. If O is aiiy open cover of a n~aiiifoldM , then ihere is a collection of Cm functions # i : M + [O, I] such that (1) tlie collection of sets ( p : # i ( p ) # O) js locally finite, (2) t i#i( p ) = I for al1 p E M , (3) for eacli i tliere is a U E O such that support #i C U. (A collccti~ri( # i : M + [O, 11: satisfying (1) and (2) is called a partition of unity; if it satisfics (3), ii is called subordinate to O.) It is riow fairly easy to prove tlie last theorem of tliis cliapter. 17. THE0REh.I. If M " is a conipact C m mailifold, tlien tliere is aii iml~edcliiig /: M + IRN for some N.

1'1200F. TIiere are a £nite number of coordinaie systems (xl, Ul), . .. , (xk,Uk) witl-i M = Ul U - - - UUk. Choose U; as in Tlieorem 14, and functions @i : M + [O, 11 whicli are 1 on and have support c Vi. Define f : M + RN, where N=nk+k,by

Tl-iisis ai-i inlmersion, because aiiy poiiit p is in U; for some i, and on U,', where qfj = 1, ihe N x it Jacobian matrix

It is also one-one. For suppose that f ( p ) = f(q). There is some i such that p E U;. T1-iei-i@i(p) = 1, SO also @i(q) = 1- This shows that we musi llave y E Ui. Moreover, @i ' xi (P) = @i - X i (Y),

Problem 3-33 sl-iows that, in fact, we can always choose N = 2rz

+ 1.

1, (a) Sliow tliat beir-ig Cm-r-elated is no.! a1.i equivalente relation. (b) 111 the pl-oof of Leninia 1, show ihat al1 charis in A' are Cm-related, as claimed.

2. (a) If M is a 1-ileti.i~space togeiber wiil-i a collection of homeomorphisms x: U + Rn whose domains cover M and which are Cm-related, show that tl-ie tt at each yoiiit is ui-iique rtrillrouf using Irivariai-ice of Domain. (b) SI-iowsin-iilarlythai aM is well-defined for a Cm manifold-with-boundary M. 3. (a) Al1 Cm fui-ictions are continuous, ai-id il-ie coinposition of Cm functions is Cm. (b) A funciion f : M + N is Cm if al-id only if g o f is Cm for every Cm func tion g : N + R . 4. How inaiiy clistii-ict Cm stl-uctures are thel-e 011 R? (Tlleie is only one up to

diffeon-iorpliisin;il-iat is i-ioi the question beil-ig asked.)

5. (a) If N c M is open and A' consists of al1 ( x , U) in A with U c N, show thai A' is maximal for N if A is maximal for M. (b) Show that A' can also be described as ihe set of al1 (xlV í l N, V í l N) for (x, V) in A. (c) SIlow ihai ihe inclusion i : N + M is Cm, and that A' is the unique atlas with this property. 6. Clieck thai tlie iwo projections Pl and P2 on S"-] are Cm related to the 2n l~omeomor~hisms _I;: and gj.

7. (a) If M is a connected Cm marlifold and p,q

E

M, tl-ien ihere is a Cm

c u m c : [O, I ] + M with c(0) = p and c(I) = q. (b) It is even possible to clioose c to be one-one. 8. (a) Sliow ihat (MI x M2) x M3 is dfleomorphic io MI x (M2 x M3) and iI-iai MI x M2 is dfleomorphic to M2 x M I . (b) The dflcrentiable structure on MI x M2 makes the "slice" maps

of M I , M2 + M1 x M2 diffel-entiablefor a11 pl E M i , p2 E M2. (c) Pl4ore gerierally, a map f : N + MI x M2 is Cm if and only if the composiiions rrl o f : N + Mi and rr2 o f : N + M2 are Cm. Moreover, ihe Cm structure \ve llave defined for Mi x M2 is the only one wiih this property. (d) If fj : N + Mi are C m (i = 1,2), can one determine ihe rank of ($1,$2) : N + Mi x M2 (71 p in terms of the ranks of _f;: at p? For fj : Ni + Mi, show thal Si x f2: Ni x N2 + Mi x M2, defined by ji x fi(pi, p2) = (Si(pi1,Sz(p2)), is Cm and determine its rank in terms of the ranks of ji.

9. Let g : S" + P"be tlie map p H [p]. SIiow that f : P" + M is Cm if and only if j' o g : S" + M is Cm. Compare the rank of f and the rank of f o g . 10. (a) If U c R" is open and f : U + R is locally Cm (every point has a iieiglil~orlioodori wllich f is Cm), tllen f is Cm. (Obvious.) (b) If f : W" + R is locally Cm, then f is Cm, i-e-,f can be exieilded to a Cm function on a ileighborhood of M". (Not so obvious.)

1 1 . If f : W" + R Ilas hvo extensions g, h to Cm functions in a neighborhood of M", illeli Qg and Q h are ihe same at points of R"-' x (O) (so we can speak of Dif at these points). 12. If M is a Cm rnanifold-wiil-i-boundary, tlien there is a unique Cm struciure ori aM sucll iliai il-ie inclusion map i : aM + M is an imbeddiilg.

13. (a) Let U C M" be an open set sucb ihat boundary U is an (n - 1)-dimensional (dflereiitiable) submanifold. Show tliat U is aii 11-dimensionalmanifoldwith-boundary (It is well to bear in mind the following example: if U = {x E EX' : d ( x ,O) < 1 or 1 < d ( x ,O) < 21, then Ü is a manifold-with-boundary, but aÜ # boundary U.) (b) Consider the figure shown below. This figure may be extended by putting

srnaller copies of tlie two parts of S' into the regions indicated by arrows, and ihen repeatirig tliis coristi-uction iudefinitely. The closure S of the final resulting figure is known as Alexnnder's Houied SSplnr. Sliow tliat S is liomeomorphic to s2. (Hinl: Tlie additional poinis in tbe closure are bomeomorphic to the Cantor sei.) If U is tlie uiibourided component of W3 - S, ihen S = boundary U, but Ü is not a 2-dimensional manifold-with-boundary, so part (a) is true only for differentiable su bmanifolds. 14. (a) Therc is a map f : EX2 + EX2

f (x,O) = (x,O) for al1 x, (2) f(x, y) c EI2 for y > O, (3) f ( ~ , yc) W2 - IN2 for y < O, (1)

S U C ~ Itliat

(4) f restricted to the upper half-plane or the lower half-plane is Cm, but f itself is not Cm.

(b) Suppose M and N are Cm manifolds-with-bounda~yand f : aM + aN is a difiomorphism. Let P = M U 1 N be obtained from the disjoint union of M and N by identifying x E aM with f (x) E aN. If (x, U) is a coordinate system around p E a M and ( y , V) a coordinate system around f ( p ) , with f ( U n a M ) = V n a N , and ( y o f)IU n a M = xlU n a M , we can define a homeomorphism from U U V c P to IRn by sending U to Wn by x and V to tlle Iower half-plane by the reflection of y. Show that this procedure does not

define a CbO structure on P. (c) Now suppose t11at there is a neighborhood U of aM in M and a dfleomorphism a : U + 8M x [O, l ) , such that a ( p ) = (p,O) for al1 p E aM, and a similar diffeomorphism p : V + aN x [O, 1). (M'e will be able to prove later that such diffeomorpllisms always exist). Show that tllere is a unique CbO structure

on P sucl-i tl-iat t l ~ eii~clusionsof M and N are CbO and such that the map from U U V to aM x (- 1,l) induced by a and B is a dfleomorphism. (d) By usiilg two difTei-entpairs (a,p), define two dflerent Cm structures on EX2, collsidcred as thc union of two copies of W 2 witli corresponding points on a W 2 iclei~tified.SIlow that the resulting CbO manifolds are dfleomorphic, but that the difKeomorphism ca?~nolbe chosen arbitrai-iIy close io the identity map.

15. (a) Find a Cm structure on W' x M' which makes the úidusion into EX2 a Cm map. Can the inclusioi-i be an imbedding? Are the projections on each factor Cm maps? (b) If M and N are manifolds-with-boundary, construct a Cm structure on M x N such that ali the "slice maps" (defined in Problem 8) are Cm.

16. Show that the fui-iction f : IR -+ IR defined by

is Cm (the formula e-'ix2 is used just to get a function which is > O for x < 0, and e-'/lXl could be used just as weil).

17. Lemma 2 (as addended by the proof of Theorem 15) shows that if Cl and C2 are disjoint closed subsets of M, then there is a Cm function f : M + [O, 11 such that Ci c f -'(O) and C2 c f -'(I). Actually, we can even find f with Cl = /-'(O) and C2 = f - ' (1). The proof turns out to be quite easy, once you know the trick. (a) It suffices to find, for any closed C c M, a Cm function f with C = /-'(O). (b) Let ( U i ) be a countable cover of M - C, where each Ui is of the form

for some coordinate system x taking an open subset of M - C onto EXn. Let Si : M + [O, 11 be a Cm function with _I;: > O on Ui and j;. = O on M - Ui. Functions like

-

axi'

will be called mixed partials of ai

axíaxk5

A, of order

= sup of all mixed partials of

Show that

"'

1,2,. .. . Let

Si,. .. ,f;. of al1 orders 5 i.

f = "Cai2' -Si i=I

is Cm, and C = f -l (O). 18. Consider the coordinate system (y1,y2) for IR2 defined by y' ( a ,b) = a y2(a,b) = a

+ b.

(a) Compute f/ayr(a, b) from the definition. (b) Also compute it from Proposition 3 (to find a I i / a y j , write each 1' in terms of y' and y2). Notice that af/ayl # 8/ / a l r even though on y and i, not just on y'.

= I r ; the operator

alay'

depends

19. Compute the "Laplacian"

in terms of polar coordinates. (First compute a/ax in terms of alar and a/a9; r a aa;) ae a (,=)J. r a then compute a2/ax2 from this). Amwer: ;

+

20. If : M" + Nm is C1 and m > n, then /(M) has measure O (provided that M has only countably many components). 21. The following pictures show, for 11 = 1,2, and 3, a subdivision of [O, 11 x [O, 11 into 22n squares, An,i,. . . ,An,2b; square An,k is labeled simply k. The numbering is determined by the following conditions: (a) The lower left square is A,,]. (b) The upper left square is

An1221f.

(c) Squares A,,k and AR,k+l have a common side. (d) Squares An,41+1, An,41+2, An,41+3i An,41+4 are contained in A,-, ,[+l.

Define f : [O, 11 + [O, 11 x [O, 11 by the condition f(t)

E An,k

for al1

k-1 22"

k 22"

- - - ,

This expression sliows that y* o (x*)-' is Cm.

ai ~i ( Y"

0

x-l) (t))-

\/Ve thus have a collection of Cm-related charts on T M , which can be extended to a maximal atlas. With this Cm structure, the local trivializations x, are Cm. In general, a vector bundle rr: E + B is called a Cm vector bundle if E and B are Cm manifolds and there are Cm local trivializations in a neighborhood of each point. It follows that rr : E + B is Cm. Recall that a section of a bundle rr : E + B is a continuous function s : B + E such that rr O S = identity of B; for Cm \lector bundles we can also speak of Cm sections. A section of T M is called a vector field on M ; for submanifolds M of Rn, a vector field may be pictured as a continuous selection of arrows tangent to M. The theorem that you can't comb the hair on a sphere just states that

there is no vector field on s2which is werywhere non-zero. We have shown that there do not exist two vector fields on the Mobius strip which are everywhere linearly independent. Vector fields are customarily denoted by symbols like X, Y, or 2, and the \rector X(p) is often denoted by X, (sometimes X may be used to denote a single vector, i ~some i M,). If we think of T M as the set of derivations, then for any coordinate system (x, U), we have n

X(P) = C ~ ' ( P )

for al1 p

E

U.

i=l

The functions a' are continuous or Cm if and only if X : U + T M is continuous or Cm. If X and Y are two vector fields, we define a new vector field X + Y by

Similarly, if f : M + R, we define the vector field JX by

The 7angeni Bundle

83

Clearly X + Y and f X are Cm if X, Y, and f are Cm. On U we can write

the symbol a/axi now denoting the vector field

If f : M + R is a Cm function, and X is a \lector field, then we can define a new fwidion 2 ( f ): M + B by letting X operate on f at each point:

It is not hard to check that if X is a Cm vector field, then every Cm function f ; indeed, iflocally

f (f )

is Cm for

then

which is a sum of products of Cm functions. Conversely, if y(f ) is Cm for e u q Cm function f , then X is a Cm vector field (since 2 ( x i ) = a'). Let F denote the set of al1 Cm functions on M. We have just seen that a Cm vector field X @ves rise to a func tion 9 : P + P. Clearly,

tlius 2 is a "derivation" of the ring F. Ofien, a Cm vector field X is identified witli the derivation f . Tlie reason for this is that if A : P + 7 is any derivation, then A = 2 for a unique Cm vector field X. In fact, we clearly rnust define X,(f> = A ( f )(P), and the operator XP thus defined is a derivation at p.

The tangent bundle is the tme beginning of the study of dflerentiable manifold~,and you should not read further until you grok it.* The next few chapters constitute a detailed study of this bundle. One basic theme in all these chapters is that any structure one can put on a vector space leads to a structure on any vector bundle, in particular on the tangent bundle of a manifold. For the present, we will discussjust one new concept about manifolds, which arises in this very way from the notion of "orientation" in a vector space. The non-singular linear maps f : V + V from a finite dimensional vector space to itself fall into two groups, those with det f > 0, and those with det f < 0; linear transformations in t he first group are called orientation preserving and the others are called dentation reversing. A simple example of the latter is the map f : Rn + R" defined by f (x) = (x1,. . .,xn-l, -xn) (reRection in the llyperplane xn = O). There is no way to pass continuously between these two groups: if we identify linear maps Rn + R" with n x n matrices, and thus with IRn2, then the orientation presewing and orien tation reversing maps are disjoint open subsets of the set of al1 non-singular maps (those with det # 0). The terniinology "orientation presewing" is a bit strange, since we have not yet defined anything called "orientation", which is being preserved. T h e problem becomes more acute if we want to define orientation preseMng isomorphisms between two dflerent (but isomorphic) vector spaces V and W; this clearly makes no sense unless we supply V and W with more structure. To provide tliis extra structure, we note that two ordered bases (vi,. . ., u,) and (vf1,.. . , vfn)for V determine an isomorphism f : V + V with f(vi) = vfi; the matrix A = (aij) of f is given by the equations

We cal1 (vi,. .. ,u,) and (vfl,.. .,u',) equally oriented if det A > O (i.e., if f is orientation preserving) and oppositely oriented if det A < 0. Tlle relation of being equally oriented is clearly an equivalence relation, dividing tlie collection of al1 ordered bases into just two equivalence classes. Either of iliese two equivalence classes is called an orientation for V. The class to which (vi,. . .,un) belongs will be denoted by [vi,. . .,vn], so that if p is an orientation of V, then (vi,... ,un) E ,u if and only if [vi,. . .,un] = p. If ,u denotes one * A cult word of the sixties, "grok" was coined, purportedly as a word from the Martian language, by Robert A. Heirilein in his pop ccience fiction novel Shnger in a Slrange Lnttd. lts sense is nicely coiiveyed by the defiriitjoii in The Anzen'can H d h g e Diclio?tag*: "To understand profoundly through intuition or empathy".

The 7angeni Bundle

Examples of equally oriented ordered bases in R, IR2, and IR3. orientation of V, tbe other will be denoted by - p , and the orientation [ e ] , ., . ,e,] for R" will be called the "standard orientation". Now if (V, p ) aild (W, u ) are two ri-dimensional vector spaces, together with orientations, an isomorphism f : V + W is called orientation preserving (with respect to p and u) if [f ( v i ) ,- .-,f ( v , ) ] == v whenever [ V I , . . ., u,] = p; if this holds for any one ( v i ,. . . ,u,), it clearly holds for all. For tlie trivial bundle P ( X ) = X x R" we can put the "standard orientation" [ ( x ,el),.. . , ( x ,e,)] on each fibre {x)x Rn. If f : e n ( X )+ e n ( X )is an equivalente, aiid X is connected, tlien f is either orientation presewing or orientation reversing on eacli fibre, for if we define the functions aij : X + R by

then det ( a i j ): X + R is coniinuous and never O . If rr : E + B is a nontrivial 11-plane bundle, an orientation p of E is defined to be a collection of condition" orientatioiis pp for rr-I ( p ) wllicll satisfy the following "~ompatibilit~ for any open connected set U C B: lf t : x-' ( U ) + U x IRn is an equivalence, and the fibres of U x IR" are given tlie standard orientation, tlien t is either orientation presewing or orientation reversing on all fibres. Notice tliat if tliis condition is satisfied for a certain t, and t' : n-' ( U ) + U x Rn is another equivalence, then t' automatically satisfies the same condition, since

-'

t : U x 1" + U x 1" is an equivalente. This shows that the orientations pp define an orientation of E if the compatibility condition holds for a collection of sets U which cover B. If a bundle E has orientation p = {,up), it has another orientation -p = {-,up), but not every bundle has an orientation. For example, the Mobius strip, considered as a 1-dimensional bundle over S ' , has no orientation. For, although the Mobius strip has no non-zero section, we can pick two vectors from each fibre so that the totality A looks like two sections. For example, we can let A be [O, i ] x {- 1, 1) with (O, a ) identified with (1, -a); then A just looks like the boundary of the Mobius strip obtained from [O, i] x [- 1 , 11. If t'o

we had compatible orientations p,,, we could define a section S : S' -, M by choosing s ( p ) to be the unique vector s ( p ) E A nn-'(p) with [s(p)] = pp. A bundle is called orientable if it has an orientation, and non-orientable otherwise; an oriented bundle is just a pair (6, p ) where p is an orientation for 6. Tliis definition can be applied, in particular, to the tangent bundle TM of a Cm manifold M. In this case, we cal1 M itself orientable or non-orientable depending on whether TM is orientable or non-orientable; an orientation of TM is also called an orientation of M, and an oriented manifold is a pair ( M , p ) where p is an orientation for TM. Tlle manifold Rn is orientable, since T W 2 E" (Rn), on which we have the standard orientation. The sphere S"-' c Rn is also orientable. To see this we

note that for each p E S"-' the vector w = pp E en(Rn) 2 TR" is nol in .i (Sn-',,) E T Rn,, (Roblem 2 l), so for V I ,.. . ,v.- 1 E Sn-lp we can define (vi,. .. ,u n - ' ) E ,up if and only if (w, i,(vi), . .. ,i, is in the standard orientation of Rnp. The orientation p = {/L, : p E S"-' ) thus defined is called the "standard orientation" of S"-'. The iorus S' x S' is aiiotlier example of an orientable manifold. This can be seen by noting that for any two manifolds Mi and M2 the fibre (Ml x M2)p of T (MI x M2) can be written as VI $ V2p where (ni)* : K,,+ (Mi),, is an isomorphism and the subspaces Vi, vary continuously (Problem 26). Since Ts' is trivial, this shows that T (S' x S') is also trivial, and consequently orientable. Any n-holed torus is also orientable-the proof is presented in Problem 16, which also discusses the tangent bundle of a manifold-with-boundary. The Mobius strip M is the simplest example of a non-orientable 2-manifold. For tlie imbedding of M considered previously we have aiready seen that on the

subset S = ((2cos 8 , 2 sin 8, O)) c M there are continuously varying vectors u,,, but tliat it is impossiWe io clioose continuously from among the dashed vectors wp = f*((O, I ) ( ~ , ~and ) ) their negatives. If we had orientations /l, for p E S, then we could simply choose wp if [up, wp] = pp and -wp otherwise. T h e projective plane IP2 must be non-oiieniable also, since it contains the Mobius strip (for any orientable bundle 6 = rr : E + By the restriction 6IB' to aiiy subset B' c B is also orientable). Non-orientability of IP2 can be seen in aiiother way, by considering the "antipodal map" A : S2 -t S2 defined by A(p) = -p. This rnap is just the restriction of a linear map A: R3 + R3 defined by tlie same formula. T h e rnap A * : S2,, -t s ~ is ~just ( (~ p ,)~ H ) A(v)), wlien S2,, is identified with a subspace of { p ) x R3. T h e map A is orientation reversing, so if vi = (p, ui) E SZP,the bases (ui, u2, p)

and

(A(ui), A(u2), A(P))

are oppositely oriented. Tliis shows that if ,u is the standard orientation of S2 and [ul, vi] E ,up, then [A,v1, A,v2] E -/,LA(,,). Thus the map A : S2+ s2is

"orientation reversing" (the notion of an orientation preserving or orientation reversing map f : M + N makes sense for any imbedding f of one oriented manifold into another oriented manifold of the same dimension). From this fact it follows easily that IP2 is not orientable: If IP2 had an orientation v = {v[,l) and g : s2 -t IP2 is the map p H [ p ] , then we could define an orientation {ji,) on S" by requiring g to be orientation preserving; the map A would then be orientation preserving with respect to fi, which is impossible, since k. = p or -p. For projective 3-space IP3 the situation is just tlle opposite. In this case, the antipodal map A : S' + s3ir orientation presening. If g : S 3 + IP3 is the map p H [P],we obviously can define orientations v, for IP3 by requiring g to be orientation presewing. In general, these same arguments show that IP" is orientable for n odd and non-orientable for n even. There is a more "elementary" definition of orientability which does not use the tangent bundle of M at all. According to this definition, M is orientable if there is a subset A' of the atlas A for M such that (1) tlle domains of al1 (x,U)

A' cover M, (2) for al1 (x, U) and (y, V ) E A', E

An orientation p of TM allows us to distinguish the subset A' as the collection of al1 (x, U) for which x, : TM lU + T(x(U)) i x(U) x IRn is orientation preserving (when x ( U ) x IR" is given the standard orientation). Condition (2) liolds, because it is just the condition that (y o x-'),: T(x(U)) + T(x(U)) is orientation preserving. Conversely, given A' we can orient the fibres of TM IU in such a way that x, is orientation preserving, and obtain an orientation of TM. Although our original definition is easier to picture geometrically, the determinant condition will be very important later on.

The 7angeni Bundle

ADDENDUM EQUIVALEMCE OF T A N G E N T B U N D L E S The fact that al1 reasonable candidates for the tangent bundle of M turn out to be essentially the same is stated precisely as foilows. 4. THEOREM*. If we have a bundle T'M over M for ea& M, and a bundle map (h,f ) for each Cm map f : M + N satisfying

(1) of Theorem 1, (2) of Theorem 1, for certain equivalences t'",

(3) of Theorem 1, for certain equivalences T'U

h:

(TfM)IU,

then there are equivalences eM :

TM + T'M

such that the following diagram commutes for every Cm map f : M + N.

PROOF. The details of this proof are so horrible that you should probably skip it (and you should definitely quit when you get bogged down); the welcome symbol 4 4 occurs quite a ways on. Nevertheless, the idea behind the proof is simple eilough. If (x, U) is a chart on M, tllen both (TM)I U and (T'M) 1 U "look like" x ( U ) x Rn,so there ought to be a map taking the fibres of one to the fibres of the other. Wliat we have to hope is that our conditions on TM and T'M make them "look alike" in a sufficiently strong way for this idea to really work out. Those who have been through this sort of rigamarole before know (i-e.,have faith) that it's going to work out; those for whom this sort of proof is a new expei-ienccshould find it painful and instructive,

* Functorites will iiotice that Theorems 1 and 4 say that there is, up to natural equivaleiice, a uiiique functor from the category of Cm manifolds and Cm maps to the category of bundles and bundle maps which is naturally equi\~alentto (E", old f,) on Euclidean spaces, and to the restriction of the functor on open submanifolds.

Let ( x , U ) be a coordinate system on M . Then we have the following string of equivalences. Two of them, which are denoted by the same symbol 2 ,are the equivalences mentioned in condition (3). Let a, denote the composition a, = ( f n l x ( u )0) 0 x* 0 (-)-l.

Similarly, using equivalence 2' for T I , we can define p,.

Then

Bx-'

o

ax : (TM)IU -t (T'M)IU

is an equivalence, so it takes tlie fibre of T M over p isomorphically to the fibre of T I M over p for ea& p E U . Our main task is to show that this isomorphism between the fibres over p is independent of the coordinate system ( x , U ) . This will be done in three stages.

(1) Suppose V maps

c U is open and y

= xlV. We will need to name all the inclusion

To compare a, and ay,consider the following diagram.

TIze 7angeni Bundle

91

O,

Each of the four squares in this diagam commutes. To see this for square we enlarge it, as sllown below. The two triangles on the left commute by condition (3) for TM, and the one on the right commutes because i o j = r .

Square @ commutes because k o y = x o j. Square @ commutes for the same reason as square - @; tlie inclusions x ( U ) + Rn and y(V) + IRn come into play. Square (4) ob~iouslycommutes. Chasing through diagram (1) now shows tllat the follo-ving commutes.

This means that for p E V, the isomorphism ay bemeen the fibres over p is the same as a,. Clearly the same is true for B, and By, since our proof used only properties (l), (21, and (3), not the explicit construction of TM. Thus By-' o (Y,, = B x - l O ax on the fibres over p, for every p E V.

(11) We now need a Lcmma which applics to both TM and TIM. Again, it will be proved for TM (where it is actually obvious), using only properties (l), (2), and (3), so that it is also true for T'M.

LEMMA. If A c IRn and B

c IRm

are open, and f : A + B is Cm, then the

following diagram commutes.

PROOF. Case 1. %e

ir a map

/:Wn + Rm with

7 = f on A .

Consider the following diagram, where i : A + IRn and j : B + IRm are the incIusion maps.

Everythirlg in tliis diagram obviously commutes. This implies tliat the two compositions Ev

T A 2( T R")I A

tnlA

&"(Rn)IA

d

C

dE"

(IRn)

old

/;

.em(Rm)

and

are equal and this proves the Lemma in Case 1, since the maps "old "old f," are equal on A.

T*'' and

&e 2. General' cae. For each p E A , we want to show that m o maps are the same on the fibre over p. NOWthere is a map j : IRn + IRm with j' = f on an open set A', wl~erep E A' C A. We tben have tlie following diagram, wliere every i comes from the fact tliat some set is an open submanifold of another,

The Gngeni Bundle and i: A' + A is the inciusion map.

0,

Boxes @, and @ obviously commute, and commutes by &e l . To see that square @ (whidi has a triangle witliin it) commutes, we imbed it in a larger diagram, in which j: A + R" is the incIusion map, and other maps have also been named, for ease of reference. T Rn

(u) h /

TA

E

(TRn)IA

i*] E

+(TP)IAr TA1 To prove that h o i, = p o K, it suffices to prove that

since v is one-one. Thus it suffices to prove j* o i* = v o p o o, which amounts to proving commutativi ty of the foilowing diagram.

Since j o i is just the inclusion of A' in Rn, this does commute.

Commutativity of diagram (2) shows that the composition

coincides, on the subsel (TA)IA', with the composition

and on A' we can replace "old /," by "old f,". In other words, the two compositions are equal in a neighborhood of any p E A, and are thus equal, which proves the h m m a . (111) Now suppose ( x , U) and (y, V) are any two coordinate systems with p E U n V. To prove that By-' 0 a, and Bx-' o a, induce the same isomorphism un the fibre of TM at p , we can assume without loss of generality that U = V, because part (1) applies to x and xl U n V, as well as to y and y1 U n V. Assuming U = V, we have the following diagram.

ry

a.-

(3) (TM)lU 4 TU

old ( y o x-'),

(y 0 h.-1)*

O-R")ly(U)

Tlle triangle obviously commutes, and tlle rectangle commutes by part (11). Diagram (3) thus shows tbat

Exactly the same result llolds for T':

The desired result B,-' o a, = B,-' o a, follows immediately. Now that we have a well-defined bundle map TM + T'M (the union of al1 Bx-' 0 a,), it is dearly an equilraience e M . The proof that eni o f, = ffi o e M is left as a masochisiic exercise for the reader. *:*

n l e Zngeni Bundle

PROBLEMS 1. Let M be any set, and { ( x i ,Ui)) a sequence ofone-one functions xi : Ui + Rn with Ui c M and x (Ui) open in Rn, su& that each

is continuous. It would seem that M ought to have a metric which makes each Ui open and each xi a homeomorphism. Actually, this is not quite true: (a) Let M = R U {s), where s 6 R. Let U] = R and x l : Ul + R be the identity, and let U2 = R - {O) U {s), with x2: U2 + R defined by

Show that there is no metric on M of the required sort, by showing that every neighborhood of O would have to intersect every neighborhood of s. Nevertheless, we can find on M a pseudometric p (a function p : M x M + R with a11 properties for a metric except that p (p, q) may be O for p S; q) such that p is a metric on each Ui and each x j is a homeomorphism: (b) If A C IRn is open, then there is a sequence A l , A2, A j, .. . of open subsets of A such that every open subset of A is a union of certain Ai's. (c) There is a sequence of continuous functions 1;.: A + [O, l], with support J c A, which "separates points and closed sets": if C is closed and p E A - C, then there is some j;: with fi (p) 6 (A nC). Hint: First arrange in a sequence all pairs (Ai, Aj) of part (b) with C Al. (d) Let Alj, j = 1,2,3,. .. be such a sequence for each open set xi (Ui)- Define gi,j : M + [o, 11 by

Arrarlge a11 gil] in a single sequence Gl, G2, G3,. .. , let d be a bounded metric on R, and define p on M by

Show that p is the required pseudometric. (e) Suppose that for every p,q E M there is a Ui and Uj wit.h p E Ui and q E Uj and open sets Bi c xi(Ui) and Bj c xj(Uj) SO that p E xi-'(Bi), q E xj-'(l?j), and xi-'(Bi) n xj-'(Bj) = 0. Sliow that p is actually a metric on M.

Tle Tangeni Bundle

97

9. (a) Show that the correspondence between T M and equivalence classes of curves under which [ x ,u], corresponds to the 7 equivalence clau of x d l o y, for y a curve in R" with y'(0) = u, makes f, correspond to $#. (b) Show that under the correspondence [ x , a ] ,n Ciaia/axilp,the map f, can be defined by Ef*re)l(g)= [ ( g f ) . 10. If V is a finite dimensional vector space over R, define a Cm structure on V and a homeomorphism from V x V to T V which is independent of choice of bases. As in the case of R", for u, w E V we will denote by u, E Vw the vector corresponding to ( w , u). 1 1. If g ; R + R is Cm show that

for some Cm function h : R + R. 12, (a) Let F , be the set of al Cm functions f : M + R with f ( p ) = 0, and 1et F, + R be a linear operator with f g ) = O for al1 f,g E F,. Show that l has a unique extension to a derivation. (b) Let W be the vector subspace of 3, generated by al1 products fg for f , g E F,. Show that the \rector space of al1 derivations at p is isomorphic to the dual space (.7;p / W )*. (c) Since (F,/ W)* has dimension n = dimension of M, the same must be true of F,/ W. If x is a coordinate system with x ( p ) = O, show that x1 + W , . . ., x" + W is a basis for F,/ W (use Lemma 2). The situation is quite different for C1 functions, as the next problem shows.

e:

e(

13, (a) Let V be the \lector space ofall C1 functions f : R + R with f ( 0 ) = 0, and let W be the subspace generated by ail products. Show that lim f ( x ) / x 2 x-o exists for al1 f E W. (b) Foro < E < 1, jet

Show that al1 $, are in V, and that they rcpresent linearly independent elements of V / W. (c) Conclude that ( V / W)* has dimension cC = 2'. 14, If f ; M + N aild f, is the O map on each fibre, then f is constant on each component of M .

15, (a) A map f : M + N is an immersion if and only if f* is one-one on each fibre of T M . More generally, the rank of $ at p E M is the rank of the linear transformation f , : Mp + N f ( p l . @) If f o g = f , where g is a difkomorphism, then the rank of f o g at a equals the rank of f at g ( a ) . (Compare with Problem 2-33(d).) 16. (a) If M is a manifold-with-boundary, the tangent bundle T M is defined exactly as for M ; elements of M p are y equivalente classes of pain ( x ,u). Although x takes a neighborhood of p E aM onto W n , rather than R", the ueciors v still run through IR", so Mp still has tangeilt vectors "pointing in ail directions". If p E aM and x : U + W" is a coordinate system around p, then

x,-' (Rn-',(,)) c M, is a subspace. Show that tliis subspace does not depend on the choice of x;in fact, it is i,(aM),,, where i : aM + M is the inclusion. (b) Let a E IRn-' x { O ) c W n . A tangent vector in W", is said to point "inward" if, under the identification of T W n with &"(Mn), the vector is ( a , u) where un > O. A vector v E Mp which is not in i,(aM)p is said to point "inward" if

x*( u ) E Wnqp1 points inward. Show that this definition does not depend on the coordinate system x. (c) Show tllat if M has an orientation p, then aM has a unique orientation a p such that [ v i ,. .- , = (ap),, if and only if [ w ,i,vi,. .. ,i , ~ , - ~ ]= pp for every outward pointing w E M p . (d) If p is the usual orientation of W",show that a,u is (- l ) n times the usual orientation of IRn-' = aWn. (The reason for this choice will become clear in Chapter 8.) (e) Suppose we are in the setup of Problem 2-14, Define g : 8M x [O, 1 ) + aN x [O, 1 ) by g ( p , t) = ( $ ( p ) , 1 ) . Show that T P is obtained from T M U T N

Tlze Z q m i Bundle

(f) If M and N have orientations p and u and f : ( a M , a p ) + (aN,au) is orientation-reuer~rg,show that P has an orientation which agrees with p and u on M c P and N c P. (g) Suppose M is s2 with two holes cut out, and N is [O, 11 x S ' . Let f be a dfiomorphism from M to N which is orientation preserving on one copy of S' aiid orientation reversing on the other. Wliat is the resulting manifold P? 17. Sliow that T P is~ homeomorphic to the space obtained from T ( s ~ ,by ~) identifying ( p , u) E (s2, i), with ( - p , - u ) E (s2, i)-,. 18. Although there is no everywhere non-zero \rector field on s2,there is one on s2- { ( O , 0, l ) ) , which is dfiomorphic to IR2. Show that such a vector field can be picked so that near ( O , 0 , l ) the vector field looks like the following picture (a "magnetic dipole"):

19. Suppose we have a "multiplication" map ( a ,b) H a - b from that inakes R" into a (non-associative)division algebra. That is,

(al + a2) b = al b + a2 b a (bi +b2) = a .bl + a .b2 h(a - b ) = (ha) b = a (hb) for h a (l,O,. ,.,O) = a and there are no zero divisors:

a,b#O

ab#O.

E

R

IR" x R" to R"

(For example, for 12 = 1, we can use ordinary multiplication, and for n = 2 we can use "complex multiplication", (a, b) (c, d) = (ac - bd, a d + bc).) Let e l , . ,e, be the standard basis of Rn,

..

(a) Every point in S"-' is a - el for a unique a E Rn. @) If a # O, then a el, . . ,, a en are linearl~independent. (c) If p = a el E S"-', then the projection of a - ez,. . , a . e, on (Sn-',i), are linearly independent. (d) Mu~tiplicationby a is continuous, (e) TSn-' is trivial. (f) TP"-' is trivial. The tangent bundles TS3 and TS' are both trivial. Multiplications with the required properties on R4 and R8 are provided by the "quaternions" and "Cayley numbers", respectively; the quaternions are not commutative and the Cayley numbers are not even associative, It is a classical theorem that the reals, complexes, and quaternions are the only associative examples. For a simple proof, see R. S, Palais, TLv Classi;cation oJReal DiuisiDn Akebrdi, Amer, Math. Monthly 75 (1968), 366-368. J. E Adams has proved, using methods of algebraic topology, that n = 1, 2, 4, or 8. pncidentally, non-existence of zero divisors immediately implies that for a # O there is some b with a b = (1,0,. . ,O) and b' with b'a = (1,0,. . ,O). If the multiplication is associative it follows easily that b = b', so that we always have multiplicative inverses. Conversely, tliis condition implies that there are no zero divisors if the multiplication is associative; otherwise it suffices to assume the existence of a unique b with a b = b a = (1, O,, . ,,O).]

.

+

.

.

20. (a) Consider the space obtained from [O, 11 x Rfl by identifying (0, u) with (1, Tu), where T : R" + Rn is a vector space isomorphism. Show that this can be made into the total space of a vector bundle over S' (a generalized Mobius strip), @) S11ow that the resulting bundle is orientable if and only if T is orientation preserving. 21. Show that for p E s 2 , the \rector pp E R3, is not in i.(s2,) by showing that the inner product (p, c'(0)) = O for al1 curves c with c(0) = p and Ic(l)l = 1 for al1 t. (Recall that

wliere denotes the traiispose; see C a h b on Man@Idr, pg. 23-1 22, Let M be a Cm manifold, Suppose that (TM)I A is trivial whenever A c M is homeomorphic to S'. Show that M is orientable. Hinl: An arc c from

The h g m t Bundle po

M is contained in some cuch A so (TM)lc is trivial. Thus one can ('transportJithe orientation of Mpo to Mp. It must be checked that this is independent of the choice of c. First consider pairs c, c' which meet only at p0 and p, T h e general, possibly quite messy, case can be treated by breaking up c E

M to p

101

E

into s m d pieces contained in coordinate neighborhoods,

Remark: Using results from the Addendum to Chapter 9, together with Problem 29, we can conclude that a neighborhood of some S' c M is non-orientable if M is non-orientable, The next two problems deal with important constructions associated with vector bundles.

-

23. (a) Suppose 6 x : E + X is a bundle and f : Y + X is a continuous map. Let E' c Y x E be the set of al1 ( y , e ) with f ( y ) = x ( e ) , define n': E' + Y by x r ( y , e )= y, and define f:E' + E by j ( y , e ) = e. A vector space structure can be defined on

by using the vector space structure on x-'( f ( y ) ) . Show that x': E' + Y is a bundle, aiid ( j ,f ) a bundle map which is an isomorphism on each fibre. This bundle is denoted by f *(C), and is called the bundle induced (from 6 ) by f . (b) Suppose we have anotlier bundle 6" = n": E" + Y and a bundle map

-

(Y,f ) from 6"

to 6 wliich is an isomorphism - on each fibre. Show that 6" 2 ' = f * ) Hid: Map e E E" to (+"'e), ?(e)) E E'. (c) If g : + y, then ( f og)*(6) 8 * ( f * ( 6 ) ) . (d) If A C X and i : A + X is the inclusion map, then i*(6) 2 6 1 A. (e) If 6 is orientable, then f is also orientable. (f) Give an example where 6 is non-orientable, but f *(6) is orientable. (g) Let = n : E + B be a vector bundle, Since n : E + 3 is a continuous map from a space to the base space B of 6 , the symbol n * ( ( ) makes sense, SIiow that if 6 is not orientable, then x*(6)is not orientable.

*(e)

e

24, (a) Given an n-plane bundle 6 = x : E + B and an m-plane bundle 7 = rr': E' + 3, 1et E" C E x E' be the set of al1 pairs (e,ef)with x ( e ) = x1(e'). Let x"(e,el) = x ( e ) = nl(e'). Show that n": E" + B is an (n m)-plane

I~undle.It js called the Whitncy sum 6 $ 7 of 6 and 7; the fibre of is the direct sum n-' ( p )$ n/-' ( p ) . (b) If f : Y + 3 , show that f 4 ( 6 $ 7 ) 2r S*(.$)$ f 4 ( 7 ) .

+

6 $7

over p

(c) Given bundles ti = ni: Ei + Bi, define X : El x E2 + B1 x B2 by rr(el,e2)= (rr1(ei),rr2(e2)).Show that this is a bundle x over Bl x B2. (d) If A: B + B x B is the "diagonal map", A(x) = (x,x), show that EZ A*(6 ZJ). (e) If 6 and q are orientable, show that $ z~ is orientable. (f) If 6 is orientable, and ZJ is non-orientable, show that $ ZJ is also nonorientable. (g) Define a "natural" orientation on V $ V for any vector space V, and use this to show that 6 $ 6 is always orientable. (11) If X is a "figure eight" (c.L Problem 5), find two non-orientable 1-plane bundles 6 and q over X such that 6 $ q js also non-orientable.

c2

e$z~

e

25, (a) If rr r E + M is a Cm vector bundle, then rr, has maximal rank at ixch point, and each fibre n-' (p) is a Cm submanifold of E. (b) T h e O-section of E is a submanifold, carried dfiomorphically onto B by rr.

26. (a) If M and N are Cm manifolds, and l i [or ~ rrN] : M x N + M [or N] is the projection 017 M [or N], then T ( M x N) h. XM*(TM)$ rrhr*(TN). (b) If M and N are orientable, then M x N is orientable. (c) If M x N is orientable, then both M and N are orientable,

27. Show that the Jacobian matrix of y,

o

(x.)-'

is of the form

Tliis shows that the manifold TM is alz~rqsork~ttabk,i,e., the bundle T(TM) is orientable. (Here is a more coilceptual formulation: for v E TM, the orientation for (TM), can be defined as

thc form of y, o (x,)-' shows that this 01-ientation is iildependeni of the choice of A-.) A differeilt proof that TM is orieiltable is $ven in Problem 29. 28, (a) Let (x, U ) be a coordinate system on M with x(p) = O and let v he Cy=la' a/axi . Consider the curve c in TM defined by

lp

E

Mp

Tlze 7ageni Bundle Show that

(b) Find a curve whose tangent vector at O is a/a(xi o rr)I,.

29. This problem requires some familiarity with the - notion of exact sequences

f

2

(c.f. Chapter 11). A sequence ofbundle maps El -+ E2 -+ E3With $ = g = identity of B is exact if at each fibre it is exact as a sequence of vector space maps. (a) If ,$ = rr : E + B is a CbO vector bundle, show that there is an exact sequence O + rr*(.$) + TE + rr*(TB) + O.

f i t : (1) An element of the total space of ~ " ( 6 is ) a pair of points in the same fibre, which determines a tangent vector of the fibre. (2) Map X E (TE)e to (e, GX). (b) If O + El + Ez + E3 + O is exact, then each bundle Ei is orientable if the other two are. (c) T(TM) is always orientable, (d) If rr r E + M is not orientable, then the manifold E is not orientable. (This is why the proof that the Mobius strip is a non-oriei-itable manifold is so similar to the proof that the Mobius bundle over S' is not orientable.) T h e ilext two Problems contain more information about the groups introduced in Problem 2-33. 111addition to being used in Problem 32, this information wiH al1 be important in Chapter 10,

.

30. (a) Let po E S"-' be the point (0,. .,O, 1). For n 3 2 define / : SO(n) + S"-' by / ( A ) = A(po). Show that / is continuous and open. Show that (po) is homeomorphic to SO(n - I), and then show that j-'(p) is homeomorphic to SO(n - 1) for ali p E S"-'. (b) SO(1) is a point, so it is connected. Using part (a), and induction on n, prove that SO(n) is connected for ali n > 1 . (c) Show that O(tz) has exactly two components.

/-'

31, (a) If T : IRn + IR" is a linear transformation, T*: IRn + Rn, the adjoint of T, is defined by (T*v, w) = (u, Tw) (for each u, the map w H (u, Tw) is linear, so it is w H (T*v, w) for a unique T*v). If A is the matrix of T with respect to the usual basis, show that the matrh of T* is the transpose A'.

(b) A linear transformation T : R" + Rn is self-adjoint if T = T*, so that (Tu, w ) = (u, Tw) for a l u, w E R". If A is the matrix of T d t h respect to the standard basis, then T is self-adjoint if and only if A is symmetric, A~ = A. It is a standard theorem that a symmetric A can be written as CDC-1 for some diagonal matiix D (for an analy& proof, see CaEculw on Man@ldr, pg. 122). Show that C can be chosen orthogonal, by showing that eigenvectors for distinc t eigenvalues are orthogonal. (c) A self-adjoint T (or the corresponding symmetric A) is called positive semid d n i t e if (Tu, u) > O for al1 u E Rn, and positive definite if (Tu, u) > O for al1 v # O. Show that a positive definite A is non-singular. Hznt: Use the Schwarz inequality. (d) Show that A' - A is always positive semi-definite. (e) Sliow that a positive semi-definite A can be written as A = B2 for some B. (Remember that A is symmetrjc.) (f) Show tliat every A E GL(n, R) can be written uniquely as A = A l .A2 where A i E O(n) and A 2 is positive definite. Hht: Consider A' . A, and use part (e). (g) The matrices A l and A2 are continuous functions of A. Hint: If A(") + A and A(") = A(")] ~ ( " 1 2 ,then some subsequence of ( A ( " ) ~converges. ) ) GL(i1, R) is homeomorpliic to O(n) x R"("("*')/~ and has exactly two components, ( A : det A > 0) and ( A : det A < O). (Notice that this also gives us anotller way of fiilding the dimension of O(n).)

32, Two continuous fuilctioils So,5 : X + Y are called homotopic if there is a coiltjiluous function H : X x [O, i] + Y such that The functions H; : X + Y defined by Hl (x) = H(x, t ) may be thought of as a path of fuilctions fi-om Ho = fa to HI = J. The map H is called a homotopy between fo and J . The i-iotation f : (X, A) + (Y, B), for A c X and B c Y, means that f : X + Y and f (A) c B. We cal1 f l ~ ,J : (X, A ) + (Y, 3)homotopic (as maps from (X, A ) to (Y, B)) if there is an H as above such that each Hr : (X, A ) + (Y, B)(a) If A : [O, 11 + GL(n, R) is continuous aild H : R" x [O, 11 + R" is defiiled by H(x, 1 ) = A(r)(x),show that H is continuous, so tllat Ijg and Hl are homotopic as nlaps from (R" ,R" - (O)) lo (Rn,R" - (O)). Conclude that a non-singular linear trailsformation T : (R", R" - (0)) + (R", R" - (O)) with det T > O is homotopic to ihe identity map. (b) Suppose f : R" + R" is Cm and f(0) = 0, while f (R" - (0)) c R" - (O). If Df(0) is non-singular, sl-iow that f : (R", Rn - (O)) + (R", R" - (O)) is

homotopic to Df (O): (Rn,Rn - (0)) + (Rn,Rn - (O)). Hint; Define H ( x , I ) = f(tx) for O < 1 5 1 and H(x, O) = Df (O) (x). To prove continuity at points (x, O), use Lemma 2. (c) Let U be a neighborhood of O E R" and f:U -4 Rn a homeomorphism with f(O) = O. Let B, C V be the open ball with center O and radius r, and let h : Rn + B, be the homeomorphism

then f

0,z:

-

(R",rw" - (O)) + (Rn,Rn (O)).

MTe will say that f is orientation preserving at O if f o h is homotopic to the identity map 1 : (Rn,Rn - (0)) + (Rn,Rn - (O)). Check that this does not depend on the choice of B, c V. (d) For p E R", Jet Tp: R" + Rn be T'(q) = p + q . If f :U 4 V is a l-iomeomorphism, where U, V c Rn are open, we will say that f is orientation preserving at p if T-J(p) O f O T,, is orientation preserving at O. Show that if M is orientable, then there is a collection C of charts ~ l h o s edomains cover M such that for every (x, U) and (y, V) in e, the map y ox-' is orientation presening at x ( ~ for ) a11 p E U n V. (e) Noiice that the condition on y o xd' in part (d) makes sense even if y ox-' is not djffereniiable. Thuus, if M is any (not necessarily dxerentiable) manifold, we can define M to be orientable if there is a collection C of homeomorphisms x : U + R" whose domains cover M, such that C satisfies the condition in part (d), To prove that this definition agrees with the old one we need a fact from algebraic iopology: If f : Rn + Rn is a homeomorphism with f(O) = O and T ; Rn S'> R" is ~ ( x ' .,. .,x") = ( x ' , . . . ,xn-l, -xn), then precisely one of f and T o f is orientation presen~ii-igat O. Assuming this result, show that if M has such a collection C of l-iomeomorphisms, then for any Cm structure on M the tailgeiit bundle TM is orientable. 33. Let M" c RN be a Cm n-dimensional submanifold. By a chord of M we meai-i a point of IRN of the form p - q for p , q E M.

(a) Prove tliat if N > 212 + 1, then there is a vector v

E sN-lsuch

that

(i) no chord of M is parallel to u, (ii) no tangent plane M, contains v .

Hint; Coilsider ceriain n-iaps from appropriate open subsets of M x M and TM to S"-'.

(b) Let IRN-l c IRN be the subspace perpendicular to U, and n : IRN + IRN-l the correspoilding projection. Show that n l M is a one-one immersion. In particular, if M is compact, then rrl M is an imbedding. (c) Every compact Cm rl-dimensional manifold can be imbedded in IR2"+'.

Note: This is the easy case of M'hitney's classical theorem, which gives the same result even for non-compact manifolds (H. Mfhitney, fizermtzabk man@ldí, Ann. of Math. 37 (1935), 645-680). Proofs may be found in Auslander and MacKenzie, Intmduction ¿o Dzffmentiabk A4un@lds and Sternberg, Latures on fiy fmential Geotne~tIn Munkres, Eln~tentaryDflia~tialZpology, there is a dflerent sort of argumerlt to prove that a not-necessarily-compact n-manifold M can be imbedded in some IRN (in fact, With N = (n + I ) ~ )T. h a l we may show that M imbeds in IR2"+' using essentiaily the argument above, together with the existence of a proper map f : M + IR, given by Problem '2-30 (compare Guillemin ,:lid Pollack, fizmnlial Topology). A much harder result of Whitn y shows that M" can actually be imbedded in IR2" (H. Wllitney, nie selj-intersectioonroja mooU1 u-t~ranfoldin 2n-space, Ai111. of h4ath. 45 (1 944), 220-246).

CHAPTER 4 TENSORS

A

II the constructions on vector buildles carried out in this chapter have a

common feature. In each case, we replace each fibre x - ' ( p ) by soirie other vector space, and then fit al1 these new vector spaces together to form a new vector bundle over the same base space. Tlie simplest case arises when we replace each fibre V by its dual space V*. Recal1 that V* deilotes the vector space of al1 linear functions A : V + R. If f r V + W is a linear transformation, then there is a linear transformation f * : W* + V* defined by

It is clear that if 1 v : V + V is the ideiitity, tlien 1 v* is the identity map of V* and if g : U + V, then (f o g)* = g* o f *. These simple remarks already sliow tliat f * is an isomorphism if f : V + W is, for (f o f ) * = 1 v* and (f 0 f-l)* = 1W+. Tlie dimensjon of V* is tlie same as that of V, fol. finite dimeilsional V. In E V*, defined by fact, if vi,. . .,v, is a basis for V, theii the elements

-'

are easily cliecked to be a basis for V*. The linear fuiiction v*i depeilds on tlle eiltire set vi,. . .,v,, not just on vi alone, aiid tlle isomorphism from V to V* obtained by sending vi to is not indepeildeilt of the choice of basis (consider what happens if vi is replaced by 2vl), On the othei. hand, if v E V, we caii define u** E V** = (V*)* unambiguously by

If v**(h) = O for every h

E

V*, then h(v) = O for al1 h

E

V*, which implies that

v = O. Thus the map v w v** is an isomorphism from V io V*'. It is called the natural isomorphism from V to V**. (Problem 6 gives a precise meaning to the word "natural", formulated only after the term had long been in use. Once the meaning is made precise, we can prove that there is no natural isomorphism from V to V*.) Now let

6 = n:E

+ B be any vector bundle. k t

and define the function x': E' + B to take each [n-'(p)]*to p- If U c B, lind I : ?r-' ( U )+ U x R" is a trivialization, then we can define a function I' : nf-'( U )+

U x (Rn)*

in the obvious way: sincc the map r restricted to a fibre, tP

: H-'(~)+

{p)x R",

is an isomorphism, it gives us an jsomorphism

(g*)-' : [ir-'(p)]*+ {p} x (R")' We can makc n': E' + B into a vector bundle, t l ~ ed u d bundle '6 of 6,by requiring that al1 such i r be local trivializations. (We first pick an isomorphism from (Rn)*to R",once and for all.)

e*

At fii-st jt n-iigllt appear tliat 2 6, since eacli n-'(p) is isomorphic to x ( p . However, tliis is true merely because t l ~ etwo vector spaces havc tlie same dime~ision. Tl-ie lack of a natural isomorpl-iism from V to V* prevents us from constructing an equivalence between '6 and 6. Actually, we will see later that in "most" cases Zr equivalei-it to 6; for the present, readers may ponder tl-iis question for tliemselves. In contrast, the bundle = (t*)' is nlu~oysequivalcnt to 6. We consti-uct the cquivalence by mapping the fibre V of 6 over p to the fibre V** of t'* oler p by the natural isomorphism. If you

e*

e**

e*

tl-iink about how is constructed, it will appear obvious that this map is indeed a n equivalence. Even if 6 can be pictured geometrically (e.g., if 6 is TM), there is seldom a geometric picture for C*. Rather, 6* operates on 6: If s is a section of 6 and o is a section of t*,then we can define a function from B to R by

P

G(P)(S(P))

s(p)E f i(p) o ( p ) E x'-l ( p ) =

,-'

(p)".

This function wjll be denoted simply by o ( s ) . When this construction is applied to the tangent bundle TM of M, the resulting bundle, denoted by T*M, is called the cotangent bundle of M ; the fibre of T*M over p is (M,)". Like TM, the cotangent bundle T * M is actuaiiy a C m vector bundle: since two trivializatjons X* and y, of TM are Cm-related, the same is clearly true for x.' and y,' (in fact, y.' o (x.')-' = y, o (x,)-'). 1% can thus define Cm, as wcll as continuous, sections of T*M. If w is a C m section of T*M and X is a C m lector field, then w ( X ) js the C m function P l+ ~ ( P ) ( X ( P ) ) . If f : M + R is a Cm function, then a Cm section d f of T*M can be defined by

d f ( p ) ( X ) = X ( f ) for X

E

M,.

The sectio~idf is called tlie differentd of f . Suppose, in particular, that X is dcldt 1 , where c(io) = p. Recall that

This means that

Adopting the elliptical notations

this equation takes the nice form

If (x,U) is a coordinate system, tlien tlie dxi are sections of T * M over U. Applyiiig the definition, we see tliat

Tlius d x l ( p ) , . . . ,d x n ( p ) is just tlie basis of M', dual to tlie basis a/axl l, a/axnIP ~f M,. Tliis means tl~atevery section w can be expressed u~~iquely on U as

. . .,

for certaiii functio~~s w ; o11 U. Tlie section w is coiitiiluous or Coaif aiid only if the functions wi are. We can also write

if wc define sums of sections and products of functions and scctioiis in the olwious way ("poii~rwise"additioii ai-id multiplication). Tlie sectioii O, such íhat there is a unique colIection of difkomorphisms #t : V + #f( V ) c M Mor Ii 1 < E with íhe following properties: (1) # :

(3) If y

E

V

X

(-E,&)

+

M, defined by #(i, p ) = & ( p ) , is Cm.

V, ttlien Xg is íhe tangení vector at i = O of the curve i

K- # f

(q).

The exarnples Fven previously sliow íhat we cannoí expecí #f to be defined for al1 i, or on al1 of M. In one case however, this can be attained. The support of a vector field X is just the closure of { p E M : X, # 0).

6. THEOREM. If X Iias compact support (in particular, if M is compact), íhen íliere are dfiomorpI~isms#t : M + M for al1 i E R wiíh properíies (11, (21, (3)*

PROOF. Cover support X by a finite number of open sets V , , . .. , Vn given by TIieorem 5 wiíli comespoiiding E I ., . . , E , and dfiomorphisms #:. Let E = m i n ( r l , .. . , E ~ ) .Notice thaí by uniqueness, # ( ( y ) = #{ ( y ) for y E Vi n 5. So we can define if y E Vi if y # support X . Clearly # : ( - E , & ) x M + M is Cm, and and each #f is a dfiomorphism. To define #, for Ii 1 >_ E, write r = k ( ~ / 2+) I'

#f

=

{

#E/2 #-E/2

O

m

O

m

O #E/2

O

- . - o #-&/2

#fct=F = #1 o # s

with k an integer, and

[#E/2

#r O

#r

It is easy to check t1iat this is the desired

if IiI,IsI, Ii

17'1

1 we can at Jeast compute

1 -[f(x( O, ... , h ,..., O)) - f(0)] 1140 11

= Jim

= lim

h40

1

- [f

h

(O,. . .,h, . . . ,O) - $(O)]

lo

Since X (O) = d/ai l by assumption, this shows tliat X,O = I is non-singular. Hence x = X-' may be used as a coordiilate system in a neighborhood of O. This is the d e s h d coordinate system, for it is easy to see tliat the equation ~ * ( a / d r') = X O x, which we have just proved, is equivalent to X = d/axl. The second use of the equation

is more co~nprelieiisi\le. The fact tjiat Xf can be defined totally in terms of the diffeon~orphisms#iI suggests that an aciioii of X on other objects can be

obtained in a similar way, To emphasize the fundamental similarity of these notions, we first introduce the notation

Lxf

for

Xf.

-

\Ve cal1 Lx f tlle (Lie) derivative of f with respect to X ; it is another function, wliose value at p is denoted variously by ( L xf ) ( p ) = L x f ( p ) ( X f ) ( p )= Xp ( f ). Now if w is a Cm covariant vector field, we define a new covariant vector field, tlie Lie derivative of w with respect to X , by

Tliis is the limit of certain members of M,*. Recall that if Xp

E

M,, then

A fairly easy direct argument (Problem 8) shows that this limit always exists, and that tlie newly defined covariant vector field L x w is C m , but we will soon compute this vector fieId explicitly in a coordinate system, and these facts will then be obvious. If Y is another vector field, we can define the Lie derivative of Y with respect to X , 1

Tlie \rector field #h+Y appearing here is a special case of the vector field a,Y defined at the beginning of the chapter, for a : M + N a difkomorpliism and Y a vector field on M . Tlius (#/,,Y), = #h* (Y$-,, (,) is obtained by evaiuating Y at ( p ) = #-A ( p ) , and then moving it back to p by #/,*.

integral curve r $1 ( P ) of X through p

Tlic definition of L x Y can be made to look more closely anaiogous to L x f and Lx w in tlie follotving way. If a : M + N is a dfiomorphism and Y is a

Tr,tor l~icldsand Dzfenntial Equatians

151

vector field on the range N, thni a vector field a*Y on M can be defined by

Of course, a*( Y ) is just (&-'),Y. Now notice that Jim

h+O

-/Z1

Y )p

yp - Yp]= h+O lim

-

(#h*Y)P 1 = lim [Yp -h k+O k

- - ( $ L ~ *Y ) , ]

Nevertheless, we will stick to the original (equivalent) definition. We now wish to compute Lx o and Lx Y in a coordinate system. The calculation is made a lot easier by first o b s e ~ n g

8. PROPOSITION. If Lx Y; and Lx wi exist for i = 1,2, then

If LxY and L x w exist, then (3) L x f Y = X f . Y $ f . L x Y , (4) Lx f - o =X f . w + f . L x w . FjnaIIy, if o ( Y ) deiiotes tlie function p then

I+

o ( p )(Yp)and L x o and LX Y exist,

PROOF. (1) and (2) are trivial. T h e remaining equations are al1 proved by the a m e trick, t l ~ eoiie used iii finding ( f g ) ' ( x ) .We will do number (3)here.

+ /Jim [f (p) - { ( m - h 140

(P))

1

#/r*

y+-,, ( p ) .

The first limit is c'learly f ( p ) L x Y ( p ) . In ihe second limit, the term iii brackeis approaches lim f ( P ) - f (#k ( P ) ) = Xf ( P ) , k+O -k while an easy argument sliows that #h,Y+-,,(p) + Yp. *3 Mre are now ready to compute Lx in terms of a coordinate system ( x , U) on M . Suppose X = a i d / a s i . We first compute L x ( d x i ) . Recall (Problem 4-1) that if f : M + N and y is a coordinate system on N, then

zyEI

We can apply this to #h*, where y is s . Then

LX (dx')( p ) = lim h+O

1

-h [(#h*)dxi( p) - dx' ( p ) ]

Now the coeffrcient of d d ( p ) is lim l 1 4 oh

["'d"'

-

$1

= lini Ir40

-I h

[

axj

{ibis step wiil be justified in a moment)

To justify (*) we note that the map A(h,q) = xi(#h(q)) is Coo from IR x M to IR; thus a2~ / d h d x j= d 2 ~ / d x j d h wliich , is wliat the intercliange of limits amounts to. It now follows tllat n aai LX dx' = -d x j . axj We couId now use (2) and (4) of Proposi tion 8 to compute Lx w i n general, but

we are really iiiterested in computing L x Y. To compute (d/axi) we could imitate the calculations of L x dx'; but there would be a complication, because #h* on \rector fields iilvolves one more composition thari #h* on covariant vector fields. Tlie trick needed to deal with this complication has already been used to prove (3), (41, and (5) of Proposition 8, and we can now use (5) to get the answa immediately;

O = L. 6; = L x [dxi

(a)] axj

= (Lx dx')

(5) + S). dx' (LX

thus,

Using (3) we obtain

Sumniing over j and tlien iiiterclianging i and j in tlie second double sum we obtain

This somewhai complicated expression immediately leads to a much simpler coordinate-free expression for Lx Y. If f : M + N is a C" function, then Yf is a function, so XYf = X(Yf) makes sense. Clearly

Tlie secoiid partial derivatives which arise here cancel those in the expression for Y ( Xf ), and we find tliat LxY=XY-YX,

alsodenotedby [X,Y].

Often, [X, Y] (which is called the "bracket" of X and Y) is just defined as XY - YX; note that this means

A straightfonvard verification shows that

so tliat [X, YIp is a derivation at p, and can therefore be considered as a member of M,. We are now in a very strange situation. Two vector fields LxY and [X, Y] liave both beeii defined independendy of any coordinate system, but they have Leen proved equal using a coordinate system. This sort of thing irks some people to no end. FortunateIy, in this case tlie coordinate-free proof is short, though hardly obvious. In Chapter 3 we proved a Iemma wllich for the special case of W says that a Cm fu~ictionf : (-E, E) + W with f (O) = O can be wntten

for a Cm fuiiction g :

(-E,

E) + W with g(0) = f'(O), namely

This lias a n immediate generalization.

9. LEMMA. lf f : (-E, E) x M + W is Cm and f (O, p) = O for al1 p E M, tlien there is a Cm function g : (-E,&) x M + W with

PROOF. Define

1 0. THEORFM. If X and Y are Coovector fields, theii

PROOF. Let f :M + R be Cm. k t X generate #,, Ii 1 < tliere is a family of Cm functions gf on M such that

E.

By Lemma 9

Then

Tl-ie equality Lx Y = [ X , Y ] = XY - YX reveals certain facts about LxY which are by no means obvious from the definition. Clearly

[ X ,Y ] = -[Y, X ] , so

[ X ,X ] = o.

Consequently,

LxY = - L y X ,

so

LxX = 0.

+

Since we obviously Iiave Lx (aYl by2) = aLx Y, + bLx Y2,it follows immediately tliat L is also linear with respect to X :

Finally, a straiglitforward calculatioil proves tlie "Jacobi identity":

Tliis equation is capable of two interpretations in terms of Lie derivatives: (a) L x [ Y ,21 = [ L x K ',l

+ [Y,Lx',],

(b) as opei-ato1.son Coafunctions, we have L [ x , = Lx o L y - L Y o Lx (whicli miglit be written as [ L x ,L Y]).

Finally, note that Lx Y is linear over constants oilly, not over tlie Coofunctions .F. In fact, Proposition 8, or a simple calculation using the definiíion of [X, Y], shows that [ f X , g Y 1 = fg[X, Y] + f f(Xg)Y- g(Yf )XThus, the bracket operation [ , ] is not a íensor-that is, [X, Y'Jp does not depend only on Xp and Yp (wliich is noí surprising-whaí can one do ío two vectors in a lector space excepí íake linear combinations of íhem?), buí on íhe vector fields X and Y. In particular, even if Xp = O, it does not necessarily foIIow that [X, Y'Jp = O-in the formula

t11e first term Xp ( Yf ) is zero, but the second may not be, for Xf may have a non-zero derivative iil tlle Yp direction even tliou~h( Xf ) (p) = 0. T h e bracket [X, Y], although not a tensor, pops up in the definiíion ofpractically a11 otlier tensol-S,for reasons that \vil1 become more and more apparent. Before procecdii~gto examine its geomeíric interpreta tion, we wil1 endeavor to become more aí ease witIl the Lie derivative by taking time out ío prove directly from tlle definiíioli of Lx Y two facís whic11 are obvious from íhe definition of [X, Y]. (1) LxX = o.

lf X generates #t, it certainly suffices to sl~owíhat (#n,X)p = Xp for a11 h. Recall that (#h, X)p = #fd+ X4,/, (p). Now X4-, (p) is just h e tailgent vector at

time r = -h to t11e curve r ío the curve

I+

#I

(p), and tlius tlie tangent vector, at time I = 0,

Y P ) = 41-/i(P). TIius @/l* X$-l,(P) is tlie íangení vecíor, aí time I = O, to the curve

But this tangení vecíor is jusí Xp .

k t o r I;ieZhand 1lz&i-entiaZ Equations

157

(2) If Xp and Y, are both O, then LxY (p) = 0. Since Xp = O, tlie unique integral curve c wit1i c (O) = p and dcldi = X (c(1)) is simply c (i) = p (an integral curve Starting at p can never get away; conversely, of course, an integral curve starting at some other point can never get to p). Then Yp = O and

so LxY(p) = o. To develop an iiiterpretation of [X, Y] we first prove two lemmas. 1 1. LEMMA. Let a : M + N l ~ ae dXeomorphism and X a vector field on M wliicli generates {#[l. Then a,X generates {a o #t o a-'}.

PROOF. We have

12. COROLLARY. If a : M + M, tlien a, X = X if and only if for al1 L.

$toa

=

13. LEMMA. Let X generate {#t) aiid Y generate {Srf}. Then [X, Y] = O if and only if #l o = $, o #t for al1 S, í .

= o #t for al1 S, tlien #,,Y = Y by CoroIlary 12. If this PROOF. If #t o is true for aII r , theii clearly Lx Y = 0. Convei-sely, suppose that [X, Y] = O, so that 1

O=Iim-[Yq-(#h*Y)q] h+o lr Giveii p

E

M, consider tlie cuive c :

(-E,

forallq.

E) + Mp @ven by

For h e derivative, ct(r), of this map into the vecíor space M, we have

=O

)

using (*) with

= 4-1 (P)

= o. Consequently c(í) = c(O), SO #t,Y = Y. By Corollary 12, #r o alI s,r. 4 4

=

o

#[ for

We kave already shown tliat if X ( p ) # 0, then íhere is a coordinate system x with X = a / d x l . If Y is another vector field, everywhere linearly independeiit of X, then we mighí expect to find a coordinaíe sysíem with

However, a sllort caIcuIation immediately gives the result

so íliere is no Jlope offi~idinga coordinaíe system satisfying (*) unless [X, Y] = 0. Tlie remarkable fact is that tlie condition [X, Y] = O is su#cient, as well as necessary, for the existente of the desired coordinate system. 14. THEOREM. If X,, . . . ,Xk are IinearIy independent Coovector fields in a neighborhood of p, and [X,, XB] = O for 1 < a,B 5 k, then tliere is a coordinate system (s, U) around p such that

PROOF. As in the pmof of Theorem 7, we can assume tllat M = Rn, thaí p = 0, and, by a linear cl-iange of coordinates, that

X, (O) = -

a = 1, ..., k.

If X, generates (4," ), define x by

x ( a l , . . ,a ) = # , ( # * ( . . ( # ( , . . . , , ak4-1 , . . . , a n ) ) . . * ) ) . As in the proof of Theorem 7, we can compute that

Thus x = x-' can be used as a coordinate system in a neighborhood of p = 0 . Moreover, just as before we see that

Nothing said so fal. uses the l~ypothesis[ X , , XB] = O . To make use of it, we appeal to Lemma 1 3; it shows that for each a between I and k , the map x can also be wri tten

and our previous argument then shows that

We tlius see tliat the bracket [X, Y ] measures, in some sense, the extent to which the integral curves of X and Y can be used to form the "coordinate lines" of a coordinate system. There is a more complicated, more diEcult to pi6ove,and less important result, which makes this assertion much more precise. If X and Y are two vector fields in a neigl~borhoodof p, then for sufficiently small h we can (1) follow the integral curve of X through p for time h ; (2) starting from that point, follow the integral curve of Y for time h; (3) then follow the integral curve of X backwards for time Ir; (4) tlien follow the integral curve of Y backwards for time Iz.

If iliere happens io be a coordinaie sysiem x with x ( p ) = O and

then these steps take us to points with coordinates (1) (2) (31 (4)

(j,O,O,...,O) ( h , h , O , ... ,O) (O,h,O,..., O ) (0,OY 0 , .

- ,O),

so tliat this "parallelogram" is always closed. Even when X and Y are (linearly iiidependent) vector fields with [ X , Y] # 0, the parallelogram is "closed up to first order". Tlie meailiiig of tl-iis phrase [an exiension of the terminology "c = y up to first order at O", which means tliat c'(0) = y f ( 0 ) j is the following. Let c ( h ) be tlie poini whicl-i step (4) ends up at,

c(/z) = *-ff

($4 (*h (#h ( O ) ) )

Tlieii tlie cunJe c is tlie constant curve p up to first order, thai is, 15. PROPOSITION. ~ ' ( 0=) O. PROOF If we define al ( i ,/z)

= * r (#h ( P ) )

a2 ( i ,11) = 4-l (*h a3 ( i l

Moreover,

h, = Sr-f

(#fi

(P ) ) )

(#-h (*k(#h

(P)))),

kctot. JieZd~and L)z&ien tial Equat ions and for any Cm function f : M + R,

while

Consequently, repeated use of the chain rule gives

Wlienever we have a curve c: (-E,&) + M with ~ ( 0 = ) p and ct(0) = O Mp, we can define a new vector c"(0) or d2c/d121, by

E

A simple calculation sliows, wi?ghe arsun@ioti ~ ' ( 0 )= 0, that this operator ctt(0) is a derivation, ct'(0) E Mp. (A more general construction is presented in ProbJem 17.) It turils out tllat for tlle curve c defined previously, the bracket [X, YIp is r'elated to tliis "second arder" derivation. Until we get to Lie groups it wiI1 not be clear how anyone ever thought of the next tlieorem. The proof, which eiids tlle cliapter, but can easily be skipped, is a n liorrendous, but clever, calculation. It is followed by an addendum containiilg some additional important points about differeiltial equations which are used Jater, and a second addendum coiicerning linearly independent vector fields in dimension 2.

16. THEOREM. c"(0) = 2[X, YIp.

PXOOF. Using the iiotation of the previous proof, since (f we have

o c ) (1)

= (foa3) (I,I )

Now

(2) 2D2,l (f a3)(0?0) = 2D1 (-Yf o a3) = 2[D1 (Yf 0 a21(O? 0) 0 2 ( Y f o a2) (0, o)] = 2XYf (p) - 202(Yf 0 a2)(0,0) = 2XYf (p) - 2[Di (Yf o al)(O,O)

+

+ D2 (yf

O

a1) (0, o)]

=2XYf(p)-2YYf(p)-2XYf(p)

Since (b) gives

by (e) by (b) and the chain rule by

(4

by (a) and the cliain rule by(c)and(f).

Trector fieldi and L)$uentiaZ Equariotls

Finally, from

Substituting (1)-(4) in (*) yields the theorem. +:+

163

ADDENDUM 1 DIFFERENTI AL EQUATIONS Al thougli we have dways solved dxerential equatioiis

with the initial condition a(0,x) = S , we could just as well kave required, for some io, that

To prove this, one can replace O by io everywliere in tlie proof of Theorem 2, or else just replace a by i w a(í - lo,x). Ano ther omissioli in our treatmen t oí' dflerential equations is more glaring: tlle differentid equations ai(r) = f (a(t)) do not even include simple equations of the form a' (i) = g(i), Jet alone equatioiis Iike ai(i) = ía (i). In general, we would Iike to solve equations

where f : (-c, c ) x U + Rn. One way to do t his is to replace f (a(i, x)) by f (1, a ( i , s))wl~ereverit occurs iii the proof. There is also a clever trick. Define

by f ( s , s ) = (1,

f (s,x)).

Then tliere is a flow (&',E2)= E: ( - b , b ) x W + R x R" with

For tlie f i i ~ coinponent t í'unction & ' this means that

a 1 (i, S,S ) = 1 -6 al

thus &'(i,s,x) = s + i . For the second component á2 we have

= f ( á l (i, S,s ) , á 2 ( t , s , s ) )

= f (S + 1, ii2(1, S, x)). Then ~ ( I , x= ) a2(i,o,~) is the desired Aow with

p (O, x ) = x. Of course, we could also llave arranged for B (ro,x) = x (by first finding á with á (io,s,X) = (S,x), not by considering the curve r I+ (i - io, S)). Finally, consider the special case of a linear differential equation

wliere g is an n x

If

c

12

mati-ix-valued function on (a, b). In this case

is any n x n (constant) matrix, then

so c .a is also a solu tion of the same differeiitial equation. This remark allows us to prove a n important property of linear differential equations, distinguishing tliem from general differential equations a' (i) =: f (r, a (i)), which may have solutions defined onIy o11 a small time interval, even if f : (a, b) x Rn + Rn is Cm. J 7. PROPOSITlON. If g is a continuous n x ( u ,b), tlien tlie solutioi~sof the equation

can al1 be defined on (a, b).

11

matrix-valued function on

PROOF. Notice tliat continuity of g implies that f (í,x ) = g(í) . x is locally Lipschitz. So for any ío E (a, b) we can solve the equation, with any given initial condition, in a iieighborhood of ío. Extend it as far as possible. If the extended solution a is not defined for al1 r wit h ío 5 í 4 b, Iet r 1 be the Ieast upper bound of tlie set of í ' s for which it is defined. Pick S with

TJien fl (r*) # O for r* < i l close enough to í l . Hence there is c with

coiricidcs witli a oii tlie iiiterval whe1.e they are defined. By uniqucncss, c I ltus a may be extended past í l as c - fl, a contradiction. Similarly, a must be defined for al1 í with a c r 5 10. +3

ADDENDUM 2 PARAMETER CURVES I N TWO DIMENSIONS If f : U + M is an immersion from an open set U c IWm into an 17-dimensional manifold M, the curve í H f (al,. . . ,aj-l,r, aj+l,. .. ,an) is caIIed a parameter curve in the ith direction. Given 7t vecior fidds Xi ,. . . ,Xn defincd in a neiglilorhood of p E M and Jinearly indepaident at p , we know that there is usuaily no immersion f :U + M with p E f (U), whose parameter c u r e s in the ith direction are the integral curves of the Xi-for we might not have [Xi, Xj] = O. However, we mighi hope to find an immersion f for which the parameter culves in ihe ith direction lie along the integral curves of tlie Xj, but have different parameierizations. A simple example (Problem 20) shows tliat even this inodest kope cannot be fulfilled in dimension 3. O n the oilier Iiand, in the special case of dimension 2, such an imbedding can be found: 18. PROPOSITION. Let X1,X2 be Iiiiearly independent vector fields in a neighboi.hood of a point p in a 2-dimensional manifold M. Then there is aii imbedding f :U + M, where U c IR2 is open and p E f (U), whose ith parameter Iines Iie aIong the integral curves of Xi.

PROOF. We can assume tliai p = O E IR2, and that Xi(0) = (ei)a. Ewry poini y in a sufficiently small iieighborhood of O is on a unique integral curve of Xl thiough a point (0,x2(y))-we proved precisely this fact in Theorem 7. Similarly, y is oii a unique iiitegral curve of X2 through a point (xi (y),O).

The map y c ( x l(y),x2(y)) is Cm, wiih Jacobian equal to 1 at O (ihese facts also foIIow from tlie proof of Theorem 7). Its iiiverse, in a sufficieiitly small neighborhood of O, is tlie required dfiomorphism. ++:

\ e can always compose f with a map of the form ( x ,y) I+ ( a @ ) ,fl(y)) for dXeomorphisms a and fl of W , which @ves us considerable flexibility IIf, for example, C c W 2 is the graph of a monotone function g , then the map

(.Y, y ) H

( x ,g ( y ) ) takes the diagonal { ( x ,A-))to C . Moreo\ler, for any particulai-

paiiimeterization c = ( e i , c 2 ) :W + W 2 of C , we can further arrange that c ( r ) maps to ( c ( r ) ,c ( r ) ) , by composing with ( x ,y) H c x ) y ) . Coiisequently, we can state 19. PROPOSITION. Let X I ,X2 be linearly ii~dependentvector fields in a ileighboi-hood of a point p in a ?-dimensional manifold M, and let e be a curve in M with c ( 0 ) = p and ~ ' ( 1 )never a multiple of Xi or X2. Then there is an imhcdding f : U + M, where U c W 2 is open and p E / ( U ) , whose ith parameter liiies lie along the i i tegral ~ curves of Xi, and for which f (r,1 ) = c ( r ) .

PROBLEMS 1. (a) If a : M + N is Cm, then a , : T M + T N is Cm. (b) If a : M + N is a diffeomorphism, and X is a Cm vector field on M, tlien a , X is a Cm vector field on N. (c) If a : R + R is a(r) = r tlien there is a Cm wcior field X on R such that a,X is not a Cm vector field.

',

2. Fii~da now11e1-eO vector field on R such tl~atal1 integral curves can be defined only on some interval around O.

3. Fii~dail example of a complete metric space ( M ,p) and a function f : M + M such that p ( f ( S ) ,f ( y ) )< p ( x , y ) foi- al1 x, y E M, but f has no fixed point.

4. Let f : ( - e , c) x U x V + R" be Cm, where U , V c R" are open, and let (xo,yo) E U x V. Pro1.e tl~atthere is a neíghborhood W of (xo,yo) and a number b > O such that for each ( x ,y ) E W tl~ereis a unique a = : ( 4b,) + U wit11 a f ( t) E V for t E (4, b) and

-

Moi-eover, if we write a(,,,,) ( t ) a ( t ,x, y ) , then a : ( 4b) , x W + U is Cm. Hi726:Consider tlie system of equations

5. \Ve soinetimes have to solve equations "depending on parameters",

wl-ierc f : (-c, c) x V x U + R", for opeil U c R" and V c Rm, and we are solving for a(,,,,) : (4, b) + U for each initial condition x and "parameter" y. For example, the equation

is such a case. (a) Define

j : (-C,C)

x

v

x

U + R"

x

W~

by T u , Y, x) = (O,f(r, y , x)). If (E', ¿i2) = ¿i : (4, b) x W + R" x Rn is a flow for (YO, XO),SO that

j

in a neighborhood of

show tliat we can write

for some a , and coiiclude that a satisfies (*). (b) Sliow that equations of the form (**)

a

a;a(r, X) = f (1, X., Mt, 4 )

can lze i-educed to equatioiis of tlie foi-m (*) (aiid tl-ius to equations

a

-41, x.) = f (a(!, x)), at

ultimately). [Wlien one pi+ovesthat a C k function f : U + W" has a C k flow a : (-b, b) x W + U, tlie hard part is to proi7e tliat if f is C i , tlien a is differei~tiable~ t i t l respect i to tlie arguments in W, aiid tliat if tlie dei-ivativewi tli 1-csl~ectto tliese ai-gumeiits is denoted by D2a, tlien (***)

DI D2a (r, s) = D2f (a(t,x)) - Dza(t, x)

(a i-esult wliicli follows directly fi-om tlie original equatioii

if f is c 2 , silice D ID? = D2D1). Since (***) is an equation for Dza of tlie foi-m (**), it follows that D2a is diffei-eiitiable if Dr f is C 1 , i.e., if f is C2. Differential~ilityof class ckis tlien proved similarly, by induction.]

14ct0r fields at2d Ihjiroztial Equnr ions 6. (a) Consider a linear dfierential equation

where g : R + R, so that we are solving for a real-valued function a. Show that al1 solutions are multiples of

where Jg(r) dr denotes some function G with Gt(r) = g (one can obtain al1 podive multiples simply by changing G). The remainder of this problem investigates the extent to which similar results hold for a system of linear dserential equations. (b) Let A = ( u i j ) be an rz x 12 matrix, and let IA 1 denote the maximum of al1 lai, l. Show that

(c) Conclude tliat the infinite series of n x n matrices

converges absolutely [in the sense tliat ihe (i, j)& entry of the partial sums converge absolu tely for each (i, j ) ] and uniformly in any bounded set. (d) Show that ~X~(TAT= - ' )T(exp A)T-'. (e) If AB = BA, tl~en exp(A

Hint: Write

C p=o

(A

+ P!

+ B) = (exp A)(exp B). N p=o

p=o

and show that l R ~ + l O as N + m. (f) (exp A)(exp -A) = 1, so exp A is always inverti ble. (g) The map exp, considered as a map cxp : lRn2 + R"', is clearly differentiable (it is even analytic). Show that expt(0)(B)= B (Notice thai for IA 1, tlie usual norm of A

(=exp(O). B). E

lRnL,we have IA 1 5 IA l 5 n lA l. )

e)

(h) Use the limit established in part to show that expf(A)(B)= exp(A) B if AB = BA. (i) Lec A : R + Rn2 be differentiable, and lec m

If B3(r)denotes t l ~ ernatrix wliose enti-ies are the derivatives of the entries of B, show that B' (t) = Ar(r) - exp(A (r)), plovided dznt A(r) A*(r)= At(r)A(r). (Tliis is clearly true if A(s)A (r) = A(r) A (S) for al1 S , r .)

(j) Show that tlie linear differential equation

lias the solution

~witided[/nt g(s)g(r) = g(r)g(s) for al1 s,r. (This certainly happens when g(r)

is a constant matrix A, so every system of linear equations with constant coeficients can be solved explicitly-the exponential of i,g(s) ds = r A can be found by put ting A in Jordan canonical form.)

7. Clieck tllat if tlie cooidinate system x is x = X-', for X = a/axi is equivaleni lo x, (alar l) = X 0 x.

x:

R"

-t

M? then

8. (a) Let M and N be Cm manifolds. For a Cm function f : M x N + R aiid y E N, let f ( - ,q ) denote the funciion from M to R defined by

If (x, U ) is a coorditiate systeni oii M, sliow that tlie function af/axi, defined bv

is a Cm function on M x N. (b) If # : (-E, E) x M + M i~ a 1-parameter group of diffeomorphisms, show tliat for every Cm function f :M + R, the limit

exists, and defines a Cm function on M . (c) If #, : ( - E , E ) x TM + TM is defined by

S ~ O M tliat ~

#, is Cm, and conclude tliat for every Cm vector field X and covari-

ant vector field w on M , tlie limit

exicts and defines a Cm function on M . (d) Treat L x Y similarly.

9. Give the argument to sliow tliat #h*Y4-,, ( P ) + YP in the proof of Proposition 8. 10. (a) Prove that

(b) How ~vouldProposition 8 have to be changed if we had defined ( L x l ' ) ( p ) as

11. (a) Show that # * ( d f ) ( Y )= Y ( f o#). (b) Usirig (a), sliow direc~lyfrom tlie definition of L x tliat for Y

E

Mp,

and conclude that L x df = d ( L x f ) .

The formula for Lx ds' , dciived in tlie text, is jiist a special case derived in an uiiiiecessarily clumsy way In the iiext part we get a much simpler proof that L x Y = [ X , Y 1, usirlg the techxiique whicli appeared in the proof of Proposition 15. (c) Let X and Y be \rector fields on M , and f : M + IR a Cm function. If X generates (#,), define / t ) = yd-r(p) ( f o #h)-

174 Show tliat

Conclude that for c ( h ) = a(h,k) we llave

12. Check t1ie Jacobi identity.

13. O n R 3 let X, Y, Z be tlie vector fields

(a) Show that tlie map

is an isomorpliism (íiom a certain set of vector fields to IR3) and tliat [U,VI H the cross-product of tlie images of U and V. (11) Sllow tliai tlie flow of u X b y c Z is a rotation of R3 about come axis througli 0.

+

+

14. If A is a tensor field of iype (): on N and # : M + N is a diffeomorpliism, we define #*A on M as follows. If v i , . . . ,vk E Mp, axid A l , . . ., hr E Mp*, then

(a) Check tliat under tlie identificatioxi of a vector fie1d [or covariani vector field] witli a tcnsor field of type [or type this agrees with our old #* Y. @) If the vector field X on M generates ( # t ) , and A is a tensor field of type ( f ) on M, we define

(y)

(i)]

Sliow that

(so that in particular). (c) Show that

Lx,+x,A = Lx, A + Lx,A. Hini: IVe already know tliat it is tme for A of type (:), (:), (:). (d) Let C : ( v )+ rJ k - I ( V ) he any contraction = contraction of

Show tliat

Lx(CA) = C ( L x A ) . (e) Noting that A(Xi,. . . ,Xk,wl , . . . ,w I )can be obtained by applying contractions repeatedly to A @ XI @ . @ Xk @ w~@ . @ wl, use (d) to show that

jl

...j, .

(f) If A has compoiients A,[ .,

n

in a coordinate system x and X =

o'a/axi, i= l

show tliat the coordinates of Lx A are $ven by

I +

n

EE

a-1 i i l

...4 . ..

. -

...l a - ] i l a + ] ...i k

15. Let D be an operator taking tlie Cm functions F to F, and the C m vector fields V to V, sucli that D : .F + F and D : V + V are linear over W and

(a) Show that D has a unique estension to an operator taking tensor fields of

type

(f) to tliemselves such tliat

(1) D is linear over W (2) D ( A @ B ) = DA @ 3 + A @ DB

(3) for any coatractioxi C, DC = CD. If we lake Df = X.f and D Y = Lx Y1 tliea tliis unique exteasion is L x . (b) Let A be a teiisoi. field of type (i), so tliat \vc cari consider A ( p ) E Elrd(Mp); then A ( X ) is a vector ficld for each vector field X . Show that if we define DAf = 0, DAX = A ( X ) , tlien DA has a uni-que extension satisfying (1), (2), and (3). (c) Show that ( D A # ) ( / ) )= - A ( P ) * ( # ( P ) ) (d) Show that

LfX = f LX - DX@df Hillf: Clieck this for functions and vector fields first. (r) If T is of type slio~rtliat

(i),

a-l

Generalize to tensors of type

"-1

u-1

e).

16. (a) Let f : W + W satisfy f' ( O ) = O . Define g ( t ) = f tllat the right-hand derivative g'+ ( O ) = lim

fz+ O+

( A )for t

2 O. Sliow

~ ( 1 2 )- g ( 0 ) =: -. f"(0) 11 2

(Use Taylor's Tlieorem.) (b) Giw.11 c : W + M witli c'(0) = O E M,, define y ( t ) = c ( & ) for t 2 0. SIIOWtllat tlic taiigeri t vectoi- cl'(0) defined by c"(0)( f ) = ( f o c)"(O) can also be desci-ibed by c"(0) = 2 y t ( 0 ) .

17- (a) Let f : M + R have p as a critical poiat, so that f,, = O. Given vectors X,, Yp E Mp, clioose vector fields X , Y witli Tp = Xp aild Y, = Y,. Define f**CXp, Y,) = 1. rY

P . .

P . .

.",(O

Using the faci that EX, Y],( f ) = O, SIIOMJ that f,,(Xp, Y,) is symmetric, and conclude that it is well-defined. (b) Sliow that

(a2

(c) Tlie rank of f / a s i a x j ( p ) )is independent of tlie coordinate system. (d) Let f : M + N llave p as a critical point. For X,, Y, E M and g : N + R define f**(X, Y ) ( g )= X p ( Y ( g 0 f )). P . .

P . .

Show that $5.:

Mp x M,

+

N/(,)

is a well-defined hilinear map. (e) If c : R + M lias O as a critical poixit, show that

takes ( l o , l o ) to the taiigent vector c"(0) defined by ctr(0)( f ) = ( f o c)"(o).

18. Let c he tlie c u n ~of Theorems 15 and 16. If around p wiih x ( p ) = O, and

sliow that

x i ( c ( t ) )= uir

A-

is a coordinate system

+ o(t2 ) ,

where o ( t 2 )denotes a function such that lim o ( t 2 ) / t 2= O.

r+ O

19. (a) If M is conipact and O is a regular value of f : M + R, then there is a neighborhood U of O E R such that f - ' ( U ) is dineomorpliic to f-'(O) x U,

by a diReomorphism #: f - ' ( O ) x U + f - ' ( U ) with f ( # ( p yt ) ) = t . Hini: Use Theorem 7 and a partition of unity to construct a vector field X on a neighborhood of f (O) such that f* X = d / d t . (11) More geiierally, if M is compact and q E N is a regular value of f : M + N: then there is a neighborhood U of q and a diffeomorphism # : f ( q )x U + f-' ( U ) with f ( # ( p , y')) = q'. (c) Ii follo\r:s from (b) tliat if al1 points of N are regular values, tlien f - ' ( q l ) and f ( q 2 )are diffeomorphic for qi, q2 sufficiently close. If f is onto N , does jt follo~rthai M is diffeomorphic to f ( q ) x N ?

-'

-'

-'

-'

let Y and Z be unit wctor fields always pointing along tlie y- and r-axes, respeciively, and let X \vi11 be a vector field oiie of whose integral curves is the s-axis, while certaixi other integral curves are parabolas in the planes J' = constant, as shown in ihe first part of the figure below. Using the second part of tlie figure, show ihai Propositioxi 18 does not hold in dimensjon 3.

20. In

CHAPTER 6 INTEGRAL MANIFOLDS PROLOGUE A matlieinatician's reputation rests oii

tlie i-iuinbei.of bad pi-oolq he has giveii. [Pioxxeer woik is clumsy]

Beauty is the first test: there is no permaneiit place in the world for ugl); mat iiematics.

A. S. Besicovi tch,

quoied iii J. E. Li~lewood, A A,faílze?nnlicin?z'sAdircellaniy

1

n ihe previous chapter, we have seen thai ihe integral curves of a vector field on a inanifold M may be defixiable only for some small time interval, even thougli the vector field is Cm on al1 of M. 14re will now vary our quesiion a little, so that global results can be obtaiaed. Iastead of a vector field, suppose ihat for eacli p E M we have a 1-dimensional subspace Ap c M p . TIie f~~nciioxi A is cdlcd a 1-dimensional distribution (this kind of distrihution has xiotliixig ~rhatsoeverto do with the disiribuiions of analysis, whicli iiiclude sucll things as the "6-function"). Tlien A is spanned by a vector field ZocuZ~;that is, wc can choosc (in niaay possilJe ways) a vecioi. field X sucli tllat O # Xq E Aq for al1 q ixi some open set around p. 14fe cal1 A a Cm distrihution if such a vectox. field X can be chosen to be Cm in a ncigliborliood of each point. For a 1-dimexlsional distx-ibution tlle iiotioa of a n i~iiegralcurve niakes no sense, I ~ u wc i dcfine a (1-dimeiisioiial) subnianifold N of M to be a n integral manifold of A if for every p E N we Iiave

i*(Np) =: Ap whei-e

i :N +M

is the ixiclusion map.

For a givcn p E M , we can always fi~ida11 iiitegral manifold N of a Cm distribution A with p E N; we just choose a vector field X with O # Xq E Aq for C] ir) a iieighboi-liood of p , find arr iiitegral curvc c of X with iiiitial condition c ( 0 ) = p, a n d theri forget about the ~mrameterizationof c, by defining N to be ( c ( t ) j . Tliis al-qrinent actually slrows that for cvery p E M there is a coordinate system (x, U) sucli tliat for each fixed sei of iiunibers u 2 , .. . ,un, the set

is an integra1 manifold of A on

U: and that tliese are the only integral manifolds

in U. TIris is still a local resuli: but because Mte are dcaling with submanifolds, rathei- thar-i cuntes ~iritha particular parameter-ization, wfe can join over1appirig iritcgral s~il~mariifolcls together. The entire manifold M can he written as a disioint uriion of con1lec ted integral submanifolds of A , which locally look like

(1-ather tlian like

or sonlething e1fer-i more coniplicated). For esample, there is a distribuiion o11 thc IOI-LISIVIIOSC iritegral 1iinriifo1ds al1 look likc tlre densc 1-diii-ierisioriaI

submariifcitc1 pici~ircclin Cllaptcr. 2. O11 t1ie other llaiicl, tliere is a disiribution o11 tlrc ioriis \,.liiclr 138s OIIC coiii~~act coririectecl integral riianifold, arid a11 otlier

ir-iicgi-alniai-iifb1cIs riorr-compact. It liíippcris tIiat thc iritcgral mariifoIc1s of tI-iesc, twfo clistril-i~iiions are also tlie in tesral cunres for certain vector fidds, but on thc

h4d1ius strip thei'e is a distributiori \vliicli is sgarined by a \rector fie1d only locally

IVe are leaving out tlie details invohred in fitting together these local integral mariifo1ds because \ve ~rilleveritualty do tliis olrer again in tlie Iiigher dimensional case. For the niornent Mre \vil1 investigate higIier- dimensioiial cases onIy locally. A k-dimensional distribution oii M is a fiiiiction p H A,, ~tliereA, c M, is a k-dimerisiorial subspace of M,. For any p E M tliere is a neighborhood U arid li vcctor fields X1,. . . ,Xk such that XI ( q ) ,. . . ,Xk ( q ) are a basis for Aq, for eacll q E U. IVe cal1 A a Cm distributioii if it is possible to choose Cm vector fields XI, . . . ,Xq witli this property, iri a iicighborhood of eacli point p. A (k-dimensional) submanifold N of M is calIed aii integral manifold of A if for every p E N we have i,(Np)= A,

where

i: N +

M

is theiriclusionmap.

Aliliough tlic dcfiiiitions givci-i so fal. al1 look the same as the 1-dimensional case, the results \vi11 look verv different. In general, integral manifolds do ~ i o f ex&!, even locally. As tlie simplest cxarnple, consider the 2-din-iensional distributioii A in IR3 for ivl-iich A, = A(,,b,,) is spanned by

l f we icler,tify TIR' witli IR" IR3, tlien A, conrists of al1 (r, S, br-),. Thus A, n1ay bc ~~icturccl as tlie plaiie with the equation

The figure be1ow s h o ~ Ap s for ~ o i n t ps =: (o, b, O). T h e plane (u, 6, c.) is just parallel to the one through (o, b,0).

through

If you can pict~rrctl-iis disti-il~ution,you can probably see that it has no integral manifolds; a proof can be @ven as fo1Iows. Suppose there were an integral manifold N o i A witll O E Al. The intersection of N and ((0, y, r ) )would he a cunfe y in the (y,:)-plane tlxrouglx O whose taiigent vectors wou1d Iiave to lie ir1 the intersection of Ala,,,=) and tlie (y, z)-p1ane. The only such vectors have tljird component O, so y 111ust 13e tl-ie 17-axis. Now corisider, for each fixed yo, the iritersection N n ((A-,?lo,c ) ). This will be a curve in the p1ane ((A-, yo, z)) t11rou~l-i(O, yo, O), with a11 tangent vectors Iiavirig slope yo, so it must be the linc ((x, yo, yox)). OUI.iiitcgi-al nianifold would have to look like the following picture. R ~ i this t submanifold cloes not urork. For example, its tangent space at ( 1,0,0) contairis \fcctoi-swi th tl-iird componcn t non-zero.

To see in greater detaii what is happening liere, consider the somewhat more general case where A(,,b,,) = Ap is

geometrically, Ap is the plane with the equation Z-Crir

f ( f l , b ) (-~ f f ) + g ( f f , b ) ( y - b).

As in tlie first exarnple, the plaiie through (o, b,c) wiil he paralle1 to the one through (u, b, O ) , since f and g depend only ori a and b. M'e riow ask wlleri tlic distribution A has an integral manifold N through each poiiit. Since Ap is never perpendicular to the (-Y, y)-plane, the submanifold is giveii locally as tlie grapli of a function:

No\%:tlie tangent space at p = (a, b, @(u,b)) is spanried by

Tlicse tangenr vectors are in Ap if and oiily if

So \ve need to fiild a funct~ona : R2 + R with

(4

a@ ax= f ,

- " g.

It is wcll-kr~ot~n tliat this is not a1ways possible. Ry usirig the equality of mked partial deri\?atives,\ve find a necessary condition on f and g :

In our previous exampIe,

so tliis iiecessary condition is not satisfied. It is also well-kriown that tlie neceshary co~~diiion (a*) is st@riFltl for the existente of the function cr satisfying (a) in

a ~i~igllborliood of any point. O. PROPOSITI ON. If f , g : IR2 + IR satisfy

in a neig1iborhood of O, and zo E IR, tlien tliere is a f~inctioncr, defined in a ncigliborliood of O E IR2, such tliat

cr (O, O) = -'o

PXOOE IVe fjrst define ~ ( xO), so that @(O, O) = o: and

riamely, wc define X

Tlleii, for eacll x, we define a ( x , y ) so that

namely, we define

This cor~structioi~ does aot use (w),and always provide us with an a. satisfying (2), aa./ay = g. IVe claiin that if (**)holds, tlieii also aa/ax =: f. To prove tliis, concider, for each fixed x, the function

This is O for = O by (1). To prove that it cquals O for al1 y, we just have to S ~ I O Mthat ~ its derivative is O. Rut its derivative at y is JI

IVc ar-e IIOW ready to look at essentially t11e rnost general case of a 2-dimensional distr-ibution in R3:

where f, g : R3 + R. Suppose tliat

N == ((x, y,z) : 6~ = a.(x,y))

is an integral manifold of A . The tangent space of N at p = (a, b, a(a,6 ) ) is spanned, once again, by

These tangent vectors are in Ap if and on1y if

aa

f (a,b,d a , 6 ) ) = a~ ( a ,b), aa g(nl b, a ( o ,b ) ) = -(a, ay

b).

In order to ohtain neccssary conditions for the existente of such a function a , we again use the equality of mixed partial derivatives. Thus (*) and the cllain ru1e imply that

Tliis condition is not very usefu1, since it still involves the unknown function a , l ~ uwe t can substitute from (*) to obtain

-(a, b,a(a,b ) )+ -(n,b,a(a,b ) ) g(a,b,@(a,b)) ay az

Now wc are looking for conditions whicli will he satisfied by f and g when there is an integral niailifoId of A Iltroqk euu-poinf, which means tliat for eacli pair ( u ,b ) these equations must ho1d no matter what a ( a ,b ) is. Thus we ohtain finally the necessary coiidi tion

In tliis more genera1 case, the necessary condition again turns out to be sufficient. In fact, there is no need to restrict ourselves to equations for a single function defined on R2; we can treat a system of partial difGerentia1 equations for n functioils o11Rm (i.e., a partial dflerential equation for a function from R" to R"). In the following theorem, we wdl use I to denote points in Rm and x for points in W"; so for a function f : Wm x R" + Rk we use

as. -

for

Dm+if.

axi

1. THEOREM. Let U x V c Rm x R" be open, where U is a neighborhood of O E R"', ar-id let J i : U x V + Rn be Cm functions, for i = 1 , . . . , 1 1 1 . Then for every A- E 1/, there is at most one function

defined in a neiglil~orhoodW of O in Rm, satisfying

(hllore precisely, ariy two sud1 furictions a1 arid a*, defined on WI and W2, agree on tlie conil~oileritof Wi n W2 wliic1i contains O.) Moreover, such a function exists (arid is autoriiatically Cm) in some lieighborhood W if and only if there is a iieighborhood of (O, x) E U x V on which

I'HOOF. Uiliqueriess ~ l i l lbe obvious from tlie ~ r o o of f existence. Necessity of tlie conditioris (34)is lcft to the readei- as a siinple exercise, and we will concern oursch~eswith provirig existerice if these coriditions do liold. Tlie proof will be like tliat of Proposition O, 14th a different twist at the end. IYe f i l ~ waiit t to define ~ ( rO,. , . .,O) so that a(0, o,. . .,O) = x

To do this, we corisider the ordinary differential equation

Tliis equation has a unique solution, defined for

Then (1) holds for 111 -= EI . Now for eacli fixed r l with

111 < E I . Define

1 1 ~ 1 < E I ,consider thc equation

7'liis Iias a unique solution for sufficiently srnall 1. At tliis point the reader must refer hack to Theor.ern 5-2, arid veri+ the following assertion: If wre choose ~1 sufficiently srnall, then for 11'1 < E I tlie solutiorls of tlie equations for P2 witli tllc initial conditioiis p2(0) = Q ( I ~0,., . . ,O) will eacli be defined for 111 ~2 for sor-ile ~2 > O. We then define

(2)

aot -(fl,l,O a12

,..., O) = f * ( f

1

1

,1,0,..., O,ot(1 ,1,0 ,..., O)) 11'1

l4'c claim that for eacli fixcd 1' witli (3)

o = g(1) = +fa@ 81

l

(El,

111 n.)

PKOOI;. 1f o E R'(v) c T'(v), we can write

= det .) A basis

A #jk for some 8 #), is either O or = =t(l/k!)#jl A Eacli At(#il 8 < j k span f i k ( v ) . If jl c . c jk, SO tlie elenlents #jl A . . A #jk for j i
O on [O,11". Let w be the 11-form w =fdx' A...A~x".

PROOF. By definition,

/

( f o c ) Jdet c'J d x ' [o,-l]l1

=

=

/

c(I0,' P')

f

A

A

dx"

by assumption

by the change of variable formula. 4 3

2. COROLLARY. Let p : [O,11' + [O,11' be one-one onto with det p' 2 0, let c be a singular k-cube in M and Iet UI be a k-form on M. Then

I- =Lpo.

PROOF. We llave

J COP

o=

J '

[O, lk =

/

(C 0 p ) * o

=

J

P*(c*o)

'

[O, jk

c * ( w ) by the Pmposition, since p is onto

The map c o p : [O, l l k + M is called a repararneterization of c if p : [O, 1 l k + [O, 11' is a C m one-one onto map with det p* # O everywhere (so that p-' is also Cm); it is called orienta tion preserving or orientation reversing depending on whetller det p' > O or det p' < O everywhere. The corollary thus shows independence of parameterization, provided it is orientation preserving; an orientarion reversiiig repai-ameterization clearly changes the s i p of the integral. Notice that there ~ f o u lbe d no such result ifwe tried to define the integral over c of a C m fuilcrioil f : M + IR by the formula

For example, if c : [O, 11 + M then

1

f ( ~ ( 1 ) ) dl

is generally

#

o

f ( c ( p ( 1 ) ) 0) 1

o

From a formal point of view, differential forms are tlie things we integrate because they transform coi.rectly (i.e., in accordance witll Theorem 7-7, so that tlle change of variable formula will pop up); funcrions on a manifold cannot be integrated (we can integrate a function f on the manifold R' only because it gives us a form f d x l A A dxk). Our definition of the integral of a k-form UI over a singular k-cube c can iinmediately IIC generalized. A k-chain is simply a formal (finite) sum of singular k-cubes multipIied by integers, e.g..

Tlle k-cllain 1c1 = 1 cl will also be deiioted siinply by c l . We add k-cllains, and multipIy them by integers, purely formally, e.g.,

Morcovcr, wc define the integral of way :

over a k-chain c =

xiaici in t11e obvious

The 1-easoil Toi. iilti.oduciilg k-chaiiis is that to evei-y k-chain c (wl~iclimay bc just a singulai A--cube) we wisll to associate a (k - 1 )-cliain ac, ~ } h i c his called tl-ie boundary of c , aild which is sul~posedto be t11c sum of the various singular

(k - 1)-cubes around the boundary of each singular k-cube in c. In practice, it

is convenient to modify this idea. The boundary of í2, for exarnpk, 4 1 not be the sum of tlie four singuIar 1-cubes indicated below on the left, but the sum,

witIl the indicated coefficients, of tlie four singular 1-cubes shown on the right. (I\'otice tliat tliis will not cliange the integral of a 1-form over al2.) For each i witli 1 5 i 5 n we first define two singular (n - 1)-cubes ICgo) and II;,,) (the (i, O)-face and (i, 1)-face of í")as foIlows: If x E [O, 11"-' , then I:,~)(X) = ~ " ( x ' , .. . ,xi-i ,O,Xi y . . . y ~ n - l )

= (x 1 ,...y xi-l

y

o, 2,.. .

I;.,,)(x) = í " ( x l , . . . , xi - 1 ,1, Si = (S 1 ,..., x i-l , l , xi ,

y

, . a

xn-l),

. ,xn-l) A-"-1)

..m,

The (i,a)-face of a singular n-cube c is defined by

Now we define

n

FinaIly: the boundary of al1 n-cllain

Ciaici

is defined by

These definitions al1 make sense only for n >_ 1. For the case of a O-cube e : [O, 1' + M, wliicli we will usually simply identify with the point P = c(O), we define ac to be tlie number 1 E IR,and for a O-chain aicj we define

xi

Notice that for a 1-cube c : [O, 11 + M we have

Mre also Iiave, for a singular 2-cube e : [O, 112 + M'

From a picture it can be cl-iecked that this also Iiappens for a singular 3-cube, a good exercise because this involves figuring out just what tfle boundary of a 3-cube looks like. In general, we have:

3. PROPOSITION. If c is any n-chain in M, then a(&) = O. Briefly,

PROOE Let i 5 j 5

i7

- 1,

and consider (I{,a))(j,B). For x

E

a2 = 0.

[O, 11"-~, we

have, from the definition a ,;('l

)(jln O, since c.1 aiid ~2 are 13otli orientatioii preservirig). However, a glaiice at tlie proof of Corollary 2 will show tliat tlie result still follows, because of tlie fact tliat support w c cl ([O, 11") n c2([O, 11"). *: Tlie romiilon iiumber

[w, for singular ii-cubes c : [O, 11"

+ M with sup-

Jc

poi-t w c c([O, 11") aiid c 01-ieiitatioii 11i-eserving,wiI1 be derioted by

lf o Is aii al-l~itrary11-form oii M, tlieii there is a cover 0 of M by open sets U. eacli coiitaiiied iii some c([O, l]"), wliei-e c is a sirigular ii-cube of tliis sort; if @ is a partitioii of uiiity subordinate to this cover, tlien

is defiiied for each 4

E

@. We wish to define

14%wilI adopt tIiis definition only when o has compact support, in wliich case the sum is actually finite, since support w can intersect only finitely many of t11e sets { p : $ ( p ) # O), which form a locally finite collection. If we haw anothei. partition of unity Q (subordinate to a cover O'), tIlen

iliese sums are al1 finite, and the last sum can clearly also be written as

so tl-iat our definiiioii does not depeiid on the pai-tition. (1We really sl~oulddenote tliis sum by

for tlle oi-ientation - p of M we clearly liave

Howevei; we usuaIly omit explicit mention of p . ) Witli ininor modifications we can define iM o evcn if M is an i?-manifoldwitli-boundary. If M c IR" is aii 11-dimensional maiiifold-~4th-bouiidaryand f : M + IR has compacr support, tlien

wllerc tl-ie 1-iglithaild side dellotes the oi.di1-iai-yii~iegi-al.TIiis is a simpIe consequerlce of Proposition 1 . Likewise, if f : M" + N" is a diffeomorphism onto: aiid o is aii 17-foi-mwitli compact support or-i N, tlieii

rlo -

..-

if f is orientation presewing ii f is orientation reversing.

AItIlough n-forms can be integrated only over orientable manifolds, there is a way of discussing integratioil on non-orientable mai-iifolds. Suppose that o is a function on M such that for each p E M we have 4

~ = )Iqpl

i.e., for any n vectors u],. . ., u,

for some E

qp E Qn(Mp),

Mp we have

Sud1 a furlction w is called a volume element-on each vector space it determilles a way of measuring n-dimeiisional volume (not signed vol ume). If (x, U) is a cooi.dii1ate system, then on U we can write =

f l d x ' ~ - . . ~ d x " l for f 2 O;

\ve cal1 o a Cm volun-ie element if f is Cm. One way of obtaining a vo1ume element is to begin witll an rr-form q and tllen define w(p) = Jq(p)i. Howeve~ not every volume elemei-it arises in this way-the form qp may not vary contii~uouslywith p. For exainple, consider the Mobius strip M, imbedded in W3. Siiice M, can be considered as a subspace of R3p, we can define w(p)(up, wp) = area of parallelogram spanned by u and w. lt is not Ilard to see thai w is a volume element; Iocally, w is of the form w = J q J for an 11-form q. But tliis canilot Ile true on al1 of M, siilce there is no 11-form q on M which is everywhere non-zero. T1-ieoi.em 7-7 Iias ail obviohs modification for volume elemeiits: 7-7'. THEOREh4. If J'; M + iV is a Cm function I~etweenrl-manifolds: (x, U) is a coordiiiate sysrem arourld p E M, aiid (y, V) a coordinate system arouiid q = f (p) E N , tlien for iion-llegative g : V + W we have

PROOF. Go tl-irouglitlie proof of Tlieorem 7-7, putting in absolute value signs in rhe right place. +f,

7-8'. COROLLARY. If (x, U) and ( y , V ) are two coordinate systems on M and Idy' ~ . . . l i d y " = J bldxi

A

~

~

~

A

g~ , hX2 0"

~

tlien

[Tliis coi-ollary sliows tliat volume elements are tlie geometric objects corresponding to tl-ie "odd scalar densities" defined in Problem 4-10.] It is now an easy matter to iiitegrate a volume element o over any manifold. First we define

Tlien foi- an n-chain e : [O, 11" + M we define

'

Tlieoirm 7-7' sliows tliat Proposition 1 holds for a vol ume element o = f ldx A A dx"] even if det e' is not 3 O. Thus Coi.ollary 2 liolds for volume elements eveii if det p' is iior 3 O. From this we conclude that Theorem 5 holds for volume elements o on aiiy manifold M, without assuming el, c2 orientatioii presen~ing(or even diat M is orientable). Coi~sequentlywe can define JM o for ai-iy voluiiie elcment o wit1-i compact suppor~. Of course, when M is orientable these consider-ations are unnecessary. For, there is a nowliere zero 11-foi-mv on M, and consequently any volume element o can be written = f Ivl, f 2 0. 1f we clloose an orientation p for M such that o ( v l , . . .,U,)) > O for vi,. . . ,U, l~ositivelyoriented, tlien we can define

I~olumeclements will be important later, but for tlie remainder of this chaptei. we are coiiccrned oilly ~ 4 t hintegrating forms over oriented manifolds. In fact, our main resuli about iiltegi-alsof forms over manifolds, an analogue of Stokes' Theor-em about the integral of forms over chains, does not work for volume elements.

Recall from Problem 3-16 that if M is a manifold-with-boundary, and p E aM, then certain vectors v E Mp can be distinguished by the fact that for any coordinate system x : U + W" around p, the vector x+(v) E Wnf(pl points "out~~ards". 14%cal1 such vectors v E Mp "outward pointing". If M has an

01-ientationp , we define iIíe induced orientation ap for aM by the condition that [ u I , . .. , E ( 8 ~ if) and ~ only if [w, vi,. . .,~ ~ - 1 E1 pp for every outward poiiiting w E Mp. If p is the usual orientation of H", then for p = (a, O) E W n we have

Since (-en)p is aii outward poiiiting vector, tliis shows that the induced orieiitation on IRR-' x {O) = a W R is (- l ) n times t1le usual one, The reason for this choice is the follotiliilg. Let c be an 01-ieiitationpreserving singuIar n-cube in ( M , p) sucli thai aM n c([o, 11") = C ( ~ , ~ ) ( [ O11"-'). , Then c(.,o): [O, 11"-' +

(aM, ap) is 01-ieilrationpi-esenring for even n , and orientation reversing for odd n. If w is ai-i (n - 1)-foi-mon M whose support is contained in tlie interior

of tl-ie image of c (this interior contains points in the image of ihat

it follows

But c(,,0) appears witll coefficient (- 1)" in ac. So

If it were not for this choice of ap we would have some unpleasant minus s i p s in the followiiig theorem.

6. THEOREM (STOKFS' THEOREM), If M is ail oriented n-dimensional manifold-with-boundary, aild aM is giveii the induced orieniation, and o is an (il - 1)-form on M with compact support, then

PROOF. Suppose first that there is an orientation preserving singular n-cube c in M - aM suc11 rhat supporr o c interior of image c. TIien

=O

since suppori o C interior of image c,

Suppose nexi that tlierc; is an orientation presenring singular n-cube c in M sucli tliat 8~ n c ([O, 11")= C(,,~)([O, 11"-'), and suppori o c interior of image c. Then once again

Jn geiieral: rIiere is an open cover O of M and a partition of unity @ subordinaie io 13 such ihai for each $ E @ the form # . o is one of the two sorts already considered. \We have

Sii-ice o lias compacr suppori, tIiis is reaIIy a finite sum, and we conclude tliat

Therefore

One of ilie simplesr applicaiioris ofSrokes7Theorein occurs wlien ilie orjeiiied iz-n~niiifold( M , p ) is coinpaci (so tliat every foim has compact support) and a& = 0. 111iliis case, if is aiiy (u - 1)-form, ihen

Tlierefore we caii fii-id aii 11-fo1.m o on M wllich is t~ofexact (eveii thougli ir iiiusr be closed, l~rcauseal1 (12 + f )-firms on M are O), simply by findiiig aii o wi t11

Such a fui.i~~ o iil~vaysexisis. Indecd we havc seen ihai il-iere is a fm-iii o sucl~ ihai foi- vi,. . .,u,, E M, we liave

If c. : [O, I]" + ( M , 11) is oi-ieiitatioii pi.esei\*iiig,trheri t l ~ efoiam c*w o11 [O, 11" is c.leaiiy g dxi A - . A dx" for some g > O oil [O, I]",

1,

so w > 0. Ii follows that JM w > O. Tliere is: moreover, no need to choose a foi-m w with (*) holding eveiywhere-we can aIIow the > sign to be replaced by >_ . Thus Mte can even ob tain a non-exact 11-form on M which Iias support contained in a coordinate neighborhood. This seemiiigly minor result already pi.oves a theorem: a compact oriented maiiifoId is not smoothly contractible to a point. As we have already emphasized, it is tlie "shape" of M, rather than i ts "size", which de termines whether or not every closed form on M is exact. RoughIy speaking, we can obtaiii more informatioii aboui tlie shape of M by analyzing more cIoseIy the extent to ivhicli closed foorms are not necessarily exaci. In particular, we wouId now Iike to ask jusr Iiow many non-exact 17-foi-msthere are on a compact oriented 11-manifoId M. Naturally, if w is not exact, then tlie same is true for w + d q for any ( n - 1)-form q , so we realIy want to consider w and w d q as equivalent. Tliei-e is, of coui.se, a standard way of doiiig tliis, by coiisidering quotient spaces. 14%will apply tliis coiistruction not onIy to 11-forms,but to forms of any degree. For eacli k, ihe collectioii z'(M) of a11 closed k-rorms on M is a vector space. Tlie space B'(M) OSal1 exact k-forms is a subspace (since d2 = O), so we can form tlie quotient vector space

+

this vecior space H k (M) is cvlled iIie k-dimensional de Rham cohomoIogy vector space of M. [de Rlzam % T/zeore~nstates tliat tliis vecior space is isomorphic to a ccrtain vectoi. space defiiied pureIy in ierms of the iopology of M (for any space M), called tile "k-dimensional coliomology gi-oup of M with real coefficients"; tlic noiatioii z', 3' is chosen to cori-espond to the notation used iii algebraic ropolog): wliere ihese groups are defined.] An element of H k ( M ) is an equivalente class [m] of a closed k-form o, two cIosed k-foi.ms w1 aiid w2 beiiig equivaleiit if and onIy if their dserence is exact. Iii terms of these vector spaces, tlie Poiiicart Lemma says that H' (W") = O (the vecior space coiitaiiiing only O) if k > O, or more generally, H' ( M ) = O if M is contractible aiid k > 0. To computc H0(A4) we iioie fiist iIiat BO(M) = O (there are no non-zero exacr O-hrms, since tliere are no non-zero (-1)-forms for them to be the difSerential 00. So H'(M) is iIie same as iIie vector space of al1 Cm functions .f : M + W ~ 4 t hdf = O. If M is coiiiiected, the condition d f = O implies tliat / is constant, so H0(M ) % R . (Iii general, tlie dimension of H'(M) is the iiuml~erof components of M.) Aside from these trivial i-emarks, we 111-esentlyknow only one other fact about H k (M) -ir M is compact and oriented, tIien H n( M ) Iias dimension >_ l . The furtlier study of H k ( M ) i.equires a careful look at spheres and Euclidean space.

O n S"-' C R" - {O) tliere is a naturaI choice of a n (n - 1)-rorm a' with E S"-lp, we define Js,l-l 0 ' > O: for VI)^, . . . ,

ClearIy iliis is > O if .. . , ( v , - ~ ) is~ a positiveIy oriented basis. In fact, we defiiied ihe 01-ientationof S"-' iii precisely iliis way- tliis orientation is just tlie induced oiientation ~ t l i e nS"-' is considered as the boundary of the unit ball { p E IW" : Ipl 5 1 ) ~ i i l tlie i usual oi-ieiiiaiion. Using the expansioii oT a rlctermiiiaiit by minors along tlie top row we see tIiat o' is tlie restriction to S"-1 of ilie form a on R" defiiied by

Tbe form a' on S"-' will now be used to find an (n - 1)-form on R" - {O) wliich is closed but not exact (thus sl~owiiigthat H"-' (W"- (0)) # 0). Colisideithe inap 1 . : IW" - {O) + S"-' defined by

ClearIy i0(p)= p if p inclusion, tllen

E

S"-'; otheiwise said, if i : S"-' + R" r

o

-

{O) is tlie

i = identity of S"-'.

seneraI, if A c X aild 1.: X + A satisfies r ( a ) = a for a called a retraction of X onto A.) Clearly, r*a' is closed: (Ii1

Ho~t-vcr,i t is not exact, for ir r"a'

11ui íve ki~owthat a' is not exact.

-

dq, then

E

A, tllen r ir;

It is a worthwhile exercise to compute by bru te force tliat xdy-ydx xdy-ydx = d6' x2 y2 v2

forn=:2,

r *o1 =

for n = 3,

r *oI = x d y ~ d z - y d x ~ d z + z d x ~ d y (,Z + }'2 + $)3/2

+

1 =-[xdy~dz-ydx~dz+zdx~dy]. u3

Since we ~ 7 i I lactually need to know r*olin general, we evaluate it in anothei. way :

7. LEA4h4A. If o is the form on IR" defined by o=

E(-

1)'-lx'

dxl A -

A

-.

dxJ A

A

dx",

aiid o' is the restrictioli i*o of o to S"-', then

IWP

PROOF. Ai any poiii t p E IWn - (O), tlie tangent space is spanned by pp aiid tlie vectors vp ili tlie tangent space oT tlie spliere S"-' (1pl) of radius 1 pl. So ir sufices to check that botli sides of (*) give the same result when applied io ir - 1 vectors eacll of wliich is one of tl-iese two sorts. Now pp is the tangent vector of a curve y lyiiig aloiig ilie straight line tliiough O and-p; this curve is takcn to tlle single point r ( p ) by r, so r+(pP)= O. O n the other hand,

So i t suffices to apply both sides of (*) to vectors in the tangent space of S"-' (1~1).Tlius (Pi-ol~lem15), i t suffices to s l i w that for such vectors vp we liave 1

But tliis is almost obílious! since the [rector vp is t l ~ etailgent vector of a circle y lyins .... in S"-' (lpl), aiid the c u n a r. o y líes in S"-' and goes I/lpl as far in the same time. *:*

8. COROLLARY (INTEGRATION I N "POLAR COORDINATES"). Ler f : 3 + IR,where = { p E Ik" : 1pl 5 J ] , and define g ; S"-' + W b!.

Tlieii

/

~ f = ~ f d x ' ~ - - . ~ dSi,x t "g o=r m PROOF. Coiisider S"-' x [O, 11 and the two projectioiis ni :

:

S"-' x [O, I ] + S"-'

S"--'

X

[O, 11 + [O,11.

Lei us use tlie abbreviation

S"-'

a' A dt = ni*orA n2*di.

a

IT (y, U) is a coordiiiate system on S"-', \vi tli a corresponding coordinate system (J',t ) = ( y o T I ,n2)on S"-' x [O, J ] , and a r = cr dy' A - - A dyn-' , theii clearly o r h d r = & o n i d j l~ . . . n d y " - ' ~ d t . From rliis i t is easy to see tliat if we defiiie h : S"-' x [O, 11 + IR by h ( p , u ) = un-' f ( u p), theii

/

SI!-

= (- )"-l 1

/

SI)- t

/zar A dt x[o,IJ

NOM'í~recaii define a dfleonioi-pl?ism @ : 3 - { O ) + S"-' x ( O , J ] l y

Then

-

(- 1)"-'

""+1

n

E ( X 'd x)l~ A

m

A

ds"

i=l

Hence

= ""-1 f .

f

dxl

A

(- l)"-'

Y"-1 d x l n -

/ J

. . . A dxn = (- I)'-'

. A dx"

$*(ha1A d l )

3-(01

= (- l)"--I

ha'

h dt

S"-' x ( 0 , l j

Stl-

t

(This lasi siep requires some justificaiion, w11icIi sliould be supplied by the reader, since tlie rorms involved do not bave compact support on the maniIoIds B - {O) aiid S"-' x (O, 1 ] where they are defined.) +:+ We are aboui i-eady io coinpuie H' ( M ) in a kw more cases. We are going to reduce our calcuIations to calculaiions within coordinate neighborhoods, which are sul~nianifoldsor M , bui not compact. It is tllererore necessary to introduce ailotIier co11eciion of vector spaces, which are interesting in their own righi.

The de Rham cohomoIogy vector spaces with compact supports 13: ( M ) are

wliere Z: ( M ) is the vector space of closed k-forms with compact support, and B:(M) is tlie vecior space of al1 k-rorms dq where q is a (k - 1)-Torm with compact support. Of course, if M is compact, then H: ( M ) = Hk(M). Notice tliat ~ , ( kM ) is nol ihe same as tlie set of al1 exact k-forms with compact support. For example, on Rn, if f 2 O is a function with compact support, and f > O ai some point, then o = f dx' A - . .~ d x " is exact (ever): closed form on Rn is) and 11as compact suppor t, but o is not dq for any form q with compact support. Indeed, if o = dq where q has compact

support, tlien by Stokes' Theorem

Tliis example shows tliat HC(Rn) # 0, and a similar argument shows that if M is any orieniable maiiifold, ilien H;(M) # O. M7e are now going to sl~ow tl~atfor any connected orientable manifold M we actually have Tliis means ilia t if we clioose a fixed o wi tli JM o # O, then for any n-form o' witli compaci support there is a real number a sucli that w' -ao is exact. The number a can be described easily: if

then

ilie problem, of course, is sliowing ihat q exists. Notice that the assertion thai R is equivalent to the assertion that H:(M) II

is an isomorpliism of H!(M) with R, i.e., to the assertion that a closed form o with compaci suppori is the dflerential of another form with compact support if JM o = O.

9. THEIOREM. If M is a coiinecied oi-ieniable ti-ma~iifold,then H," ( M ) i-- R.

PROOF. \Ve wi11 establish tlie theorem in three steps: (1) The theorem is true for M =: W (2) If the t11eorem is a u e for (n - 1)-manifolds, in particular for S"-', then i t is true for W n .

(3) If t11e tlieorem is true ior IR", then it is true Tor any connected oriented 11-manifold. S~efi1. Le t w be a 1-form on W wi th compact suppor t such that JR w = O. There is some fui~ctionf (not iiecessarily with compact support) such that w = df. Since suppori o is conipact, df = O outside some interna1 [-N, N], so f is a

constan1 cl on (-m, -N) and a constant c2 on (N, m ) . Moreover,

TIierefore ci = c2

wIiere f

-

c aiid we Iiave

- c Iias compaci suppori.

Skp 2. Let w = .f dxlA - A dx-" be an 11-hm witl~compact support on W" such tliat IR,, w = O. For simplicity assume that support w c ( p E W" : Ipl < 1). Mre ki~owr11ai tliere is an (rt - 1)-form q on W n suc11 tliat w = dq. In fact, from Prol~lem7-23, we tiave an expIicit formula for q,

Using tlie subsriru tion u = 1 pl l this becomes

Define g: S"-' + W by

0 i i

tlie ser A = { p E Wn : 1 pl > 1) we I1a17e f = O, so on A we Iiave

h4oi~eover,11y CoroIIary 8 hreIiave Ioi ilie

=

J..

(11

- 1)-Iorm ga' on S"-',

o = o.

Tlius, Ily rlie I~yporliesisfor S ~ c j i2, go' = d l

Ior some

(il

- 2)-form 1 o11 S"-'.

I x t lr : R" + [O, I ] l ~ caiiv CbO Iunctioli witli h = 1

iirigliboi-Iioodof O. Tlieii

jii-*A

011

is a Cm form on W" and

A aiid h = O in a

the foi'm q - d(hlm*h)Iias compact support, since on A we have

Sley 3. Choose an n-form o such that JM o # O and o has compact support coiitaiiied iil an open set U c M, with U diffeomorphic to Rn. If o' is any otlier 12-Sorm with compact support, we want to show that there is a number c aiid a form q witli compact suppor t sucli tliat

Using a partiiion of unity, we can write

wliei-e eacli $;o' lias coinpact support contained iii some open set Ui C M with Ui diffeomorpliic to R". Jt obviously suffices to find ci and with $irn/ = t i a + dqil Sor eacli i . In otlier words, we can assume m' Iias support contained in some open V c M wliicll is difíeomorphic to Wn. Usiiig tlie coiiiiectedness of M, ii is easy io see iliat there is a sequence of open sets u = vi, ..., V, = if diffcomoi-pliic to R", with I/i n

n

W ~ I C I P al1

# 0. Clioose Sorms w j witli support

c

and Jv. oi # O. Siiice we are assuiniiig the tlieorem for W" we haw I

q j liave coiiipact support (C

desired resuli. *3

K).

From tliis we clearly obtain the

The method used in tl-ie last step can be used to derive another resu1t. 1 O. THEOREh4. IT M is any connected non-orientable n-manifold, then H ; ( M ) = O. PROOF. Clioose a n 11-Torm o ~ 4 t compaci h suppori contajned in an open set U difiomorphic to IR", such that Ju w # O (this integral makes sense, since U is orieiitable). It obviously suffices to sIlow that o = dq for some form q with compact support. Consider a sequence

u = v , , . . . , vr = v of coordinate systems (Vi, x i ) where cacIi xi o xi+l -1 is orientation preserving. Choose tIle forms mi in S1el3 so that, using the orientation of Vi which makes xi : 6 + IR" orieniaiion pi-esen~iiig,we llave iYii > O ; then also JVirl mi > O. Consequently, the numbers

iK

c =

w

/

It ~ O ~ I O W S that oi

=:

Cw

are positive.

o

+ dq

wliere c > 0.

Now if M is unorientable, tllere is such a sequei-icewhere V, = VI but is orieiitaiion rezielrnizg. Taking o' = -o, we llave -w = cw

+ dq

A-,

0x1-'

Tor c > O

*

(-c

- ] ) o= dq

Tor

- c - 1 # O. +3

We can also compute H " ( M ) Toi- non-compact M 11. THEOREh4. If M is a connected non-compact n-manifold (orientable or not), ihen H n ( M ) = 0.

PROOF. Consider firsi an 11-form o witli support contained in a coordinate i~eigliboi-lioodU wIiic1-i is diffeomorphic io IR". Sii-ice M is not compact, t11ere is an infiniie sequence u =: U,,Uz,U3,U4,...

of sucli coordinate neighborhoods such that Ui n Ui+1 # 0, and sucli that the sequence is eventually in the complement of any compact sei.

Now clioose ir-forms mi witll compact support contained in Uj n Ui+i, such tliat Jui q # O. There are constants cf and forms with compact support c Ui sddi tliai

Since aiiy poiiit p

E

M is eventually in the complement of the UiYs,we have

where the riglit side makes sense since the Ui are eventually outside of any compact set. Now it can be sliown (ProbIem 20) that there is actually such a sequence U1,U2, U3,. . . whose union is al1 of M (repetitions are allowed, and Uj may intersect several Uj for j < i, but the sequence is still eventually outside of any compact set). Tlle cover 13 = ( U ) is tllen locally finite. Let ($u) be a partition of unity suboi.di~iareto O. If o is an 19-form oii M, then for each Ui we have seen that $vio =: dqi Hence

wl~ereV i has support contained in Ui U Ui+1 U Ui+2 U - . -

.

SUMMARY OF RESULTS

(2) If M is a connected n-manifold, then

=

H;(M)

R O

ir M is orientable ir M is non-orientable

e Hn(M'-{~

if M is compact if~isnotcompact.

\/Ve also know that Hn-' ( R n- ( 0 ) )# 0, but we have not listed tllis result, since we will eventually improve it. In order to proceed furiher with our computations we nced to examine the behavior of the de Rham cohomology vector spaces under Cm maps f ; M + N . If o is a closed k-form on N , then f * w is also closed ( d f *o= f * d o = O), so f * takes Z k ( N ) to Z k ( M ) . O n the other hand, f * also takes B k ( N ) to B k ( M ) , since f * ( d q ) = d( f *q). This shows that f * induces a map ( N ) / B ~ ( N+ ) z~(M)/B~(M),

zk

also deno ted by f * :

f*:H ~ ( N+ ) H~(M). For example, consider the case k = 0. Ii N is connected, then H O ( N )is just the coIIection of constant functions c : N + R. Then f " ( c ) = c o f is also a constani funciion. If M is connected, then f * : H O ( N )+ H ' ( M ) is just the identity map under tlie natural identificatioii of H O ( N )and H O ( M )with R . If M is disconnected, with components M,, a E A , then H O ( M )is isomorphic to the direct sum R where each W , % R ; U EA

ilie map f * takes c E W into the element of @ R . with a" component equal to c . V N is aIso disconilected, with components Ng, E B , then

takcs the elemeiit {cg) of @gEB Rg to

(41, where c; = cg when / ( M u ) c Ng.

A inore interesting case, and the only one we are pi-esently in a position to Iook at, is the map f * : H n ( N ) + Hn(M) when M and N are both compact connected oriented n-manifoIds. There is no natural way io make H n ( M ) isomorpl-iic to IR,so we reaIly want to compare

for o an 11-form on N. Clioose oiie wo wi th JN number a sucll tliai r F

o0

#

O. Then there is some

Siiice o H lM o is an isomorphism of H n ( M ) and R (and similarly for N) it folIows tliai for evay form o we Iiave

TIie ilumber a = deg f, w1iicIi depends onIy on f, is called tIle degree of J: If M and N are not compact, bu t f is propei- (the inverse image of aily compact set is compact), tlien we have a map f * : H,"(N) + H,"(M) and a number deg f, sucli that

h r al1 forms o on N with compact support. Until one sees the proof of the nexi iheorem, it is almost unbelievabIe that tlis number is always an inlegm.

12. THEOREM. Let f : M + N be a proper map between two connected oriented n-maiiifolds ( M , P ) and (N, u). Let q E N be a regular value of f. For each p E f -' (q), let signp f =

(

1

if f*p : Mp + N4 is orientation preserving (using the orientations pP for Mp and v4 for N4)

-1

if

is orientation reversing.

TIien degf=

1

sipp/

(=0iff-'(~)=0).

~€f-'(4>

PROOF. Notice first that regular values exist, by Sard's Theorem. Moreover, f -'(q) is finite, since it is compact and consists of isolated points, so the sum above is a finite sum. Let f -' (q) = ( p l , . . . , pk}. Choose coordiiiate systems (Ui, xi)' around pi such tliat all points in U j are regular values of S, and the Ui are disjoint. \Ve want to clioose a coordinate system ( V, y ) around q such that f -' ( V) = U1 U . - U Uk. To do this, first choose a compact neighborliood W or q, and let

W'

c M be the compact ser

w'=

f-'(w)-

(u1 u . ' . u U k ) .

TIien f ( W') is a dosed set which does not contain q. We can therefore choose V C W - f ( W'). T h i s eiisures tliat f -' (V) C U1U . -UUk. Finally, redefine Ui to be Ui n f -'(VI. Now clioose w on N to be w = g dyi A A (13'" wliere g 2 O has compact support contained in V. TIieil support f *w c U1 u . - u Uk. SO

Sinc,e f is a diffeomorpliism li-om each Ui to V we have

S, f*w=S, w

=-S, w

ir f is orieiitation preserving if f is orientation reversiiig.

Siiice f is orientatioii pi-esenriiig [or reversing] precisely wlien s i p , f = 1 lor -11 tllis proves the theorem. *t,

As an immediate application of tIie iheorem, we compute the degree of tlie "antipodal niap" A : S" + S" defined by A ( p ) = - p . ' We have already seeii that A is orientation presenring or reversing at aU points, depending on wliether rl is odd or even. Siiice A-' ( p ) consists orjust one point, we condude tha t deg A = ( - ] ) " - l . We can draw an interesting concIusion from this result, but we need to intsoduce ailotller imporiaiii concept first. Two fuiictions f,g : M + N between two Cm manifoIds are called (smoothly)homotopic ir there is a smooth function

tlle map H is calIed a (smooth)homotopy between f and g. Notice that M is smootlily contraciible to a poiiit po E M ifand only if the identity map of M is liomotopic to tlie coiistaiit map po. Recall tliat Tor every k-form o on M x [O, 11 we defiiied a (k - 1)-form ío on M such thai ir*# - io*o -- d ( 1 o ) + 1 ( i l o ) . M7e used tliis fact to sllow tliat al1 cIosed forms oii a smoot1~1ycontraciible maiíifold are exact. We can iiow prove a more general result. 1 3. THEOREM. If J; g : M + N are smoothly homotopic, then the maps

f*:H ' ( N ) + H ' ( M ) g*: H ' ( N ) + H ' ( M ) are equal, f* = g*.

PROOF. By assumprioii, ihere is a smootIi map H : M x [O,11 + N with f =:HoiO g =H oii. Aiiy element of H' ( N ) is tlie equivaleiice class [o] o i some closed k-form o on N. Tlien g*w - f * o = ( H o i i ) * o- ( H O i O ) * o =: i l * ( H * w )- io*(H*w)

d ( 1 H *o)+ í ( d H * o ) = d ( í H * w )+ 0. =:

But tliis means that g * ( [ o ] )= f * ( [ o ] ) .+a:

14. COROLLARY. Ir M and N are compact oriented n-manirolds and t11e maps f,g : M + N are homotopic, then deg f = deg g.

15. COROLLARY. T i vector field on S".

11

is even, then there does not exist a nowhere zero

PROOF \/Ve have aIready seen that the degree or the antipodal map A : S" + S" is ( - ] ) " - l . Since tlie identity map has degree 1 , A is iiot homotopic to the identity ror I I even. But if there is a nowhere zero vector field on S", then we can construct a I-iomotopy between A and the identity map as roIIows. For each p, there is a unique great semi-circle yp irom p to A ( p ) = - p whose iangeni vector at p is a muItipIe of X ( p ) . Define

For n odd we can expIicitly construct a nowhere zero vecior fieId on S". For p = (xi ,. . . ,xn+i ) E S" we define

this is perpendicular to p = ( x i,x2, . . . ,x , + ~ ) and , therefoore in S",. (On S' this gives the standard picture.) The vector field on S" can then be used to give

a Iiomotopy between A and the identity map. For another applicaiion or Theorem 13, consider the retraction r : R"

Ir i : S"-' + R"

- ( O ) + S"-'

Y(P)

=P/IPI.

- ( O ) is tl-ie inclusion, tlien

r o i : S"-' + S"-' is the identity 1 or S"-'.

Tlle map

is, of course, not the identity, but i t is I-iomotopic to the identity; we can define

ihe llomotopy H by

A 1-etraction with tllis property is called a deformation retraction. Whenever 1. is a deformation i-etractioil, the inaps (r oi)* and (i o).)* are the identity. Thus, for tlle case of S"-' c R" - ( O ) , we have

and

r.* o i* = ( i o r)* = identity of H k ( R " - {O)) i*

So i* aiid

i."

o

r.* = ( I . o i)* = identity of H k ( s " - ' ) .

ai-e inverses of eacll other. Thus

Hk(S"-')

Iii particular, we Iiave H"-'(R" tlle closed form r*aJ.

' ; .

H k ( R " - ( O ) ) for al1 k. R . A generator of H"-' (R" - { O ) ) is

-{O))

We are now going to compute H k (R" observa tion. The manifold

M x (O)

- ( O ) ) for al1 k . We need one further

cM

x R'

is clearly a deformation retraction of M x R'. So H k ( M ) for al1 I .

H k ( M x IR')

Now ~ ' IA 'IB) - 0

onAnB

and

H' (A

H'(R3 - (O)) = O.

n 3 ) = H'([R3 - (O)] x R)

So ~ J A- ~ J B= d h for some O-form h on A n B. Unlike the previous case, we cannot simply consider VA - d h , since this is not defined on A . To circumveni this difficulty, note that there is a partition of unity ($A,$B) for the cover (A, 3 ) of R3 - (0): $A+$B

+

=1

d $ ~d $ = ~O support $A c A s u p p o r t $ ~C B. Now, if denotes

$ ~ honAnB O onA-(AnB),

and similarly for $AA, then $ ~ h is a Cm form on A $AA i s a C W f o r m o n 3 .

dh - d $ A~ = 'IA ($A - 1) d h d$A = 'IA - dh d($Ah) = 'IB d($Ah).

VA - d ( $ ~ h )=: VA - $B

+

+

+

So we can define a Cm form on R" - (O) = A on A, and q~ + d ( $ ~ h )on 3. Clearly,

so w is exact. Tlie general inductive step is similar. Q

+

U 3 by

A

letting it be

VA - d ( $ ~ h )

\/Ve end this chapter with one more calculation, which we wiU need in Chapier 11.

17. THEOREM. For O

l . Hinl:Show that a smooth f : S1+ S" is not onto. (d) If M is simply-connected and p E M , then any smooth map f : S1+ M is smoothly contractible to p. (e) If M = U U Ir where U and V are simply-connected open subsets with U n V conilected, then M is simply-connec ted. pliis gives ano ther proof that S" is simply-connected for n > 1.) Hint:Given f :S' + M , partition S' into a fmite number of intervals each of which is taken into either U or V. (f) If M is simply-connected, ihen H 1( M ) = O. (See Problem 7.) 9. (a) Let U c R2 be a bounded open set such that W' - U is not connected. SIlow ihat U is not smoothly contractible to a point. (Converse of

'

Prooblem 7-24.) Hinl:If p is in a bounded component of R2 - U , show that tliere is a curve in U which "surrounds" p. @) A bounded connected open set U c IR2 is smoothly contractible to a point if and only if i t is simply-connected. (c) This is false for open subsets of R 3 .

1 O. Let o be an n-form on an oriented manifold M". Let @ and q be two partitions of uiiity by functions with compac t support, and suppose thai

(a) This implies tliai (b) Show that

JM

9

w converges absolutely.

xJMmw=

E t:JMqmw.

#e@

#E@

*e*

and show the same result with o replaced by lo 1. (Note that for each are only finitely many xdiicl-i are non-zero on support 9.) (c) Sliow iliat Ceo*JM @ lo1 < m, and that

9, there

\Ve define tliis common sum to be j, u. (d) Let A , c (n,iz + 1) be closed sets. Let f : R + R be a CbQfunction witIi iAII f = (-])"/ti and support f c U, A,. Find two partitions of unity @ and q sucli tliat &E, 9 f d x and ,E&' i,il. f dx converge absolutely to different values.

IR

IR

1 1. Followiiig Problem 7-1 2, define geometric objects corresponding to odd 1-elab'veteiisors of type (f) and weight w (w any real number). 12. (a) Let M be ((x, E R2 : I(x, y) 1 < I ), together witli a proper portion JJ)

of its boundary, aiid let w = A- dy. Show that

elen ihough bot1i sides inake sense, using PI-obleni 10. (No computatioiis iieeded-iioie tliai equaIity would hold if we had the entire boundary.) (11) Similarly, fiiid a countei-example to Stokes' Theorem when M = (0, 1) aiid w is a O-form whose support is not compact. (c) Exaniiiie a partition of uiiiiy for (O,?)by fuiiciions with compact support io see just wliy ilie proof of Stokes' Tlieorem breaks down iii this case.

13. Suppose M is a compact orientable 77-manifold (with no boundary), and 19 is an (n - 1)-form on M. Show that de is O at some point. 14. Let M I , M2 C R" be compact n-dimensional manifolds-with-boundary with M2 c Mi - aMl. Sl-iow that for any closed (n - 1)-form w on Mi,

15. Account foi- the factor I/lp 1" in Lemma 7 (we have r,(vp) = ( I / ~ p l ) ~ , ( ~ ) , but this only accounts for a factor of l/lpln-', since there are n - 1 vectors vi , ,vn-1). 16. Use the formula for r*dxi (Problem 4-1) io compute **ot. (Note that

tl-ie map i o r : R"

- (0)+ R" - (O) is just

r, considered as a map into R" - {O}.)

1 7. (a) Let M" and N m be oriented manifolds, and let w and q be an n-form and an m-form with compact support, on M and N, respectively We will orient M x N by agreeing that vi, . . . ,u,, wr, . . ., w, is positively oriented in (M x Mp $ Ng if V I ,. .. ,Un and wi , . . . , Wm are positively oriented in M, and N,, respectively. If ni: M x N + M or N is projeciion on the ith factor, show that

where g ( p ) = JN

-)V.

~ I ( P-,1 = q

h ( p Y01.

(c) Every (m + n)-form on M x N is h n1* m A n2*TJ for some w and q .

18. (a) Let p ER"-(O}. Let wi,...,wn-2 some h E R. Show that

E

IR? andletu

E

IR> be (hp), for

(b) Let M C IR" - (O) be a compact (n - 1)-manifold-with-boundary which is ilie uiiion of segrnents of rays through O. Show that jM r*o' = 0.

(c) Let M c IRn-(O] be a compact (n- 1)-manifold-with-boundary whicl-i in tersects every ray througli O at mosi once, and let C(M) = (hp : p E M, h 2 O}.

Sliow tl-iai /M

r*o' =

/

r*ol.

C(M)flSz

Tl-ie latiel. integral is il-ic nieasure of tlie solicl arigle subtended by M. For ibis reason we often denote r*ol by d e n .

19. Fol. all {-Y,y,-) E lR3 except tliose wiih A* = O, y = O, r E ( - 0 0 , O], we deline $(x, y,z ) to be ilie angle be tween il-ie posiiive z-axis and the ray from O through (x, y, r ) .

+

(a) # ( x , y , z ) = arctan(Jx2 y 2 / z ) (with appropriate conventions). (b) If v ( p ) = IpI, and 8 is considered as a function on W 3 , @ ( Sy, , z ) = arctan y / x , then (u, 8 , #) is a coordinate system on the set of all points ( x ,y , z ) in IR3 except those with y = O, x E [O, cm) or with x O, y = 0, z E ( - c m , O ] . (c) If u is a longitudinal unit tangent vector on the sphere S 2 ( r ) of radius r, then d#(u) = l . If w points along a meridian through p = ( x ,y , z ) E SZ( r ) ,

-

t hen

(d) If 8 and 4 are taken to mean the restrictions of 8 and # to [certain portions ofl S', then o' = h d8 A d 4 , where h : S2+ IR is

h ( x , y , z )=

-m

(tiie minus si91 comes fmrn the orientation).

(e) Conclude that

a' = d ( - cos # d e ) .

(f) Let r2: R2- (O) + S' be the retraction, so that d9 = rz*i*o,for the form o on It2. Sliow that r2*d@= de. If n : R3 + R2 is the projection, then the form d9 on [par t of] IR3 is just n*d@, foi- tlie form d9 on [part ofJ R2. Use this to show that

(g) Also prove tliis directly by using the result in part (c), and the fact thai

r,(vp) = v,/lpl for v tangent to S2(1pl). (h) Conclude that

(i) Siniilarly, express d 0 , on R"

- (O) in terms of

do,,,

on R"-'

- (O).

20. Prove iliai a connected manifold is the union UI U U2 U U3 U . , where ille Ui are coordinate neighborhoods, with Ui í l Uj # 0, and the sequence is eventually outside of any compact set.

2 1. Let f ; M" + N" be a proper map beiween oriented n-manifolds such that f, : Mp + NI(,) is orientation preseniing whenever p is a regular point. Sllow that if N is connected, then either f is onto N, or else al1 points are critica1 points of f. 22. (a) Sliow tliat a polynomial map f : C + C, giveii by f(z) = z" +ai 2"-' + - . +a,, is proper (n 2 1). (b) Let fl(z) = 71ín-I (n - I ) a ~ z " - ~ .+an-1. Show that we have f' ( E ) = lim [f ( c + w) - f (i>]/w, wliere w varies over complex numbers.

+

+

w+o

(c) M7riteJ'(x+iy) = u(x, y )+iv(x, y ) for real-valued func tions u and v. Sllow that

av -(x, ay

au

y) - i-(x, ay

Hinl: Choose w to be a real h , and then to be ih. (d) Conclude that If '(x i y ) l2 = det Df (x, y),

+

y).

where f ' is defined in part (b), whde Df is the linear transformation defined for any dflerentiable f : R2 + R2. {e) Using Problem 2 1, give ano ther proof of the Fundamental Theorem of Alge bra. (f) There is a still simpler argument, not using Problem 2 1 (which relies on many theorems of this cliapter). Show directly that if f : M + N is proper, then the number of points in f (a) is a locally constant function on the set of regular values of f. Sliow thai tliis set is connected for a polynomid f:C + C, and conclude that f takes on all vdues.

-'

23. Let M'-' C R" be a compact oriented manifold. For p E Rn - M, choosi an ( u - 1)-sphere E aiaund p such that aU points inside C are in R" - M. Let rp : R" - ( p ) + E be the obvious retraction. Define the winding number w (p) of M around p to be the degree of rp1 M.

(a) Sllow tllat this defii~itionagrees with that in Problem 3. @) Show tllat tllis definition does not depend on the choice of C. (c) Sllow that w is constant in a neighborhood of p. Conclude that w is constant on each component of IRn - M. (d) Suppose M coiitains a portion A of an (ti - 1)-plane. Let p and q be points

close to tl-iis plane, but on opposite sides. Show that w(q) = w(p) z t l . (Show tliat rq 1 M is homotopic to a map which equals rp1 M on M - A and which does not take any point of A onto the point x in the figure.) (e) Show tliat, in general, if M is orientable, then W n - M has at least 2 components. Tlie next few Problems show Iiow to provc tlie same result even if M is not orientable. More precise conclusions are drawn in Chapter 11. 24. Let M and N be compact n-manifolds, and let f, g : M + N be smoothly honiotopic, by a smootli liomotopy H : M x [O, 11 + N.

(a) Let q E N be a regular value of H. Let #/-'{y) of points in f (q). Show tliat

-'

#f

-'(q) = #g-i (q)

denote the (fiiiite) numbei.

(mod 2)

HUZI:H-' (y) is a compac t l -manifold-\vi tli-bouiidary The iiumber of poin h iii its boundary is clearly even. (Tliis is one place where we use tlie strongeiform of Sard's Theorem.) (b) Show, more geiicrally, tliat tliis result holds so long as y is a regular value of I~otlif and g. 25. For two maps f, g : M + N we will write f smoothly Iiomotopic to g. (a) If

f

-

g to iiidicate that f is

g , theii thei-e is a smooth homotopy H': M x [O, 11 + N sucli tliat ' { p , ) = f . ) for I in a neighborliood of O, H' { j ~ 1,) = g ( p ) for iri a iieigliborhood of l .

01) -. is an equivalence i-elatioli. 26. If f is smootlily Iioniotopic to g by a smooth homotopy H sucl-i ihat p w H { P , I ) is a diffeomorpIiism for eacli I, we say that f is smoothly isotopic to g.

(a) Reing smootlily isotopic is an equivalence relation. (b) Let 4 : IW" + IW he a Cm function wliich is positive on tlle interior of the uiiit hall, aiid O elscw1iei.e. I'or p E S"-', let H: W x Rn + W" satisfy

(Lach solution is defined for al1 r, by Tlieor-em 5-6.) Show that each s w H{t, s) is a diffcoinoi-pliism, wliich is smootlily isotopic to the identity, and leaves al1

points outside the unit hall fixed.

171iegm[ion

2 95

(c) Show tl-iat by choosing suitable p and t we can make H(t, O) be any point in the interior of the unit bdl. (d) If M is connected and p , q E M, then there is a diffeomor~hismf : M + M such that f (p) = q and f is smoothly isotopic to the identity. f Skp 3 of Theorem 9. (e) Use part (d) to give an alternate ~ r o o of (f) If M and N are compact n-manifolds, and f : M + N, then for regular values ql, qz E N we have

(where #fml(q) is defined in Problem 24). This number is cdled the mod 2 degree of f. (g) By replacing "degree" with "mod 2 degree" iii Problem 23, show that if M c IR" is a compac t {n ])-manifold, then R" - M has at least 2 components.

-

27. Let ( X ' ) be a CbDfamily of CbDvector fields on a compact manifold M. (To be moi-e precise, suppose X is a Cm vector field on M x [O, 11; then X f ( p ) will denote T ~ * X ( ~ , From ~ ) . ) the addendum to Chapter 5, and the argument wl-iich was used in the ~ r o o of f Theorem 5-6, it follows that there is a CbDfamily (4, ) of diffeomorphisms of M [not necessarily a l-parameter group] ,with & = identity, which is generated by (X'), i.e., for any CbDfunction f : M + IR we

{X'f )(p) = lim

f( 4 i + h ( ~ )-) f (4' (P)) ii

h+O

For a family o, of k-forms on M we define the k-form ¿I,-- lim

Ut+h

h+O

- a' 11

(a) Show that for q{t) = 4,*ofwe haw

(b) Let o o and o1 be riowhere zero 11-forms o11 a compact oriented n-manifold M, and define o' = (1 - t)wo + /o1.

Sliow that the family

4, of di&omorpliisn~s generated by (X') satisfies ~ I * W= wo

if and only if

for al1 t

(c) Using Problem 7-18, show that this holds if and only if

(d) Suppose that JM 00 = JM 0 1 , there is a dfleomorphjsm J : M

that wo - o1 = d l for some l . Show that t M such that wo = fi*ol. SO

28. Let f : M' + W n aiid g : N' + R" be Cm maps, where M and N are compact oriented manifolds, n = k + I + 1, and f ( M ) n g ( N ) = 0. Define

a l r g :M x N + S"-' C W" - (O)

We define the linkingnumber of f and g to be

wliere M x N is oriented as in Problem 18. (a) [ ( f ,g ) = (-l)k'+ie(g, f ). @) Let H : M x [O, J ] t W" and K : N x [O, 11 + W" be smootli homotopies with

such that

( H ( P , I ): p

E

M } n { K ( q , r ): q E N ) = 0 for e v e r y ~ .

Show that

[ ( f , g ) = [tf 2). (c) For f , g : S' + W 3 show that

where

A ( u , u ) = det

( f l )'(u)

( f 2>'(u> ( f3 ) ' ( ~ > ( g l)'(u) (g2)'(v) (g3)'(v> s l ( v > - f l ( ~ )g 2 ( v ) - f 2 @ ) g 3 ( v ) - f 3 ( u )

(tlie factor 1/ 4 x comes from the fact that ls,o' = 4 n [Problem 9-14]). (d) Show that L( f , g ) = O if f and g both lie in the same plane (first do it for (x,y)-plane). Tlie next problem shows how to determine L ( f ? g ) without calculating. 29. (a) For ( a , b , c ) E W 3 define

For a compact oriented 2-manifold-with-boundary M le t

c

and ( a ,b , c )

M,

E

a ( a ,b , C ) = JM

dQ(a,b,c).

Let ( a ,b , c) aiid (a', b', c') be points cIose to p E M , on opposite sides of M. Suppose ( a ,b,c) is on tlie same side as a vector wp E W 3 p - M, for which the

triple w,, (vi ),, ( ~ 2 is) positively ~ oriented in R~~ when (vi,,) (u2), is positivel y oriented in Mp. S how that

fi ( a , b, c ) - S2 (a', b', c') = -471 lim ta,b,c)+ p (a',b7,c')+ Hint: First sliow that if M = a N , then L?( a ,b , c ) = -471 for ( a ,b , c ) E N and S2 ( a ,b , c ) = O for ( a ,b , c ) N .

-M

(b) Let f :S' + W' be an imbedding such that f ( S 1 )= aM for some compact oriented 2-manifold-~4th-boundary M . (An M with this property always exists. See Fort, Zpologv qf 3-A..lnn23[dr, pg. 138.) Let g : S' + lT3' and suppose The figure on the left shows a non-orientable surface whose boundary is the "trefoil" knot,

l ~ i i tthe surface on the right-including

the hemisphere behind the plane of the paperi r orientable.

that wben g(t ) = p E M we llave dg/dr $ M,. Let u+ be tlie number of intersecrions wl-iere dg/dt poiiits iil the same direction as tlie vector wp of part (a), and the number of other intersections. Show that

(c) Show that

S1iow tliar

17

=

for tlle pairs shown below.

30. (a) Let p , q E R" be distinct. Choose open sets A, 3 c 8" - ( p , y) so tl-iat A and 3 are diffeomorphic to Rn - (O), and A fi 3 is diffeomorphic to IR". Using an argument similar to that in the proof of Theorem 16, show that

H'{R" - ( p , q)) = O for O < k < 11 - 1, and that H"-' {R" - (p, q)) has dimension 2. @) Find the de Rham col-iomology vector spaces of R" - F where F C Rn ís a finite set.

31. We define the cupproduct u : H k ( M ) x H'{M) + Hk+'(M) by

(a) Show t11at u is well-defined, i.e., w A q is exact if w is exact and r~ is closed. (b) Sllow that u is bilinear. (c) If ci E H ~ { M )a n d p E H'{M), then c i u p = (-1) k l p u a . (d) If f : M + N, and ci E H k { N ) , E H'{N), then

(e) Tlie cross-product x : H' { M ) x H' { N ) + Hk+'{ M x N ) is defined by

Sliow tllat x is weI1-defined, aild tllat

(f) If A : M + M x M is the "diagonal map", given by A ( p ) = ( p , p ) , show that L Y V=~At(a x p). 32, O n the n-dimensional torus

n times

let d e i denote ni*d9, where n i : T" + S' is projection on the ithfactor. A de'" represent different elements of H k ( T " ) ,by (a) Sliow that al1 dei' A finding submanifolds of T" over which they have different integrals. Hence dim H k ( T " )2 Equality is proved in the Problems for Chapter 1 l . (b) Show that every map f : S" + T" has degree O. Hint: Use Problem 25.

6).

CHAPTER 9

+

RIEMANNIAN METRICS

1

n previous chapters we have exploited nearly every construction associated with vector spaces, and thus with bundles, but tliere has been one notable exception-\ve have never mentioned inner products. The time has now come to make use of this neglected tooI. An inner product on a vector space V over a field F is a bilinear function from V x V to F, denoted by (v, w) i-, (v, w), which is symmetric,

arid non-degenerate: if v # 0, theil there is some w # O such that

For us, the field F will always be R. For each r with O 5 r 5 17, we can define an inner product ( , ), on lT3" by

this is non-degenerate because if a # 0, then n

((a

1

,. . .,a n ) ,(a 1 , . . . , a r , -a r+ 1,

. ,-an)Ir

- C ( a i 1 2 > O. i=l

In particular, foi- i. = II we obtain the "usud inner product", ( , ) on R", (a, b) =

E aibi.

For this inrler product we have (a, a ) > O for aily a # O. In generd, a symmetric bilinear function ( , ) is called positive definite if

A positive definite bilinear function ( , ) is clearly i~on-degenerate,and consequently an iniier product. 30 1

Notice that an inner product ( , ) on V is an element of T ~ ( v ) ,so if f : W + V is a lii~eartransformation, then f * ( , ) is a symmetric bilinear function on W. TIiis symmetric bilinear function may be degenerate even if f is one-one: e.g., if ( , ) is defined on R2 by (a, b) = a ' b1 - a 2 b.2 : aiid f: IR + IR2 is

f (a>= (a, a > However, f * ( , ) is clearlg iiondegeiierate if f is an isomoi.pl~ismozzlo V. Also, if ( , ) is positive definite, then f * ( , ) is positive definite if and only if f is or~e-one. For aily basis V I , . . , v, of V, witli corresponding dual basis U*], . . . ,U*,,,we can write

In this expressioii. so symmetry of ( , ) implies tl-iat tl-ie matrix {gij) is symmetric, Tlie matrix {gij) lias anorhcr important iilierpretation. Since an inner producr ( , ) is lii-iearin rlie seco~ldargument, wc can define a liiiear fuilcrional$, E V*, for eacli v E V, by $v{w) = (u, w). Sirlce ( , ) is linear in tl-ie fii-st argument, tlie inap v I-+ 4, is a liiiear transfor~nationfi-om V io V*. No~-i-degeiiei.acyof ( , ) implies that $, # O if U # 0. Tlius: if V is fii-iiiedimensional, an i1111er111-oduct ( , ) gives irs al] isomor-phism CY : V + V*, witl-i (v,w) = Q(v)(w). Clearly, rl-ie matrix (gij) is jusr tbe matrix of CY : V + V* witl-i 1-espect to tl-ie I~ases{vi} [or 1' ai-id { v *} ~for V *. Thus, 11o1-i-degeileracyof { , ) is cquiualei-ii to the cor-idition thar Positive d ~ f i r - i i t ~ ~of - i ~(s s, ) correspo~~ds to rhe more complicated coi-idition thar the n-iatrix (gij) be "positive definitc", meaning that II

''>O

gija a i= 1

for al1 a l , . . . , a n witl-i at least one a' # O.

Given aiiy posiliue deJilzib iiiner product ( , ) oii 1/ we define the associated norm ll Il by (the positive square root is to be taken). llvll = JT;;;)

In IR" we denote the norm corresponding to ( , ) simply by

Tlie principal pi-operties of

11 11 are tlie followiilg.

l . THEOREh4. For al1 u, w

E

V we have

(1) ll0vll = 101 + llull. (2) 1 ( u ,w)l 5 11 v 11 11 w 11, with equality if aild only if v and w are linearly dependent (Schwarz inequality). (3) Ilv wll 5 llvll 11 w 11 (Triangle iiiequality).

+

+

PRQQE (1) is trivial. (2) If v aiid w are linearly dependent, equality clearly holds. If ilot, then O # hv - w for al1 h E R, so O < l l h ~ - w 1 1 ~ =( A v - w , h v - w ) = h211v112- 2 h ( v , w )

+ llw112.

So the right side is a quadratic equation in h with no real solution, and its discrimiiiaiit must be iiegative. Tlius 4 ( v , w ) -~411vl1211wl12 o.

Tlie fuiic.tioi.i 1 11 has certaii-i uiipleasai-it properqies-for example, the functioii 1 1 017 IR" is ilot dflereiitiable at O E IRn-wliich do riot arise for the function 11 112. This lattei fui-ictioii is a "quadratic function" on V-in ter-ms of a basis ( v i ) for V it can be wi-itten as a "homogeiieous polynomial of degree 2" in the

More succii-ictly, n

An invariai-it definition of a quadratic furlction can be obtained (Problem 1) from the following obsentatioi-i.

2. THEOREhil (POLARIZATION IDENTITY). If 11

11 is the norm associ-

ated to an inner product ( , ) on V, then

PXOOF Compute. Theorem 2 shows that t\vo iilnei- products which induce the same norm ar-c theiiiselves equal. Similarly, if f : V += V is 1101-m131-esen~ing, that is, 11 f ( v ) 11 = llvll fol. al1 v E 1/, then f is also iniler product presenfing, that is, (f (u), f (w)) = (v, w) for al1 v, w E V. \Ve wil1 now see that, "up to isomorphism", diere is only one positive definite ii-iner product-

3. THEOREhil. If ( , ) is a positive defiiiite iiiiier product on ai-i n-diinei-isional vector space V, then there is a basis vi,. . . ,v, for V such that (vi, vi) = 6,. (Such a basis is called orthonormal with respect to ( , ).) Consequentl!; there is an ison~oi-pliismf : Rti + V such that

In other words,

f*(

7

>={

9

>.

PRO0.F Let Wr,. . . , wtl 13e ai-iy basis for V. \Ve obtain the desired basis by applviiig tlie "Granl-Scl-imidt orthonormalization proc.ess" to this basis: Since wi # O, we can define

aiid clcady

Ilui

11 = l . Suppose that we have constructed u], . . .,u k so that (vi, v j > = 6,

I si,jsk

and

span V I , . . . ,vk = span W I ,. . ., wk.

T h e n wk+] is linearly independent of V I , .. . ,vk. Let

It is easy to see that

So we can define

and continue inductively. *3 A positive definite inner pi-oduct ( , ) on V is sometimes called a Euclidean 71zeh-i~011 V. Tilis is because we obtain a metric p on V by defining

T h e "triangle iilequality" (Theorem l(3)) shows that this is indeed a metric. 147e also caH llvll the length of v. \die have onfy one more algebraic trick to play. Recalf that an inner product ( , ) on V prcwides an isomorphism a : V 4 V* with

Using the natural isomorphism i : V + V4*,defined by

1dre can ilow use

/3 to define a bilinear functioii ( , )* o n V* by (A,p}*= /3(A)(p) = i a - l ( A ) ( ~ )= P ( ~ - - ~ ( A ) ) .

Now, tlie symmetry of ( , ) can be expressed by tbe equation

Letting #(u) = A,

u(w)

= p,

this can be written 1

( A )= Au-l

(A)),

which shows that ( , )* is also symmetric,

Consequently ( , )* is aii inner product on the dual space V* (in fact, the one which produces j3). To see what this al1 means, choose a basis (ui) for V, jet (u*i) be the dual basis for V*, and 1et n

Then (gij) is the matrix of

u : V + V*

with respect to (vi) and (u*¿)

V* + V

witli respect to (u*i) and ( q )

(gjj)-' is thc matrix of u-': so (gG)-' is the matrix of

B:

V* + V** witli respect to

and ( ~ * * i ) .

Tlius, if we let g i j be the entries of the inverse matrix, (g") = (gjj)-l, so tliat

theii

=

g " ~ i @ u j , ifwe considerui

E

V**.

Onc can clieck directly (Pi-oblem 9), without the invariant definition, tliat this equation defines ( , )* indepelidently of the choice of basis.

Notice that if (

, ) is positive definite, so that

then, letting a(v) = h, we have

so ( , )* is also positive definite. This can also be checked directly from the defiiiition in terms of a basis. In the positive defiilite case, the simplest way to describe ( , )* is as follows: TIle basis u*], . . . , v*, of V* is orthonormal with respect to ( , )* if aiid only if v i , . . . ,u, is orthoilormal ~ 4 t hrespect to ( , ). Similar tricks can be used (Problem 4) to produce an inner product on al1 tlie vector spaces T k ( v ) , T,(V) = T k ( v * ) , and a k ( v ) . Howwer, we are intcrested in oiily oile case, which we will i ~ odesci-ibe t in a completely invariant way. The vector space Qn(V) is 1 -dimensional, so to produce an inner product on it, we need only describe whidl two elements, o and -o, will have length 1. Let V I , . . ,v, aild W r , . . . ,w, be two bases of V which are orthonormal with respect to ( , ). If we write

So the transpose matrix A' of A = (ujj) satisfies A - A' = 1, which implies that det A = fl . It follows fi-om Theoi-ein 7-5 that for aily w E S2"(V) we have

It clearly follows that

We have thus distinguished two elemen ts of R" (V); they are both of the form v * A~ . - 'AV*, for (vi) an ortho11ormalbasis of V. \IZ1e will cal1 these two elements

the elements of norm 1 in Rn(V). If we also have an orientation p, then we can furtlier distinguish the one which is posiiive when applied to any (vi,. . ., U,) Hfith[u1,. . .,Un] = p; \ve will cal1 it the positive element of norm 1 in n n ( V ) . To express tlie elements of norm 1 in terms of an arbitrary basis wi,. . . ,w., we cl-ioose an ortbonormal basis vl, .. . , U, and write

Problem 7-9 implies that

then

det(gjj) = det(At A) = (det Al2. In particular, det(gij) U o l w ~posiiiu~. ) ~ ~ Consequently, the elements of norm 1 in R" (V) arc

c

\Ve noiv apply our new tool to \~ectorbuiidles. If = a : E + B is a vector l~uiidle,we define a Riemannian metric on [ to be a function ( ) whicli assigns to eacli p E 3 a positive definite iiiner product { , ), on a-'(p), and which is coiltinuous iil tbe cense that for aily two continuous sections si ,$2 : 3 + E, the functioil ( ~ 1 3 ~ 2=) p (SI(].'),S~(P))~ is also coiltinurius. If 6 is a Cm vector builclle over a Cm manifold we can also speak of Cm Riemannian metrics.

[Another approach to the definition can be given. Let Euc(V) be the set of al1 positive definite inner products on V. Ifwe replace each n-'(p) by EUG(X-] (p)), and Jet EucO) = ~uc(rr-'(p)),

u

PEB

then a Riemannian metric on can be defined to be a section of E M ( { ) .Tbe only problem is that Euc(V) is not a vector space; the new object E~ic(c)that wc obtaiii is not a vector bulidle at all, but an iiistance of a more general structure, a fibre buiidle.] 4. THEOREM. Let { = n : E + M be a [Cm] k-plane bundle over a Cm maiiifold M. Tlieil tliere is a [Coo] Riemailnian me tric on

c.

PROOE Tliere is aii opeii locally fiiiite cover O of M by sets U for which there exists [Cm] trivializations ru: =-'(U) + U x

IR^.

On U x IRk, tliere is aii olwious Riemannian metric, ((p,4,(p,b))p = ( ~ , b ) . For u, w

E

n-'(p), define

(u, w)pO = ( I U ( ~~u ) ?(w))p. Then ( , ) u is a [Cm] Riemannian metric for CIU. Let ( # u ) be a partition of uiiity subordinate to 0.We define ( , ) by (u,

=

E #u(p) (u, w)p

u, w

E

x-~(~).

UEO

Then ( , ) is coiitiiiuous [Cm] and eacli ( , )p is a symmetric bilinear function oii n-' ( p ) . To sliow that it is positive definite, iiote that

eacli #u (p)(u, u):

2 O, aiid fol. some U strict inequality liolds.

'+ :

[The same argument shows that aily vector bundle over a paracompact space lias a Riemannian metric.] Notice that the argunient in the filial step would not work if we had merely picked iioii-degeiierate iiiiier piaducts ( , ) 111 fact (Pi.oblem i),there is no ( , ) on TS' wliicli gives a symmetric I~ilinearfuiiction on each sZP which is iiot positive defini te 01. iiegative defiiiite bu t is still non-degenerate.

As an application of Theorem 4, we settle some questions which have ti11 now remained unanswered. 3.

COROLLARY. If 6 = x : E + M is a k-plane bundle, then {

PROOF. Let ( , ) be a Riemannian metric for

c*.

c. Then for each p E M, we

have an isomorphism U p : x-i

( p ) + [x-1 (p)]*

defined by ol,(v)(w) = ( u , W

p

v, w

E

x-l (p).

Continuity of ( , ) iinplies that the union of al1 ap is a homeomorphism from E lo E' = U,,,[x-' (p)]*. O

6. COROLLARY. If { = x : E + M is a 1-plane bundle, then { is trivial if aiid only if is orieiitable.

PROOF The "only if" part is trivial. If t has an 01-ientation p aiid ( Riemaiinian metric o11 M tlien there is a unique

,)

is a

with p

= 1,

[s(p)I = Pp

Clearly s is a section; \ve then define an equivalente f : E + M x R by

ALTEMATIVE PROOF \Ve know (see the discussion after Theorem 7-9) that if is orientable, theil there is a nowhere O sectioil of

c

so that

t* is trivial. But { t*.+3

All these coi~sidei.atioiistake on special significai-ice wl-ien our bundle is the taiigeiit bui~dleT M of a C* manifold M . Iii this case, a C* Riemanniaii inetric ( , ) for T M , which gives a positive definite inner product ( , ), oi-i

each Mp, is called a Riemannian metric on M. If (x, U) is a coordinate system on M, then on U we can write our Riemaniliail metric ( , ) as

wherc the Cm fuiictioils gij satisfy gij = gji, since ( , ) is symmetric, and det(gli) > O since ( , ) is positive definite. A Riemannian metric ( , ) on M is, of coui.se, a covariai-ittensor of order 2. So for every Coo map f :N + M there is a covai-iant tensor f*( , ) on N, which is c1early symmetric; it is a Riernai-iniai-imetric on N if and only if f is an immersion is one-one for a11 11 E N). The Riemaililian metric ( , )*, wl~ich( , ) induces on the dual bundle T4M, is a coi1travariant teiisor of order 2, and we can write it as

Our- discussion of innei- products irlduced on V* shows that for each p, the matrix (g"(p)) is the inverse of the matrix (gjj(p)); thus

Similarly, foi- each p E M the Riemanniail metric ( , ) on M determines two c1cments of S2"(Mp), the elemei-its of norm l . \Ve have seen that they caii 11e written I Jdet(glj(p)) d x l ( p ) A . - . A dxn(p). If M has ai1 or.ieiitation p , then pp aUows us to pick out the positive element of r1oi.m 1, and we ol~tainan i?-fosm on M; if x : U + R" is orzenfaiio~zpres~ving, tlien o11 U tliis form caii be written

Even if M is iiot oi.icntabIe, we obtaii-i a "voIun1e e1ement" on M, as defined in Chaptei- 8; in a coor-diiiate system (x, U) it can be written as

Tliis volume elen~eiitis deiioted by d V, eveil thougll it is usual1y not d of ai-iytliii-ig(even wl~enM is orientable and it can be considered to be an n-form),

and is calied tlie volume element determined by ( , ). We can then define the volume of M as F This certainly makes sense if M is compact, and in tlie non-compact case (see Problem 8-10) it eitl~erconverges to a defiilite iiumber, or becomes arbitrarily large over compact subsets of M , in which case we say that M has "infinite volume". If M is ail 11-dirneii.sioiialmanifold (-witli-boundary) in R", with tlie "usual Riemannian metric" n

then

gij

= dij,

SO

d V = Idxl

A

- - A dxnI,

and "volume" becomes ordinary volume. Tliere is an eveii more importaiit construction associated with a Riemannian metric on M , ~rhicliwill occupy us for the I-est of tlie chapter. For every Cm cunte y : [a,b] + M,we have tangent vectors

and can t1ierefoi.e use ( , ) to define tlieir length

14Jccaii then define the length of y from a to b,

If y is mesely pulc~wrsesttzooilr, n~eaiiiiigtha t there is a parti tion a = ro < . . t,, = b of [u, b] sucli tliat y is smootli on eacli [ ~ i - i ,li] (with possibly different

left- and right-hand derivatives at t i , . . . ,tn-]), we can define the length of y by

IYhenever tliere is no possibility of misuiiderstaiiding we will denote L,b simply by L. A little ai-pmeni shows (Problem 15) tliat for piecewise smooth curves in IRn, with the usual Riemannian metric dx' @ d x i 7

tliis defiiiition agrees ivith tlie definition of length as the least upper bound of tlle lengths of inscribed polygonal curves. We can also define a function S : [a,b] + IR, the "arclength function of y" by

Co~~sequeiitly d y / d r has constant length 1 precisely when S ( t ) = t thus precisely when s(1) = E - a. Then

+ constant,

We can reparameterize y to be a curve on [O, b - u] by defiiiing

p(r) = y(t - a ) . For the new curve

y

we have

iiew s ( l ) = L;(?) =

= old s(i f u ) - old s ( u )

= t.

If y satisfies S ( [ ) = r we say that y is parameterized by arcfength (and then often use S iiistead of r to denote the argument in the domain of y).

Classically, the norm 11 11 on M was denoted by d s . (This makes some sort of sense aren in modern notation; equation (s)says that for each cuwe y and corresponding s : [ a ,b] + IR we have

on [ a ,b].) Consequently, in classical books one usually sees the equation

Nowadays, this is sometimes interpreted as being the equivalent of the modern ' d x j , but what it always actually meant was equation ( , ) = x y , j = l g i j d ~ @

Tlie symbol dx'dx j appearing here is n o l a classical subai tute for dx' O d x j the value ( d ~ ' d s j ) (of~ d) x ' d x j at p should not be interpreted as a bilinear function at ali, but as the quadratic function H

d x i ( p ) ( u ) d x j ( P ) (u) *

v

E

Mp ,

and we would use the same symbof today. The classical way of indicating d x i @ d x j was very strange: one wrote

E gij dx'dxj

where d x and Sx are independent infinitesimals.

i,j=l

(Classically, the Riemannian metric was not a function on tangent vectors, bu1 tlie inner product of two "infinitely smaU dispiacements" d x and Sx.) Consider now a Riemannian metric ( , ) on a cotlneckd manifold M. If p, q E M are any two points, then there is at least one piecewise smootli cuwe y : [ a ,b] + M from p to q (tliere is even a smooth curve from p to q). Define

d ( p ,q ) = inf { L ( ~: y) a piecewise smooth curve from p to q ) . It is clear that d ( p , q ) > O and d ( p , p ) = 0 . Moreover, if r E M is a third point, then for any E > O, we can choose piecewise smooth curves

y, : [u,b] +- M from p to q with L ( y i ) - d ( p , q ) < E y2: [ b , ~+ ] M from q to r with

L ( y 2 )- d ( q , r ) < E .

If we define y : [a,c] + M to be y, on [a,b] and y2 on [b,c],then y is a piecewise smooth cuwe fmm p to r and

+

L ( Y ) = L(Y,) L(y2) < d ( p , q ) + 4 % 4 + 2 ~ . Since tliis is true for al1 E > O, it follows that d ( p ,r ) 5 d ( p , 9 ) + d ( q ,4. [ I f w did not allow piecewise smooth cuntes, there would be difficulties in fitting together y, and y2, but d ~rouldstill turn out to be the same (Problern 1 T).] The function d : M x M + R has aíl properties for a metric, except that it is not so clear that d ( p ,q ) > O foi- p # q. Tliis is made clear in the following 7. THEOREM. The function d : M x M + R is a metric on M , and if p : M x M + IR is the original metric on M (wliich makes M a manifold), then ( M ,d ) is homeomorphic to ( M ,p).

PROOF. Both parts of the theorem are obviously consequences of the folíowing 7'. LEMMA. Let U be aii opeii neighborhood of the closed ball 3 = { p E R" : jpl 2 11, let ( , ), be the "Euclidean" or usual Riemannian metric on U, n

aiid let ( , ) be any otlier Riemannian metric. Let 1 1 = 11 11, and 11 corresponding norms. Then there are numbers m, M > O such that

Il be the

aiid consequelitly for any cunte y : [a,b ] + 3 we have

??lLe(y)5 L ( Y ) PROOE Define G : B x S"-< R by

< ML,(Y).

G ( p , a ) = llap llp. Then G is contiiiuous and posiiive. Since 3 x S"-' is compact there are numbers m, M > O such that on B x S"-'. il7 < G < M Now if p

E

B and O # bp E Rnp, let a

E

S"-' be a = b/lbi. Then

1?7lbl < lblG(p,a) < Mlbl; siiice

l b j G ( p , ~= ) lb1 . l

l ~ ~ =l lIiObIlOpIIp ~

= IIbIIp,

tliis gives the desired inequality (wliich clearly also holds for b = 0). O

Notice that tlie distance d ( p , q ) defined by our metric need liot be L ( y ) for any piecewise smooth curve from p to q. For example, the manifold M might be IR2 - {O), and y might be - p . Of course, if d ( p , q ) = L ( y ) for come y,

then y is clearly a shortest piecewise smooth curve from p to q (there miglit be more than one shortest curve, e.g., the two semi-cirdes between the points p and - p on S ' ) . ln order to investigate the question of shortest curves more tl~oroughly,we liave LO employ tecliiiiques fi-om the "calculus of variatioils". As an introduction to such teclii~iques,we consider first a simple problem of this sort. Suppose we are giveli a (suitably differentiable) function

Wc seek, amoiig al1 functions J.: [ a ,b ] + IR with f( a ) = a' and f ( b )= b' oiie

whicli will maximize (or minimize) the quantity

For examplc, if-

then we are looking for a function f on [u,b] which makes the curve t ( E ,f ( 1 ) ) between (a, u') and (b, b') of shortest length

H

As a second example, if

then we are trying to minimize the area of tlie surface obtaiiied by revolviiig tlie graph of f around the A--axis,which is given (Problem 12) by

To appl-oach this sort of problem we recall first tlie metliods used for solving t1.ie n~uchsimpler problem of detei-ininiiig t l ~ einaximum or niinimum of a function f : R + R. To solve tliis problem, \ve examine tlie critical points of f, i.e., those points x for wliicli fl(x) = O. A critica1 poilit is not iiecessarily a maximum or minimum, or even a local maximum or minimum, but critical points arc the only caiididates for maxima or mii.iin.ia if f is everywliere difTereiitiable. Similarly, for a functioii f : IR2 + IR we consider poiiits (x, y) E R2 for which

This is tlie same as saying that the cuntes

have derivative O at O. \Ve might try to get more information by considering the condition

o = (f

0

c)'(O)

for every cuiw c : (-E,&) + R2 with ~ ( 0 = ) (x,y), but it turns out that these conditions follow from (*), because of the chain rule. To find maxima and minima for

we wish to proceed in an analogous way, by considering curves in Ihe sel ufall funclions f : [u,b] + R. This can be done by considering a "variation" of f, that is, a function

a : (-E,E) x [u,b] + R such that

@(O, 1) = f U). The functions 1 H a(u,I) are then a family of functions on ( - E , E) which pass through f for u = O. We will denote this function by &(u). Thus & is a function from (-E,E )to the set of functions f : [u,b] + R. If each ¿?(u) satisfies &(U) (u)= u', & (u)(b)= b', in other words if

for al1 u E (-E,&),then we cal1 a a variation of f keeping endpoints fixed For a variation a we now compute

As is usual in classical notation, the arguments of functions are either put in indiscriminately or left out indiscriminately-in this case, not only are the arguments t and (1, f ( r ) , f f ( t ) ) omitted (resulting in the disappearance of tht fuilction f for which we are solving), but the dependeiice of SJ on a is not indicated (which can make diings pretty confusing). If f is to maximize or minimize J , then S J ( a ) must be O for every variation cx of f keeping eiidpoints fixed. As in the case of 1-dimensional calculus, there is no reason to expect that the condition S J ( a ) = O for al1 tr will imply that f is eveii a local maximum or miilimuin fos J , and h7eempliasize this by introduciiig a definition. \Ve caB f a critica1 point of J (01-an extremal for J) if S J ( a ) = O for al1 variations a of f keeping endpoints fixed. The particular form (**)into which we have put SJ iiow allo~tsus to deduce an important condition. 8. THEOREM (EULER'S EQUATION). The C2 function f is a critica1 point of J if and only if f satisfies

JF - 7 ax

.

f'

(

dr

r

ay

,

, ( r ) ) =o.

)

PROOI;: Clearly f must niake t l ~ eintegral in (**) vanisli for e u q l

which vaiiishes at a and b. So the theorem is a consequence of the followiiig simple 8'. LERIMA. If a continuous function g : [u,b] + R satisfies

for every Cm fuiictioi~q on [a,b]with ?](a)= q(b) = 0, tben g = 0.

PROOE Choose q to be #g where # is positive 011( a ,b ) and # ( a ) = # ( b ) = 0. +3 As an example, consider tlie case where F ( r , s , g ) = cquatioil is

m.

Tlie Eules

hence 0 = (1

+ fr2)f"

- ftj-'/ = (1 - f t + f12) f",

wliich implies that f" = O, so f is linear. Notice that we ~ ' o u l dhave obtained the same result if we had considered t l ~ e case F(t,x, y ) = 1 Y2,for then the Euler equatioii is simply

+

This is analogous to the situation in 1-dimensional cafculus, where the critica] points of f i are the same as tliose of f, since

For tlie case of tlie surface of revolution, wliere F(l, x , Y ) Euler equation is

=

xJ1

+ Y',

[he

this Ieads to the equatioii

whic.11 we will a1so write in the cIassica1 form

To solve this, we use oiie of the Ko standard tricks (leaving justification of the details to t1ie reader). We let

Then

so our equation becomes

1 2

- log(l + p 2 ) = log y + constant y = constant

J;+pi

and thus (see Problem 20 for tlie definition and properties of tlie "liyperbolic cosine" function cosh and its inverse)

Replacing c by 1 / c , we write this as

The graph of cosli x =

ex

+ eYX 2

is sliown below; it is syn-imetric about the y-axis, decreasing for iricreasing for x 2 0.

A-

5 0, and

So our surface must look like the one drawn below. It is, by the way, not trivial to decide whether there a7-e constan ts k and c which will make the graph of (*) pass through (u, a') and (b, b'). Problem 2 l investigates the special case where a' = b'.

It is easy to generalize these considerations to the case where f : [a, b] and

t

Pn

In this case \ve consider a : ( - E , & ) x [a,b] t Pn with G(0) = f, and compute that

Tlius, any critica1 point f of J must satisfy the n equations

We are ~iowgoing to apply these results to the problem of finding shortest paths in a manifold M. If y : [u,b] t M is a piecewise smooth cunte, with y ( a ) = p and y ( b ) = q , we define a variation of y to be a function

for come E > O, such that

(1) ~ ( 0 7 = ~ ~) ( 0 , (2) there is a partition a = ro < E]
O on C. Since C is compact, there is E > O such that g > 2~ on C. Then f is one-one on the E-neigl-iborhoodof Xo. 43

+

20. THEOREM. Let M

cN

be a compact submanifold of N . Then M has a tubular neigliborhood T : U + M in N: which is equivalent to the normal bundle of M in N.

PROOF, Choose a Riemanniaii men-ic ( , ) for N , with tlie corresponding norm 11 11, and metric d : N x N + R . Lei

It follows easily from Theorem 13, and compactness of M, tl-iat exp is defined o11 E, for sufficieiitly small E > O. We claim that for sufliciently sm al1 E, the map exp is a diíTeomorpliism from E, onto U,. This will clearly prove the theorem. Let V c E be the set of a non-critica1 points for exp. Then V 3i M (considered as a subset of E via thc O-section), and VI = V n El is compact; since exp is one-one on M c I/i, it follows from Lemma 19 that for sufficiendy small E tlie map exp is a difTeomorphism on E,. It is clear also that exp(E,) c U,. To prove tliat exp is onto U,, choose any q E U,, and a poirit p E M closest to q . If y : [O, 11 + N is the geodesic of 1engtl-i < E witli y(0) = p and y(1) = q , it is easy to see that y is perpendicular to M at p (compare the second proof of Gauss' Lemma). This means that q = exppdy/dt(0) where dy/d1(0) E E,. 43 O11c of tlie ii-iiei-estingfeatures of Theorem 20 is that al1 the paraphernalia of Riemanriian metrics and geodesics are used in its proof, while they do not even appear in tlle statement. Theorem 20 will be rieeded only in Cl-iapter 11, wl-iere we will also need the following modification.

Rienzalzn ian Adel?.ics

347

2 1. THEOREh4. Let N be a manifold-with-boundary, with compact boundary aN. Theii aN has (arbitral-illlsmall) opeii [and dosed] neighboiqlioodsfor which there are deformation re tractjons onto a N . PRO0.F Exactly the same as the proof ofTlieorem 20, using only inward poiiiting normal vectors. +3

PROBLEMS 1. Let V be a \lector space over a field F of characteristic # 2, and let h : V x V + F be srmmetric and bilinear. (a) Define q : V + F by q(v) = h(v, u). Show that if e l , . . . ,#n is a basis for V*, then n

for SOme U i j . (b) Show that

(e) Suppose q : 1/ + F satisfies q(-u) = u, and that h(u, u) = q(u+ v) - q(u) q(v) is bilinear. Show that Conclude that q(0) = 0, and q(2u)

-

4q(u). Tl-ien show that q(v) = h(u, u).

2. Let ( , ) be a Euclidean metric for V*. Suppose #i, Sri E V* satisfy #i A - A #k = Sri A - A # O, and 1et W4 and W* be the subspaces of V* m

m

spanned by the

ei and Sri.

(a) Show tl-iat w E W4 if and only if w A #i A . - . A #k = O. Conclude that w4= w*. (b) Let u], .. . ,uk be aii or~lionoi-malbasis of WQ = W y. If #i = & U j i U j , s11ow that the signed k-dimensioiia1 volume of the parallelepiped spanned by ,. . . ,#k is det(aij). (Tlie sign is if #i,. . . , Iias the same orientation as 01, . . .,Ok, and - otherwise.) (c) Usirlg Problein 7-9, sliow that tliis volume is the same for Srl ,.. . ,@k. (d) Conversely, if W4= W*,and the si\gned volumes of the parallelepipeds are A . - .A Srk. the same, sliow that #] A - A #k =

el

+

If we ideiitify V witll V**,so tliat we llave a wedge product vi A - - - A vk of vectors vi E V, tlieil we liave a geometi-ic coiiditioii for equality with wl A . A wk. 111L p n s sur la GWniArie des Q l i c e s de Rieinliiiii, É. Cartan uses this condition to d@n~nk( Y * ) as formal sums of equivaleiice classes of k vectors; he deduces geornen-ically ihe coi~espondiiigconditions on the coordinates of vi, wi . 3. Let V be an it-dimensioi-ial vector space, and ( , ) an inner product on V wliich is iiot riecessarily positive defiiiite. A basis v i , .. . , u , for V is called orthonormal if (vi, vj) = f6,.

(a) If V # {O}, then there is a vector u E V with (u, u) # 0. (b) For W C V, let W' = (u E V : (v, W ) = O for al1 w E W}. Prove that dim W' 2 n - dim W. Hini: If {wi} is a basis for W, consider the linear functionals hí : V + R defined by h i ( ~ = ) (U,wi). (e) If ( , ) is non-degenerate on W, then V = W @ w', and ( , ) is also non-degenerate on w'. (d) V has an orthonormal basis. Thus, there is an isomorphism f : R" + V with f * ( , ) = ( , ), for some r (the inner product ( , ), is defined on page 301). (e) Tbe index of ( , ) is the largest dirnension of a subspace W c V such that ( , ) 1 W is negative definite. Show that the index is n - r, thus showing that r is unique ("Sylvester's Law of Inertia9'). 4. Let ( , ) be a (possibly non-positive definite) inner product on V, and let vi,. . . , u, be an orthonormal basis (see Problem 3). Define an inner product ( , l k on nk(v) by requiring that

be an orthonormal basis, wit1-i

(a) Sliow tliat ( , l k is independent of the basis vi,. . .,uk. (Use Problem 7-16,) (b) Show thai

(c) If ( , ) has index i , theii ( u * ~A

.

. A u*,,

u*] A

"

. A U*,)"

= (-lli.

(d) For tliose wlio k n w about @ and iIk. Usiiig the isomorphisms ( g k v ) *and i I k ( v * ) 4 ( i I k v ) * ,define inner products on and iIkv by using tlie isomorphism V +- V* given by the ii-iner product on V. Show that tl-iese ii-ii-ierproducts agree witl-i the oi-ies defined above.

5. Recall tlie definition of u1 x

x

u,-1

in Problem 7-26.

(a) Sliow tliat (VIx - - - x v,-1, vi) =: O. (b) S ~ O M tli¿~t ' Iu] X * - X U,,-1 1 = l/deto,wliere gij = (ui, uj). H&L: Apply tlie result oi-i page 308 to a certain (11 - 1)-dimensional subspace of R".

c

B be a vector bundle. An indefinite metric on 6 is a continuous choice of a non-positive definite inner product ( , ) p on each 71-'(p). Show that the index of ( , ) p is constant on each component of B.

6. Let

=: 71 : E +=

7. This problem requires a li ttle knowledge of simple-connec tedness and covering spaces. (a) There is no way of continuously c hoosing a 1-dimensional subspace of S%, for each p E s2.(Consider the space consisting of the two unit vectors in each subsp ace .) @) There is no Riemannian metric of index 1 on s 2 .


O for r near O. If c is c',then v ( t ) approaches a limit as t + O (even though v(0) is undefined). Show that if c is then there is some K > O such that for al1 r near O we have

c',

Hnt: In Mq we clearly have

Since expq is locally a difTeomorp11ismthere are O < K I < K2 such that

for al1 tangent vectors v at points near q. (c) Conclude that

L ( c I [O, I11) lirn = lim h-O d ( p , ~ ( h ) ) h-0

lh

Ju'(t)2

+ I$(U(t), u(h)

1)

1

2

di

-

1.

(d) If e is c', show that L(c) is the least upper bound of inscribed ~iecewise geodesic curves.

34. (a) Using the methods of Problem 33, show that if c is the straight line joining u, w E Mp, then L (exp o e) lim = 1. v,w+o

L,(c)

(b) Siinilarly, if yviWis the unique geodesic jojnjng exp(v) and exp(w), and Yv. w = exp o cu,,, then lim L ( Y ~w), = 1. v,w+o L(cuiW) (c) Conclude that d (exp u, exp w ) ljm = 1. v,w+O flu-wlf 35. Let f : M + N be an jsometry. Show that f is an isometry of the metric space structures determined on M and N by their respective Riemannian

36. Let M be a manifold with Riemannian metric ( , ) and corresponding metric d. Let f : M + M be a map of M onto itself which preserves the metric d. (a) If y is a geodesic, then f o y is a geodesic. (b) Define f t : Mp + MAp) as follows: For y a geodesic with y(0) = p, let

Show that ff f'(X) ff = ff X 11, and that f'(c X) = cf '(X). (c) Given X, Y E Mp, use Problem 34 to sliow that

- IIxI12+ IfU12 -

flxll - IlYlI

lim [ d ( e x p t X , e x p r ~ ) I 2 IlrX lf . 111 y 11

r-0

Conclude tliat (X, Y), = (f '(X), f '(Y))J(,), and then that f'(X + Y) = f '(X) + f'(Y1. (d) Par t (c) sho~vstfiat f ' : M, + Ml(,) is a diffeomorphism. Use this to show that f is itself a diíTeomorphism, and hence an isometiy. 37. (a) For u, w

E

Rn ulth w # O, show that lim IIv+lwII-IIvlI 1-0

1

--( v , ~ ) llvll

The same result tlien Iiolds in aiiy vector space with a Euclidean metric ( , ). Hin~:If u : Rn + R is the norm, tlien tlie limit iS Du(v)(w). Alternately, one can use the equatiori (u, u) = ijuli . iivii - cose wliere 8 is the arigle between u and v . (b) Coiiclude tliat if w js liriearly independent of u, then

lim

iIv+fwll - llvll - lltwll

&

(e) Let y : [O, 11 + M be a piecewise CI critical poirit for lerigth, aiid suppose that yl(to+) # yl(lo-) for some 10 E (O, 1). Clioose r l < 10 and consider the varjatioii cr for wliicli 6 ( 2 1 ) is obtairied by following y up to 11 , then tlie unique geodesic fi-om y(fI) to y(lo + u), and firially the rest of y. S110~7that if 11 is

1

close eiiough to 10 then d L (iu (a))/du I,=o # O, a contradic tiori. Tlius, critica1 paths for leng-th carinot h a ~ khks. e

38. Consider a cylinder Z

c IR3

of radius r . Find tlie metric d induced by tlie Riemannian metric it acquires as a subset of p. 39. Consider a cone C (without the vertex), and let L be a generating lbe. Unfoldiiig C - L onto lR2 produces a map f : C - L + lR2 whicli is a local

isometi-); but which is usuaUy not oiie-orie. Iinrestigate tlie geodesics on a cone (tlie number of geodesics between two points depends 011 the angle of the cone, arid soine geodesics may come back to their initial point).

40. Let g : S* + Pn be tlie map g ( p ) = [ p ]= (p, - p ] . (a) Sliow tliat there is a uiiique Riemaririian metric (( , )) on P" such tliat g*(( , )) is tlie usual Riemanriian metric on S" (tlic one that makes the iriclusion of S" into lRnil an isometry). (11) Sliow tl-iat every geodesic y : IR + Pn is closed (tliat is, tliere is a riumber a sucb that y(t +a) = y(t) for al1 E), and tliat every two geodesics isltersect exactly once. (c) Sllow tbat tliere are isornetries of P"onto itself taking arly tangent vector at orie point to any tangent vector at any other point. Tliese recults sbow that P ' provides a model for "elliptical" non-Euclideaii georiieti-)! Tlie sum of the angles in aiiy triangle is > n.

41. Tlie Poincark upper halFp1ane 3f2 is the maiiifold ( ( x ,13 m:itli the Riemannian metric

E

lR2 : y > 0)

(a) Compute that k = 0. al1 otl-ier rij

(b) Let C be a semi-circle in X 2 with center at (O, e) and radius R. Considering it as a curve t t-+ (1, y(t)), show that

(c) Using Problem 27, show that all the geodesics in X 2 are the (suitably parameterized) semi-circles with center on the x-axis, together with the straight lines parallel to the y-axis. , (d) Show that these geodesics have infinite length in either direction, so that the upper haif-plane is complete. (e) Show that if y is a geodesic and p 4 y, then there are infinitely many geodesics through p which do not intersect y. (f) For those who know a little about conformai mapping (compare with Problem IV7-6). Consider the upper haif-plane as a subset of the complex numbers C. Show that the maps

are isometries, and that we can take any tangent vector at one point to any taiigent vector at any other point by some $,. Conclude that if length AB = length ~ ' 3and ' length AC = length A'C' and the angle between the tangent vectors of p and y at A equais the angle between the tangent vectors of p' and y' at A', then length BC = length B'C1 and the angles at B and 3' and at C and C' are equal ("side-angle-side"). These results show that the Poincaré

upper lialf-plane is a model for Lobachevskian non-Euclidean geometry. The sum of the angles in any triangle is < x.

42. Let M be a Riemannian manifold sucli tliat every two points of M can be joined by a unique geodesic of minimai length. Does it necessarily follow that the Riemannian manifold M is complete? 43. Let M be a manifold with a Riemanliian metric ( , ), and choose a fixed point p r M . Suppose that every geodesic y ; [a,b ] + M with initial vaiue y (a) = p can be extended to ail of R. Show that the Riemalinian manifold M is geodesically complete. 44, Let p be a point in a complete non-comlacl Riemanilian manifold M. Prove that there is a geodesic y : [O, m) + M with the initiai value y(0) = p , having the property that y is a minimai geodesic between al-iy two of its points. 45. LRt M alid N be geodesically complete Riemannian manifolds, and give M x N the Riemannian metric described in Problem 26. Show that the Riemannian manifold M x N is aiso complete. 46. This problem presupposes knowledge of covering spaces. Let g : M + N be a covering space, where N is a Cm manifold. Then there is a unique Cm structure o1-i M wliicli makes g an immersion. If ( , ) is a Riemannian metric oli N, tlien g*( , ) is a Riemannian metric on M , and ( M , g *( , )) is complete if and only if (N, ( , )) is complete.

47. (a) If M" C is a submanifold of N, show that the normal bundle v is indeed a k-plane bundle. (b) Usiiig tlie i-iotioii of Whitney sum @ introduced iil Problem 3-52, show that

48. (a) Show that tlie normal bundles v i , vz of M" c defined for two different Riemannian metrics are equivaient. (b) If t = rr : E + M is a smooth k-plalie bundle over M*, show that the normal bundle of M c E is equivaient to 5 . 49. (a) Given an exact sequence of bundle maps

as in Problem 3-28, wllere tlie bundles are over a smooth manifold M [or, more generally over a paracompact space], show that E2 2 E l $ E3. (b) If t = r r : E + M is a smooth bundle, conclude that T E R. rr*(t) $ x*(TM).

50. (a) Let M be a non-orientable manifold. According to Problem 3-22 there is S' c M so that ( T M ) I S ' is not oricntable (the Problem deals with the case where (TM)IS' is aiways trivial, but the same conclusions will hold if each (TM)IS' is orientable; in fact, it is 11ot hard to show that a bundle over S' is trivial if and only if it is orientable). Using Problem 47, show that the normal bundle v of S? M is not orientable, (b) Use Prol~lem3-29 to condude that there is a neighborhood of some S' c M which is not orientable. (Thus, any non-orientable manifold contains a "fairly smail" non-oi.ientable open submanifold.)

A+

CHAPTER 10

T

LIE GROUPS

liis cliapter uses, and illuminates, many of the results and concepts of tlle pixeding chapter-s. It will also play an important role in later Volumes, where we are concerned ~ 4 t geometric h problems, because in the study of these pi-oblems the groups of automorphisms of various structures play a central role, aiid tliese groups can be studied by the methods now at our disposal. A topological group is a space G which also lias a group structure (the product of u , b E G beiiig denoted by a b ) such that the maps from G x G to G

(a,b ) M a b

u

H

from G to G

a

al-e coiitinuous. It cleai-lysuffices to assume instead that the single map ( a ,b ) w nb-' is continuous. 147~will mainly be iiiterested in a very special kind of topological group. A Lie group is a group G which is also a manifold with a Cm structure such that

are Cm functioiis. It clearly suffices to assume tliat the map (x, y) w xy-' is Cm. As a mattei. of fact (Problem 1), it eveli suflices to assume that tlie map (x, 1)) M xy is Cm. Tlle simples1 example of a Lie group is R", with the operation . The circle S' is also a Lie group. Oiie way to put a group structure on S' is to consider it as tlie quotient group R/Z, wliere Z C R denotes the subgroup of iiitegers. The f~iiictions M cos2nx and x M sin 2nx are Cm functions on R/Z, and at each poilit al least one of them is a coordinate system, Thus the map ( A - , y ) W .Y - }' W x)'-' m

+

m

m

R x R + R + S'=R/Z. wkich can be expressed in coordiliates as one of thc two maps (x, 1 1 ) (x,y)

cos 2x(x - 11) = cos 2nx cos 2n y + sin 27rx sin 2ny M sin 2n(x - y) = sin 27rx cos 27r y - cos 27rx sin 2xy, M

is Cm: consequently the rnap

(A-,y) w

xy-' from S' x S' to S' is also Cm.

If G and H are Lie groups, then G x H, with the product Cm structure: and the direct product group structure, is easily seen to be a Lie group. In particular. the torus S' x S' is a Lie group. The torus may also be desuibed as the quotient group

the pairs (a. b) and (a', b') represent the same element of S' x S' if and only if a' - a E Zalid b' - b E Z. Many important Lie groups are mati-is groups. The general linear group GLOI,R) is the group of al1 non-singular real 17 x 17 matrices, considered as a subset of R"'. Since the function det : R"' + R is continuous (it is a polynomial map), the set GL(i1, R) = deth'(R - (O)) is open, and hence can be given tlie Cm struciuir wliich makes it an open submanifold of R"'. Multiplication of mati-ices is Cm, siiice tlie entries of A 3 are polynomials in the entries of A and B. Sinoothness of tlie inverse map follows similarly fi-om Cramer's Rule: (~-')~ =idet Aij/det A , where A'J is the matrix obtained from A by deleting row i and column j. Oiie of the most importaiit esamples of a Lie gi-oup is the orthogonal group O(II),consistiiig of al1 A E GL(r1, R) witll A A' = 1, wliere A' is tlie transpose of A. Tlis condirion is equivaleilt to tlie coiidition tliat the rows [alid columns] of A are 01-tlioiiormal, which is equivalent ro tlle coiidition that, with respect ro the usual basis of R", the inatrix A i-epresentsa linear ti-ansformation wliich is aii "isomerry", Le., is norm presenling, and thus inner product presenling. Problem 2-33 presen ts a proof tliat O (17) is a closed submanifold of GL(i1, R), of dimension i l I r ? - 1)/2. To sliow tliar O(I?)is a Lie group we must sliow rliat the map (x, y) i-, xy-' which is Cm on GL(i1, R), is also Cm as a map from O(]?)x O(i7) ro O(II).By Propositioii 2-1 1, i t suffices ro sliow rllat ir is coiitinuous; but this is true because the inclusion of O(i7) + GL(n, R) is a homeomorphism (sii~ceO ( I ~is) a sii11niaiiifold of GL(n , R)). Later in the cliapter we will havc anotlier way of proving that O(i?) is a Lie group, aiid in particular, a maiiifold. Tlie argument in the previous paragraph shows, generally, tliat if H c G is a subgroup of G aild also a submanifold of G, rheii H is a Lie group. (Tliis gives aiiotlier prooftliat S' is a Lie groiip, for S' c R2 caii be coiisidered as tlle gioup

motions, Le., isometries of R". A little arpmenr shows (Problem 5) tliat even. element of E(]?)can be writren uniquely as A T where A c O ( n ) , and r is a translation, T(X)

= Ta(X) = X

+U .

Cd¢st~.iicturewhich makes it diffeomorphic to O ( n ) x R*. Now E(n) is iiot tlie direcr pi-oduct O(i7) x IR" as a gi-oup, since translations aiid ]Ve can give E(]?)the

ortl~ogoiialtrallsformations do not generally commute. In facr,

SO

Ara A-' = r ~ ( a ) ,

Ar, = %(o} A .

Consequen tly:

wliich shows that E(i7) is a Lic gmup. Clcarly tlie compoiient of e r E(r7) is the subgroup of Al Ar with A E SO(i?). For any Lie group G, if a E G we define [he left and right translations. L,: G + Gaild R , : G + G , b v

L,(b) = ub Ru ( b ) = bu. Notice tliat L, and R, are bot1.i diffeomol-pliisms,witll iiiverses La-, and R,-1. resliect ively. Corisequently, tlle maps

are isomorphisms. A vector field X on G is called left invariant if

La,X

=X

for al1 a

E

G.

Recall tliis means thai

It is easy to see tliat tbis is rrue if we merely liavc

La, X, = Xa

for al1 a

E

G.

Co~lscquently,giveli X, E G,. tl1e1-e is a uliique left invarianr \rector field on G wllich lias rlle value Xp at e .

X

2. PROPOSITION. E1ler-y left invariant vector field X on a L e group G is Cm.

PROOE It suffices to prove tbat X is Cm

a neigliborhood of e , since the diffeomorphism L, theti takes X to the Cx \lector field L., X around a (Problem 5-1). Ler (x, U ) be a coordinate system around e . Choose a neighborI~oodV of e so rliat a!b E V implies ab-' E U.Tlien for a E V we have

Since the map (a,b)

H

iii

ab is Cm on V x V we can write

lbr some Cm fuiictioti f' on x(V) x x(V).Tllerl

xxi(a)= xe(A-' n

=

o

n

a(xi o L,)

Ccj j= 1

L.) where X, =

ax j e

ci

a

j =1

wliich shows tliat x x i is Cm. This implies that X is Cm. O

3. COROLLARJ'. A Lie group G aiways has a trivial tangent bundle (and is consequently orietitable).

PROOE Cl~oosea basis Xle, . . . , X,,for G,.Lct Xl,. . . ,Xn be the left invarialit vector fields wirll these values ar e . Tlien XI ,. . . , Xn are clearly everywliere litiearly il~dependetit,so we can define an equivalente

A left invarianr vector field X is just one that is La-related to itself for all a. Consequently, Proposition 6-3 shows that [X, Y] is left invariant if X and Y are. Henceforrh we will use X, Y, etc.: to denote elements of G,, and X, Y, erc., to denote the left invariant vector fields with R(e) = X, y(e) = Y, etc. We can then define an operation [ , ] on Ge by

The \lector space G,, together with this [ , ] operation, is called rhe Lie algebra of G, and will be denoted by k ( G ) . (Sometimes the Lie algebra of G is defined insread [o be the set of left invariant \rector fields.) We wiIl also use the more customary notation Q (a German Fraktur g) for J ( G ) . This notation requires some conventions for particular groups; we write QI(II,IR) o(n)

fortheLiealgebraofGL(n,R) for the Lie algebra of O(n).

In general, a Lie algebra is a finite dimensional vecror space V, with a bilinear operation [ , ] satisfying [X, X] = o [[X,Y], Z ]

+ [[Y,Z], X] + [[Z,X], Y] = O

'yacobi identity"

for al1 X, Y, Z E V. Since the [ , ] operation is assumed alternaring, it is also skew-syrnmetric, [X, Y] = -[Y, X]. Consequently, we cal1 a Lie algebra abelian or commutative if [X, Y] = O for all X, Y. Tl-ie Lie algebra of IR" is isomorphic as a vector space to IR". Clearly L(IRn) is abelian, since t11e vector fields a/axi are left invariant and [a/dxi, d/axj] = 0. The Lie algebra 6 ( S 1 ) of S' is 1-dimensional, and consequently must be abelian. If are Lie algebras with bracket operations [ , 1; for i = 1,2, then we can define an operation [ , ] on the direct sum V = 6 @ V2 (= VI x V2 as a set) by [(Xi, X2), (Y19 y211 = ([Xl, Ylli, [X2, y212). It is easy to check that this makes V into a Lie algebra, and that L ( G x H) is isomorpl~icto J ( G ) x L ( H ) with this bracket o~eration.Consequently, the Lie algebra L(S1x - - x S' ) is also abelian. The structure of gi(n,IR) is more complicated. Since GL(n,R) is an opcil sulnnanifold of IR"', the tangent space of GL(n,R) at the identity I can br

identified with Wn2.If \*e use the standard cooi-dinatesxij on Wn', then an i7 x n (possibly singular) matrix M = (Mij) can be identified with

Let fi be tbe left invariant \rector field on GL(n, W) corres~ondingto M. 1% compute the function M x k i on GL(n,B) as follows. For every A E GL(ri,W),

NOMIthe function xk'

o

LA: GL(li, W) + GL(r1,R) is the linear function

witli (constant) partial derivatives

So if N is another n x

17

matrix, we have

Eom this we see that

[M, ElI = ~ k,'

al

( M -NN M ) ~ '

axki

;

thus, if we identify 91( n , W ) with Wn2,the bracket operation is just

[ M , N= ]M N - N M . Notice that in any ring, if we define [a,b ] = a b - b a , then [ , ] satisfies the Jacobi iden ti h. Since O(ii)is a submanifold of G L ( n ,W ) we can consider O ( n ) l as a subspace of G L ( n , R ) ] , and tlius identify o ( n ) with a certain subspace of Wn2. This subspace may be determined as follows. If A : ( - E , E ) + O ( n ) is a curve with A ( 0 ) = 1, and we denote ( A ( t ) ) , by A j j ( i ) , then

which shows that

A i j f ( 0 )= - A ~ ~ ' ( o ) . Thus

O(il)C ~

w"' can contain only matrices M wliich are skew-symmetric,

Tliis subspace has dimeiisioii ~ ( 1 1 - 1)/2, wliich is cxacrly the dimensioii of O ( n ) , so O ( U )must ~ coilsist exactly of skew-symmetiic matrices. If we did not hiow [he dimeilsioil of O(iz), we could use tl-ie foltowing line of reasonii-ig. For each i , j with i < j, wc can define a cunre A : W + O(n) by

~ 4 t hsin t and - sin t at (i, j ) and ( j , i ) , 1 'S on tlie diagonal except at (i,i) and ( j ,j),and O's elsewhere. Then the set of all ~ ' ( 0 span ) the skew-symmetric matrices. Hence O ( I I )must ~ consist exactly of skew-symmetric matrices, and O(n) must have dimension n(n - 1)/2. \Ve do not need any new calculations to determine the bracket operation in o(11). In fact, considel. a Lie subgroup H of any Lie group G, and let i : H + G be the inclusion. Since i, : H, + G, is an isomorp hism into, we can identify f& witli a subspace of G,. Any X E H, can be extended to a left invariant vector field X" on H and a leR invariant vector field on G. For each a E H C G, we have left translations L,: G + G La : H + H , and L a o i = i o La. rY

LI

i, X (a) = i, La, X = L,,(i* X) = X (u).

Iti, other woids, X" and f are i-related. Co~isequentl~ if Y aiid [X, Y]are i-related, which means that LI

rY

E

&, then [X, Y]

LI

Tlius, H, c Ge = Q is a subalgebra of Q,thar is, H, is a subspace of Q wliich is closed uiider the [ , ] operation; moreover, He with this induced [ , ] operation is jusr 1) = L ( H ) . This coi-respoiideticebetween Lie subgt-oups of G and subalgebras of 9 turns out to work in the othei- direction also. 4. THEOREh4. Let G be a Lie group, and 11 a subalgebra of 9. Then there is a utiique connected Lie subgroup H of G whose Lie algebra is $.

PROOF. For u E G, let A, be tlie subspace of Gu consisting of al1 ?(o) for X E 11. The fact that 11 is a subalgebra of g implies that A is an integrable distribution. Let H be rhe maximal integral manifold of A containing e. If b E G, then clearly Lb*(Aa) = Aburso Lb* leaves the distribution A invariant. It follows immediately that Lb permutes the various maximal integral manifolds of A among rliemselves. Iti particular, if b E H, tlien Lb-[ takes H to the tnaximal iiitegrai manifold containitig Lb-, (6) = e , SO Lb-] ( H ) = H. This iinplies rhar H is a subgi-oup of G. To prove tliat it is a Lie subgroup we just iieed to show tliat (u, b) I-+ ub-' is Cm. Now tliis map is clearly Cm as a map into G. Using Theorem 6-7, ir follows that it is Cooas a map into H. The pt-oof of uniquetiess is left to the reader. *3

There is a very dificult theorem of Ado which states that every Lie algebra is isomorphic to a subaigebra of GL(N7R) for some N. It then follows from Theorem 4 that e v q Lie aigebra ti tiomo@ic to /he he aigebra ofsome Liegroup. Later on we will be able to obrain a "local" version of tliis result. We will soon see to what extent the Lie aigebra of G determines G. 1% continue the study of Lie groups along the same iSouteused in the stud!. of groups. Having considered subgroups of Lie groups (and subaigebras of tlieir Lie algebras), we next consider, more generaily, homomorphisms between Lie groups. If # : G + H is a Cm homomorphism, then #,,: Ge + He. For anv a E G we clearly have so if X E Ge, and 5 value #,, X at e, t hen

=

#.,X

is the left invariant wctor field on H with

Thus X" and 5 are #-i-elated. Consequently, tlie map algebra homomorphism, tliat is,

#,,

: Q

+ 1) is a Lie

Usually, we will denote #,, simply by #, : g + I). For example, suppose tliat G = H = R. There are an enormous iiumbeiof homomorphisms # : R + R, because R is a wctor space of uncountable dimciision wer Q, and ewry linear transformation is a group homomorphism. But if # is Cm, then the conditioi-i

implies thar

which ineans that #(t) = ct for some c (= #'(O)). It is iiot hard to see tliat eveii a coi-itii-iuous# must be of rhis form (one first shows that # is of this form on tlie

rational numbers). ]Ve can identify L(R) with R. Clearly the map #, : IR + R is just rnultiplication by c. Now suppose that G = R, but H = S' = R/Z. A neighborhood of the identity e E S' can be identified with a neighborhood of 0 E R, giving rise to an identification of l ( S 1 )with R. The continuous homomorphisms # : R + S' are clearly of the form

once again, #, : R + R is multiplication by c. Notice that the oiily continuous homomorphism # : S' + R is the O map (since (O) is the only compact subgroup of R). Consequently, a Lie aigebra homomorp hism Q + 11 may not come from any Cw homomorphism # : G + H. However, we do have a local result.

5. THEOREM. Let G and H be Lie groups, ai-id a: Q + b a L e aigebra homomorphism. Then there is a neighborhood U of a E G and a Cm map #: U + H such that #(ab) = # ( n ) # ( b ) and sucli that for every X

E 9

when a, b,ab

E

U,

we have

Moreover, if tliere are two Cw liomomorphisms #, S r : G + H with #, $*e = a, and G is connected, then # = @ .

=

PROOE Let f (German fiaktur k) be the subset f c Q x b of al1 (X, @(X)),for X E Q. Since is a homomorphism, f is a subalgebra of Q x b = L(G x H). By Tlieorem 4, there is a unique connected Lie subgroup R of G x H whose Lie algebra is f. If xl : G x H + G is projection on tlie first factor, and w = x11 K , rhen w : K + G is a Cm homomorphism. For X E Q we have

so w, : K(e,el + G, is aii isomorphsm. Consequeiitly, there is an opcn neighl~orhoodV of ( e , e ) E K such that w takes V diffeomorphicaily onto an open neigliborliood U of e E G. If x2: G x H + H is ~rojectionon the second factor, we can define # = x2 O u-' on U.

Tlle first condition on # is obvious. As for the second, if X

E Q,

then

Given #,S, : G + H , define tlie one-one map 8 ; G + G x H by

The image Gr of 8 is a L e subgroup of G x H and for X

E

g we clearly have

so L ( G r ) = f. Thus G' = K,which iinplies that $ ( u ) = #(a) for al1 a

E

G. Q

6. COROLLARY. If two L e groups G and H have isomorphic L e algebras. then they are locally isomorpliic.

PXOOE Given an isomorpliism 0 : g + I), let # be the map $ven by Tlieorem 5. Since #, = is aii isoinoi-phism, # is a dfleomorphism in a neighborhood of r

E

G. e3

Xenlark: For tllose wl-io know about siniply-coilnected spaces it is fairly easv (Problem 8) to conclude that two simply-coriiiected Lie groups with isomoi-phic L e algebras are actually isomorpliic, aild that al1 connected Lie groups with a given Lie algebra are covered by [he same simply-connected Lie gi-oup. 7. COROLLARY. A connected Lie group G with an abelian Lie algebra is itself abeliaii.

PROOE By Corollary 6, G is locally isomorpbic to IR", so a b

= bu for a , b

in a neighborhood of e. It follows that G is abeliaii, sii-ice (Problem 4) neighborhood of e geiierates G. +$ 8. COROLLARY. For every X # : R + G such tl-iat

E

aii~

G,, thei-e is a uiiique Cm I-iomomorphism

FIRSTPROOE Define 0 : R + 6 ( G ) by @(a) = (xx.

Clearly 0 is a Lie algebra homomorphism. By Tlieorem 5, on some neigliboiliood (-E,&) of O E R there is a map # : ( - E , & ) + G with #(S

+ 0=# ( ~ ) # ( l )

lsl, l t l , 1s

+11

E

and

It 1

To extend # to R we write every t with 1

= k ( ~ / 2+) Y

E

uniquely as

k an integer, Ir 1 < & / 2

and define

.

[ # ( ~ / 2 )appears k times]

# ( ~ / 2 ) # ( ~ / 2 ) #(r) m

#(i) =

#(-~/2)

# ( - ~ / 2 ) - #(Y)

[#(-&/S)

appears -k times]

Uiiiqueiiess also follows from Theorem 5. SECOJVD (DIRECV PROOE If .f : G + IR is C m , aiid # : homomorpl-iism, then

R

k 3 0 k < 0.

+ G is a C m

= lim S(#(t)#(ld) - f( $ 0 1) h+O h

Thus # iriust be an integral cunre of if # : R + G is aii integral curve of

X", whicb proves uniqueness. Con~rersely~

X,tben

is aii iiitegi-al curve of X wliich passes through clearly true for i

E-

#(S

#(S)

at time i = O. The same is

+t):

so # is a hoinomoi.pliisrn. MTekno\rr that integral cunres of X" exist locally; they can lx extended to al1 of R usiiig [he method of the first proof. *3

A homomorphism # : IR + G is called a l-parameter subgroup of G. We thus cee that there is a unique 1-parameter subgroup # of G with given tangent wrtnr d#/di (0) E G,. l e have already examined the 1-parameter subgroups ui R. More interesting things liappen when we take G to be IR - {O), with multiplicatioil as tlie group operation. Then all Cm homomorphisms # : IR + IR - {O), with #(S + t ) = #(s)#(I), must satisfy

The solutions of tliis equation are

Notice that IR - (O) is just GL(1, IR). A l Cm l~omomorphisms# : IR + GL(n, IR) must satisfy tlie analogous differential equation

where iiow denotes matrix multiplication. The solutions of these equations can be written formally in the sarne way

where exponeiiriation of matrices is defined by

This follows from t he facts in PI-oblem5-6, some of which will be briefly recapitulated here. If A = (ujj) and IAl = max lau 1, then clearly

+ + * . m

(11

1A I ) ~ + *

(N

+ K)!

+O

asN+m.

so tlie series for exp(A) converges (the (i,j)'hentry of the partial sums converge), and convergence is absolute and uniform in any bounded set. Moreover (see Problem 5-6), if AB = BA, then exp(A

+ B) = (exp A)(exp B).

Hence, if # (1) is defined by (m),then #'(I) = lim

exp(t#'(O)

+ 17#'(0)) - exp(t#'(O) 1 h

h+O

= lim h+O

=

lim

[exp(l7#'(0)>- 11 exp(t#'(O) h 1!

h-to

2!

h

exp (1#'(O) 1

so # does satisfy (*). For any L e gi-oup G, we now define the "exponential map" exp: g + G as follows. Given X E 9, let # : R + G be the unique Cm homomorphism witfi d#/dt(O) = X. Then exp(X1 = #(]l. We clearly have

9. PROPOSITION. The map exp: Ge + G is Cm (note that G, R" has a iiatural Cm structure), and O is a regular poirit, so that exp takes a neighborhood of O E Ge diffcoinorpliically oiito a rieigfiborhood of e E G. If S,: G + H is any Cm homomorphism, tlien

exp o S,*

=

S, O exp.

expl G-H

lexp

S,

PROOE The tangent space (G, x G)tx,,) of the Cm manifold G, x G at the point (X,u) can be identified with G, @ G,. We define a vector field Y on G, x G by 4

\

Y,,.,

=

o @ ?(a). (S')Lx S i

-

Tlien Y ]las a flow cx : R x (G, x G) + G, x G, which we know is Cm. Since exp X

= projection

it follows tliat exp is Cm. If \ve ideiitifv a \lector u tangent \lector v at O. So

E

on G of a(], O @ X),

(G,)o witli G,, tlien tlie curve ~ ( i = ) t v in G, has

So exp+ois tIle identity, aild lle~iceone-oiie. Tlierefore exp is a diffeomorphism i i i a neigl~borhoodof O. Given $: G + H, and X E G,, let #: R + G be a Iiomomorpliism with

Con~equeiitly~ exp($*X> = S, O # ( i ) = $(exp X). e3 10. COROLLARY. Every o~ie-oneCm Iiomomorpliisrn #: G + H is ail

immersion (so #(G) is a L e subgroup of H).

PROOE If 13ut rlien

= O for some iioii-zeio X E Q, then also #*,(X) = 0. = exp #+e(lX) = #(exp(lX)),

contradictiiig tlie fact tllat # is oiie-one. *=r

11. COROLLARY. Every coiltinuous I-iomomorphism #: R + G is Cm.

PROOF, Let U be a star-sllaped open neighborhood of O is one-one. For any a E e x p ( ~ U ) if, a = exp(X/2) for X a = exp(X/2) = [exp(x/4)12,

exp X/4

E

E

G , on which exp U, then

E

exp(; U).

So a has a square rooi iii exp(; U). Morewer, if a = b2 for b E exp(4u), then b = exp(Y/2) for Y E U, so exp(X/2) = a = b2 = [exp(y/2)12 = exp Y. Since X/2, Y E U it follows that X/2 = Y, so X/4 = Y/2. This shows that every a E exp(; U ) has a unique square root in the set exp(; U). Now dioose E > O SO that #(í) E exp(; U) for 11 1 E. Let #(E) = exp X, X E exp(f U). Since [#(&/2)12= #(E) = [exp X/212, it follows from ihe abwe that # ( ~ / 2 ) = exp(X/2). By induction we have #(&/Y) = e ~ p ( X / 2 ~ ) . Helice #(m/2" E) = #(~/2")" = [exp(X/2")lm = exp(n7/2" X).

12. COROLLARXi. Every continuous Iiomomorphism # : G + H is Cm

PROOE Clioose a basis Xl, .. . ,X, for G,. The map t i-, #(exp tXi) is a coiitiiluous l~on~omorphism of R to H, so tl-iere is Yi E He such thai #(exp iXi) = expiYj. Thus, # ((exp11Xi ) - . - (exp i, x,~)) = (exp ti YI ) - . (exp hYn). (*> Now tlle map S,: Rn + G given by is Cm and clearly

so is a diKeomorpliism of a tieighboi-liood U of O llood V of e E G. Then on V,

E

R" onto a neighbor-

# = (# O S,) O S,? aiid (*) sliows tllat # o S, is Cm. So # is Cm at e, and tllus everywhere. 9

1 3. COROLLARXr. If G and G' are Lie groups wllich are isomorphic as topo-

logical groups, then they are isomorphic as Lie groups, that is, there is a d f i o morphism between them wllicll is also a group isomorphism.

PROW Apply Corollary 11 to the continuous isomorpliism and its inverse.

+:+

Tlle propert jes of tlie particular exponential map exp : V' (= 1

,R )

+

GL(n, R)

may 11ow be used to show tllat O(n) is a Lie group. It is easy to see thal exp(~'= ) (exp M)'. h/loreo\~er,sirlce exp( M

+ N) = (exp M) (exp N) when M N = NM, we have (exp M)(exp -M) = I .

(exp M)(exp M)' = 1; Le., exp M E O(n). Co~lversely,any A E O(!]) sufficiently close to I can be written A = exp M for come M. Let A' = expN. Then I = A A' = (cxp M)(exp N), so es11 N = (exp M)""' = exp(- M). For sufficiently small M and N tliis implies tliat N = -M. So exp M' = A' = exp(- M); lieilcc M' = -M. It follows tliar a ncighborhood of I in O(n) is an n(n 1)/2 dimensional su bmailifold of GL(i1, R). Since O(iz) is a subgroup, O (11) is itself a submanifold of GL(ir ,R). Just as in GL(n, R), tbe equation exp(X Y ) = exp X exp Y liolds wlie~ievci [X, Y] = O (PI-ol~lein13). 111 general, [X, Y] measures, up to first order, the cstent to which tliis equatioii fails to hold. In tbe following Theorem, aiid i11 its piaof, t~ iiidicate diat a functioii c. : R + G , has tlie property that c ( t ) / t is l)ouiided for small t, we wiD denote it 11y O(t3). Thus O(f3)will denote differeni fi~~ictions at differeiit times.

-

+

14. THEOREM. If G is a Lie group and X, Y E G,, theii

+ Y ) + if '[ ~ , YI + 0 ( t 3 ) J

(1) exp t~ exp / Y = e x p { f ( x

(2) exp(-tX) exp(-t Y) exp tX rxp t Y = exp{f2 [ ~Y], + O(t 3)i

(3) exp tX exp t Y exp(-tX) = exp(ty

+ r2[x,Y] + ~ ( t ~ ) } .

(i)

Ff(a) = Fa( f ) = L. x(/) = X(J

o

L ~ =)

l o

f a exp UX).

Similarly, m

Irf(a) For fixed

S,

=

-

f ( a exp uY). m

let #(i) = f(expsX expiY).

Then

(iii)

f=.l,

d #'(i)=-f(expsXexpiY)= di

Applying (iii) io

yf

insiead of

m

#"(/) = [Y (Yf )](exp sX exp /Y).

Now Taylor's Theorem says thar

Suppose that f (e) = O. Then we have

Similarly, for any F.

sx exp I Y exp u Y)

/ gives m

(iv>

f(exp

-

Substituiingin (v) for F = f , F = y~ and F = Y ( Y f ) gives (vi) f'(expsXexpiY)=s(yf)(e)+i(Yf)(e)

Now for small i we can wriie exp iX exp t Y

= exp Z ( i )

for some Coo function Z witll values i1.i G,. Applying Taylor's formula to Z gives Z ( r ) = ízI i2z2 0 ( t 3 ) ,

+

+

for some Zi ,Z 2 E G,. If f (') = O , then clearly f ( A ( t ) O({'), SO by (vi) we have

+ 0 ( t '))

= f ( A( 1 ) )

+

Siiice we can tnke tfle .f?s to be cooi-dinate fuilctioi~s,comparison of (vii) and (viii) gives

w11i cli givw 1

thus proving (1). Equatioi~(2) follows immeciiately from (1).

To prwe (3), again cl-ioose f with f ( r )= O. T l ~ e nsimilar caiculations give (ix) f (exp t X exp t Y exp(-t X))

then we also have

Comparing (ix) and (x) gives the desired result.

e3

Notice tliat for-mula (2) is a special case of Tl~eorem5-16 (compare aiso with Problems 5-1 6 and 5-13). T l ~ ework involved in prmii-ig Tl-ieorern 14 is justified by its role ii-i the following beautiful theorem. 15. THEOREh4. If G is a Lie group and H c G is a closed subset which is also a subgroup (algebraically),the1-i H is a Lie subgroup of G. More precisely tl~ereis a Cm sti.ucturc on H , triith die relntiue topoiogy, tllat rnakes it a Lie subgroup of G. PROOE We attempt to recoiistruct the Lie a1gebi.a of H as follows. Let B be the set of al1 X E G, such tliat expiX E H for al1 t .

c Ge

Assulioll I . Let Xi r Ge with Xi + X and let t j + O witl~each ti # O. Suppose exp tiXi E H for a11 i. Then X E 11. Pro!$ le[

We can assurne ti > 0, since exp(-tiXi) = (exp li Xi)-'

ki (1) = largest integer 5

-.1 ti

E

H. For t > 0:

exp(ki (t)tiXj) = [exp(ii~

ki(r) j ) ]

E

H,

ki(t)tiXi + t X . Thus exp tX r H , since H is closed and exp is continuous. We clearly also haw expiX E H for i c O, so X E 1). QE.D. ]Ve now clairn that 1) c G, is a vector subspace. Clearly X E 1) irnplie~ sX r 1) for al1 s E R . If X , Y E 11, we can write by (1) of Theorern 14 exp tX exp t Y = exp{t(X + Y ) + iZ(i)i

+

wliere Z(i) + O as i + O. C1loose positive tj + O aiid let Xi = X Y + Z(ii). Then Assertion 1 irnplies that X + Y r b. Alternatively, we can write, for fixed i.

+

taking 1irnits as n + ca gives exp t ( X Y ) E H. [Sirnilarly, using (2) of Tlieorern 14 we see tl~at[X, Y] E b, so that 1) is a subalgebra, but we will ilot even use this fact.] Yow let U be an open neigliborllood of O r Ge on whicla exp is a d8'Teornoi.pl~isrn.Then exp(b n U ) is a submanifold of G. It c.lear1y suffices to sllow thai if U is srnall enough, tlleil

H n exp(U) = exp(1)n U). Clloose a subspace 1)' C G, cornplernei~taryto 11, so tliat G, = 1) $ 11'. Assutioil 2. Tl-ie rnap # : G, + G defined by

#(X

+ X') = exp X exp Xf

X E Ij, X ' E 1)'

is a dXeornorpllisin i i i sorne neigl-iborliood of O. Rooj: Cl~oosea basis X1,. . . ,Xk, . .. ,X, of G, witli XI ,. . ., Xk a basjs for [l. Tlieii # is given by

Siiice tlie rnap .x:=, a i X i it suftices to show thai

H

( a l,. , .,on)is a diffeomorphisrn of G , onto

W".

@(al,.. . ,a,) = exp is a diffeomorphism in a neigliborhood of O E

IR". This is clear, since

Assc~~olr 3. Tlicre is a i~eighborlloodV' of O in 11' such that exp X' $ H if

0 # X'

E

V'.

Plo$ Clioose aii iiiner product on lj' aiid let K c 11' be the compact set of al1 X' E I)' with 1 5 IX'I 5 2. If the assertion were false, there would be Xi' E b' ~vitliXj' + O and esp Xi' E H. Clioose integers 17i wcitli

Choosirig a suheqiience if necessary, we can assirrne Xi'

-+

X'

E

K. Since

it follows fi-orn A . T . T ~ .1~ that z o ~ ~X' E 11, a contradiction. Q1E.D.

\Ve cari now cornpletc tlie proof of the tlieorem. Choose a neighborhood U = W x W' of G , or-i which exp is a d~eornorphisrn,with W a neighborhood of O E 11 W' a neighborhood of O E 11' siicl~that W' is contained in V ' of Assn-tion 3, and # of Asscriioli 2 is a daeornorpliism on W x W'. Clear1)-

To pr-we the reverse iriclusion, let a E H n exp(U). Then

X r W, X' r W'.

n = exp X esp X'

Since n , exp X

E

H we obtain exp X'

E

H, so O = X', and a

E

exp(b flU). +$

Up to iiow?we liave concentrated on tlie left invariant vector fields, but rnan? properties of Lie groups are better expressed in terrns of forrns. A forrn w is called Iefi invariant if L,*w = o for al1 a E G. This rneans that

Clearly? a left invariant k-forrn o is deterrnined by its value o ( r ) E SZ' (G,). Heiice: if o',. . . , o" are left iiivariant 1-forrns such that w 1(E), . . .,w n ( e )span G,*. theri e\-eI?I left invariaut k-forrn is

foi- certain conslanh al. 1f o' (e), . . . , o" ( r )is t1ie dual basis to Xi , . . ., Xn t11er-i any Coovector field X can be written

X =

E

G,.

for Cm functions j.'.

f'Tj

Tl~ei~

w i ( x )= f ' , so o' is Cm. It folloius that aiiy left invariant forrn is Cm. If o is leíi invariant, tllen for a E G we have

L,*dw = d ( LO*#)= d o , so clw is also left in~~ariailt. The for.rnula on page 215 irnplies that for a lefi invarian t 1-forrn w and left in\-ariant \rector fields y and Y we llave

--

d w ( X ,Y ) = X " ( w ( p ) )- r(o(2)) - o ( [ X ,Y ] ) N

-

= -o([X,Y]).

tlie bracket being the operation in Q.

Thc ir~ter-plg.betweeii left ii-i\.ai.iant arid 1-ight in\7ariarit \lector fields is tlie sul~jijectof 1'1-oblern 1 l. Her-e \ve coiisider the case of for-nis.

16. PROPOSITION. Let *: G + G be @ ( a )= a - ' . (1) A for-rn w is left invariant if and only if $*w is rjglit invariant. E Q ~ ( G ~then ) , $*oe= (-])'oe. (2) 1f (3) If w is left and right invariant, then d o = 0.

(4) If G is abelian, then 0 is abelian (converse of Corollary 7).

If o is left invariant, tlien

so @*o is riglit invariant. T h e converse is sirnilar: (2) It cleasly sufices to prove this for k = 1. So it is enough to show that SI,,(X) = -X for X E Ge. Now X is the tangent vector at i = O of the curve 1 i-, exprX. So **e X is tlle tangent vector at r = O of 1 i-, (expix)-' = exp(-f X); tl-iis tangent \lector is just -X.

(3) If w is a left and right invariant k-forrn, then

Sir-ice @*o ar-id o are 110th left invariant, we have

T l ~ eforni do is also lefi a i d nght invariant, so

But

$-'(do) = d(**w) = d((- 1 l k o ) = (- 1)k dw.

(4) If G is akliail, then al1 left invaiialt 1-fol-rns w are also rigl-it invariant. So d o = O for al1 left invariant 1-foi*rns. It follows fiorn (x) tllat [X, Y] = O for al1 x , Y E n.

A ! ~ e ~ ? ? a k p r o d(4). o f B y Tlieorern 14, if G is abelian, then for X, Y h ave

H ence 2 [X, Y]

E

G, \ve

+ o(t3)p2 = ;[Y,x ] + 0 ( 1 ' ) / 1 ~ .

Letting r + O, we obtain [X, Y] = [Y, X]. $+ Since do is left invariant for any left invariant o , it follows tliat for a basis o ' , . . . ,o" of iiivariaiit 1-forms we can express eacli dwk in terms of the o' A d . First clioose Xi , . . ., X, E G, dual to o' ( e ) , . . ., & ( e ) . There are constants C$ such that ti

clearly we dso liaw n

Tlie iiumbers C$ are called tlie constants ofstrucnire of G (witli respect to h e basis Xl ,. . . , X, of g). fi-om skew-sylniiietry of [ , ] aad tIicJacohi idcntity we obtain

Frorn (*) on page 394 we obtain

Ii iiiriis out that (2) is esactly wliat we obtaiii korn the relation d 2 0 k = O. Coiidition (2) is tlius an ir-itegrability coiidition. lo fact, we can prwe (Probleni 30) that if C$ ale coiistaiits satisfyiiig (1) aiid (2), tlien we can find everywlicrr liiiea1.ly indcpciideiit 1-forms o ' ,. . . ,o" iii a neigliborhood of O E W" sucli tliat

Morewer, the existente of such o' jrnplies (Prohlern 29) that we can define a rnultiplication (a,b) I-+ab in a neighborf~oodof O which is a group as far as it can be and which has the o' as left invariant 1-forrns. Frorn this latter fact and (a suitable local version of) Theorern 5 we could irnrnediately deduce the following Theorern, for which we supply an independent proof.

17. THEOREM. Let G be a Lie group with a basis of left invariant 1-forrns o', . . .,o" aiid constants of structure c;. Let M" be a differentiable manifold aiid let O', . .. ,O" be everywhere linearly independent 1-forrns on M satisfying

Tlien for every p E M there is a neigllborhood U and a diffeornorphisrn f : U + G such that 0' = f*J.

PROOF. k t

T C ~:

M x G + M and xz : M x G + G be the projections. Let

Then d(ek - ók)= -

EC; ([ei oj] - [ói A

A ój])

icj

By Proposition 7-14., M x G is foliated by 11-dimensional rnanifolds whose tangent spaces at each point are anniliilated by al1 8* - o*. Choose a E G and let r he tlie foliurn through ( p , a). Now .. . ó', .. . ,ónare linearly independent everywliere; so on r(p,alrwhich is tlie set ofvectors in ( M x G)(p,a) where - ók = O, tlie sets o ' , . . . ,8" and o',... ,ón are each linearly indepeiident. Hence T C ~: r + M and x2: r + G are eacli diffeornorphis~ns i i i sorne neigliborhood of (p,a). Tl~isrneans that r contains the graph of a diffeornorpliisrn f fi-orn a neigliborliood U of p to a iieighborhood of a.

e', ,en,

ek

Let

f:U + M x

G be the rnap

It is also possible to sap by how rnucll any two such rilaps differ.:

18. THEOREM. Let M be a coi~nectedmanifold, let G be a Lie group, and let .j;, .j i : M + G be two Cm niaps sucll tliat

for al1 lefi in\.ai-iant 1-for-rnsw . Tl~eiiJ j and J" differ by a left translation, tlial is, ther-e is a (uriique) a E G sucl~tllat

PEDISTliLclAr PROOF. C m 1. M = IR and tlze 1x10ntaps y,, y2: IR + G soiisfi y1(0) = y, (0). ]Ve rnust show tliat y, = y2. For every left iiivariant 1-forrn w we llavc.

1r follows tliar

If wc r-egai-d y, as givei~?and wi-ite this equation out iri a coordinate system. tl-ii~iit bccoiiies an or-diiiaiy difl'cr-critialeqiiation for y2 (of t l ~ etype considered

in the Addendurn to Cl~apter5), so it lias a unique solution with tlie iiiitial coridition y2(0) = y, (O). But this solution js clearly y2 = y,.

Cme 2. M = IR, but 11le nzajis y,, y2 are a r b i l r a ~ .Choose a

E

G so that

If o is a left invariant 1-forrn, then (L.

0

Y,)*(#) = y,*(La*w) = y,*(o) = y,'(o).

Since La o y, (O) = y2(0),it follows frorn Cme 1 that La o yl = y2.

C a s ~3. Ge~ia.01cme. Let po

E

M. Choose a

E

G so tliat

For aiiy p E M there is a Cm curve c : IR + M i v i t l ~c(0) = po and c(1) = p. Let yj = fi o C. Then

By C a e 2, we llave y2(t ) = a . y, ( t ) iii

particular for t = 1, so $2(])) = a -

for al1 f .

(P).

E L E G A A T P R O O F : Lec : G x G + G be projection on the ithfactor. Clioose a l~asiso ' , . . . ,o'' for tlle left iinrariant 1-forrns. For (a, b) E G x G, let

Tlieil A is aii integr-able distribution on G x G. In fact, if A(G) c G x G is tlie diagoiial subgi-oup {(a,a ) : a E G), tlieii tlie rnaxirnal integral rnanifolds of A are t l ~ elefi cosets of A(G). Now define h : M + G x G b?.

By assurnptioii,

Since M is connected, it follows that Iz(M) is contained in sorne left coser of A(G). 1 n otliei- words, tl1ei-e are a , b E G witll

19. COROLLARY. If G is a connected Lie group and f : G + G is a CE map preserving left invariant forrns, then f = L, for a unique a E G . \Yliile left ini-ai-iant 1-forrns play a fundamental role in thc study of G, t h c left iilvariant 12-for-rns arvz also very irnportant. Clearly, al1 left invariant 11-form: ai-c a constant rnultiple of anv non-zero one. If o" is a left invariant n-forrn. then o" deter.rnii1es al1 orieiitation on G, aiid if f : G + R is a Cm fui-iction ivitll coimpac t support, we can define

Siiice a'' is usually kept fixed in any discussioi-i, this is often abbreviated to

T h e latter iiotation has adiyantages iii cer-tain cases. For exarnple, left invariai~cc of o" irnnlies that

f ( n ) da = /G

jG j'(ba) da.

m

in other words,

S, S, fa" =

1 note

ga',

wherr glo) = .f(b«);

that Lb is an orientation presening difkornoi-phisrn, so

~cllichproves the rorrnula]. \Ve can, of c.ourse, also consider right iiivariai~t n-f'oi.rns. Tliese gcncr-ally turn out to be quite different frorn t l ~ eleft invariant 12-ioi-ms (see the csarnple in Pr-oblern 25). But in one case they coincide.

20. PROPOSITION. If G is conipact niid connected and w is a left irivariai~t 12-hl-in,then o is also rigllt invariant. 1'1~00.E Suppose o # O. For each a E G, t l ~ elorrn Ra*o is left invariant. so

tliere is a unique real nuinber f ( a )with

Siiice R,* o RbA= (Rab)*,we have

S o f ( ~ C) IR is acoinpact connected subgroup of R-(O). Hence f ( G )= { 1). +$

]/Ve can abo consider Riernannian rnetrics on G. In the case of a cornpact group G there is ahvays a Riernannian rnetric on G which is both left and right invariant. In fact, if ( , ) is any Riernannian rnetric we can choose a bi-invariant 17-forrn an and define a bi-invariant (( , )) on G

We are finally ready to account for sorne terrninology frorn Chapter 9. 2 1. PROPOSITION. Let G be a Lie group with a bi-invariant rnetric. (1) For any a E G, the rnap I, : G + G giwn by Ia(b) = ab-'a is an isornetq. wllich reverses geodesics through a , i.e., if y is a geodesic and y (O) = a , then Iu (Y (t)) = Y (-0. (2) T h e geodesics y with y(0) = e are precisely the l-pararneter subgroups of G, i.e., the rnaps f J-+exp(tX) for sorne X E g. PROOE (1) Since I,(b) = b-', the rnap l e , : G, + Ge is just rnultiplication by tion l6(2)), so it is an isornetry on G,. Since

-I

(see tlle proof of Proposi-

for any a E G, the rnap J,, : G, + Gu-1 is also arl isornetry. Clearly Ie reverses geodesics through e . Since la = R,-l : it is clear that lais an ison-ietry reversing geodesic through a . (2) Let y : IR + G be a geodesic with y(0) = r . For fixed t, let

T l i e i ~y is a geodesic and Y (O) = ~ ( 1 ) So .

It follows by induction that Y ( n l ) = Y (0"

If 1'

== 12'1

a n d 1''

==

for any integer n.

nfft for integers n f and n", then

so y is a hornornorphisrn on 0. By continuity, y is a 1-pararneter subgroup. T l ~ e s eare the only geodesics, since there are 1-pararneter subgroups with any tangent vector at t = 0, and geodesics tbrough e are deterrnined by theirtarigent vectors at t = 0. *:* ]Ve conclude this chapter by iiitroducing sorne neat forrnalisrn which aHows us to write the expression for dok in an invariant way that does not use tlic constants of structure of G. If V is a d-dimensional vector space, we define a V-valued k-form on M to be a furiction o such that each o ( p ) is an alternating maP k times

If u], . . .,ud is a basis for V, tlieri tliere are ordinary k-forrns o ' , . . ., od sucli tliat for- A'], . . . ,Xk E Mp we have

we will write sirnply

For- anv V-valued k-forrn w we define a V-valued ( k

+ 1)-forrn d o by

a simple c.alculation slio~vsthat tl~isdefinition does not depend o n tl-ie choice of basis u', . . . vd for V.

Líe G'7.aups

403

Sirnjlarly, suppose p : U x V + W is a bilinear rnap, where U and V have bases u l , . . . , u , and u ] , . .. ,ud, respectively. If w is a U-valued k-forrn

and q is a V-valued 1-forrn

+

is a MI-ualued (k 1)-forrn; a calculation shows that this does not depend on the choice of bases u], . . . ,u, or v i , . . .,ud. M'e will denote this W-valued (k + ¡)forrn by p(w A q). Tliese concepts have a natural place in the study of a Lie group G. Althougli there is iio natural way to choose a basis of left invariant 1-forrns on G, there U a natur-al q-valued 1-forrn on G, narnely the forrn o defined by

U$ng tlie bilinear rnap [ , 1: CJ x + g, \ve have, for any R-valued k-form q a11d any n-valued ¡-forrn h on G, a new g-valued (k + 1)-forrn [q A A] on G. Now suppose tbat X i , . . . ,X, E Ge = g is a basis, and that o', . . . ,wn is a dual basis of leíi invai-iant 1 -forrns. T h e forrn o defined by (*) can clearly be writteii n

Tlien

O n the other hand, n

Cornparing (1) and (2): we obtain the equations of structure of G:

T h e equations of structure of a Lie group ~ 4 1 1play an irnportant i-ole in Voiume 111. For the pi-esent we rnerely wish to point out that the terrns d o and [oA o]appearing in this equation can also be defined in an invariant way. Foitlle terrn d o wejust rnodify tlle formula in Theorern 7-13: If U is a vector field on G and f is a g-va1ued fuiiction on G, then (Problern 20) we can define a g-valued function U ( f ) on G. O n the other hand, # ( U ) is a g-valued function on G. For vector fields U and V we can then define

Recall that the value at a E G of the right side depends only on the values U, and Va of U and V a t a . I f w e clioose U = R, V = Y for sorne X 7 Y E G r , then

Fa)= O - O - o-( a-) ( [ X Y, ] , ) rY

rY

do(a)(yQ7

= - w ( e ) ( [ X ,Y ] , ) -o(e)([X, Y ] ) = -[X, Y ] E

= - [ o ( a ) ( y a ) ,o ( a ) ( y a ) ]

--

since [ X , Y ] is left invaimiani by definition of [ , ] in G , by definition of o.

It foHows that foi. any lector fields U and V we have

Pi-oMem 20 gives an iinrariant definition of & A V ) and shows tliat tliis equation is equivalent to tlie equations of structure.

MIARNING: In sorne books the equation which we have just deduced appears ) ,( ~ ) ] T h.e appearance of the factor f Iiere Iias as d w ( U , V) = - L2 [ ~ ( U o noi1rin.g to d o with the f in the other forrn of the structure equations. It comes about because sorne books d o not use tIie factor ( k l ) ! / k !l ! in the definition of A. This rnakes their h A q equal to f of ours for 1-forrns h and v. T h e n the defiiiition of d ( x dx') as d o i A d x i rnakes their d o equal to f of ours for 1-forrns o.

+

PROBLEMS 1 . Let G be a group which is also a Cm manifold, and suppose that ( x ,y) J-+ ñ-1. is CE.

(a) Find 1 - ' w h e n f : G x G + G x G i s J ( x , y ) = ( x , x ) l ) . (b) Show that ( e ,e ) is a regular point of f. (c) Conclude that G is a Lie group. 2. Ler G be a topological group, and H c G a subgroup. Show that tlir of H is also a subgroup. closure

3. Let G be a topological group and H c G a subgroup. (a) If H is open, then so is every coset gH. (b) If H is open, then H is closed. 4. Let G be a connected topological group, and U a neighborhood of e Let U" denote al1 products al a,,for ai E U.

E

G.

(a) Show that Un+' is a neighborl~oodof Un. (b) Conclude that U, Un = G. (Use Problem 3.) (c) If G is locaily compact and connected, then G is a-compact. 5. Let f : Rn + Rn be distance presenring, with f(O) = 0.

(a) Show that (b) Sliow that (c) Show that (d) Show that translation.

J takes straight lirles to straight lines. f takes planes to planes. f is a linear transforrnation, and hence an elernent of O ( n ) . any elernent of E ( I z )can be written A z for A E O ( n ) and z a

6. Show tliat tbe tangent bundle T G o f a Lie group G can always be made into a Lie group.

7. \Ve have cornputed that for M E gl(n, IR) we have

(a) Show that this means that

(It is ñctually clear n that M defined in this way is left invariant, for LA* = LA since LA is linear.) (b) Fi~indthe i.ight invariant lector field with value M at J.

8. Let G and H be topological groups and #; U + H a rnap on a corinected open neighbrhood U of e E G such that #(ab) = #(a)#(b) when n, b,nb E U.

(a) For each c E G, consider pairs (1/, S,), where V c G is an open neigliboi-hood of c with V V-' c U: and where S,; V + H satisfies $(o) S,(b)-] = #(ab-' ) for n, b E V. Define (Vi, S,]) y (V2,S,2) if S,] = S,2 on sorne srnaller neigliborhood of c. Show that the set ofall 7 equi\lalence classes, for al1 c E G , caii be rnade into a covering space of G. (b) Conclude that if G is sirnply-connected, tl~en# cari be extended uniquel\to a liornornorphism of G into H.

9. In Theorem 5, show that # and S, are equal elten if t h q are defined only o11 a neigliborliood U of e E G, provided that U is connected. 10. Sliow that Corollary 7 is false if G is not assurned connected.

11. If G is a gi.oup, \ve define the opposite group G 0 to be tlie sarne set 134th tlie rnultiplication + defined by a +b = b .a. If g is a Lie algebra, ~ 4 t hoperation [ , 1, we define th e opposite Lie aIgebra f1° to be tlie sarne set with t l ~ eopei-atiori [X, YI0 = -[X, Y]. (a) G0 is a group, and if S,: G + G is a I-+ a-': then S, is an isornoi.phism from G to GO. (b) no is a Lie algebra, and X I-+ -X is an isornorphisrn of g onto gO. (c) 6 ( G 0 ) is isornorphic to [6(G)1° = gO. (d) Let [ , ] be the operation on G, obtained by using right invariant \lectorfields instead of left invariant ones. Then (n, [ , ]) is isornorphic to 6 ( G 0 ) , and hence to gO. (e) Use this EO give aanother proof that g is al~elianwllen G is abelian. 12. (a) Show that

e X p (-a O

:)=( -coso sin

a

cosa sino

(11) Use the matrices A and B below to show tllat exp(A + B) is not generally equal to (exp A)(exp B).

13. Let X, Y E G, with [X, Y] = 0. (a) Use Lemrna 5-13 to show that (exp sX)(expt Y) = (exp t Y)(expsX). (b) More generally, use Theorem 5 to show that exp is a hornornorpllism on the subspace of G, spanned by X and Y. In articular, exp(X + Y ) = (exp X)(exp Y).

+

14. Problem 13 implies that exp i ( X Y) = (exp iX)(expt Y ) if [X, Y ]= O. A more general result liolds. Let X and Y be vector fields on a Cm manifold M with corresponding local 1-parameter families of local diReomorphisms {#r): Suppose that [X, Y ]= 0, and let vi = #[ o = $[ o #[.

(b) Using Corollary 5-1 2, show that

In other words, (vr) is generated by X

+ Y.

15. (a) If M is a diagonal matrix ~ ? i tcomplex h entries, show that det exp M = etrace M (b) Show that the same equatioii holds for al1 diagonalizable M with complex entries. (c.) Conclude that it holcis for al1 M witli complex eritries. (The diagonalizable matrices are deiise; compare Problem 7-1 5.) (d) Using Pi-opositioil 9, sliow tliat for tlie homomorphism det: GL(n,R) + R - {O), the map ciet,: ~l(12, R) + J ( R - (0)) = R is just M J-+trace M. (e) Usc tliis fact to give a fancy proof tliat trace M N = trace NM. (Look at trace(MN - NM) = trace[M, N].) (0Prove the result iii part (d) ciii-ectly: ivitl~outusii~g(c). (Sincc det, and tracc are hornomorphisrns, it suffices to look at matrices with only one non-zero entry.) (g) Now usc tliis i-esult and Pr-oposition 9 to give a fancy proof of (c).

16. (a) Let U be a iieighborliood of the ideiitity (1, O) of S' (considered as a subsct of R2). Show that no matler liow small U is, there are elements a E U which have square roots outside U in acidition to their square root in U. (b) Show that foi. each n 2 1, there is a neighborhood U of e E G such thal eitery elemeiii in U has a unique nth root in U. (c) For G = S ' . show tl~attliere is no neighborl~oodU which has this propcrt); for al1 n.

17. (a) Let (x, V ) be a cooi-diiiatesystein arouiid e

E

G with x i ( e ) = O. Let

for Cm functions

J'. Show that

(b) If a,p : (-E, E) + G are difherentiable, show that

(c) A s o deduce this result fiorn Theorern l4(I). (Not even the full strength of (1) is needed; it suffices to kiio\v that exp tX exp t Y = exp{t ( X Y) O(t)). Tlie argurnent of p a n (a) is essentially equivalent to the initial parc of the deduction of (1)s)

+ +

18. Let G be a Lie group, and let H C G be a subgr.oup of G (algebraically), such that every a E H can be joined to r by a Cm path lying in H. Let 6 c G, be tlie set of tangent vectors to al1 Cm paths Jying in H. (a) Show tliat 11 is a subalgebra of G,. (Use Theorern 14.) (b) Let K C G be tlie connected Lie subgroup of G with L e algebra 6. Show that H c K. Hint: Join any a E H to r by a Cm curve c, and show that the tangent \rectors of c lie in tlie distribution constructed in the proofofT1ieorern 4. (c) Let t i , . . . ,ck be curves in H with {cit(0)) a basis for b. By considering tlie rnap / ( t i , ..., t k ) = c ~ ( i 1 ) . . . c k ( i k show ), that K c H. Thus, H is a Lie subgroup of G. It is even true that H c G is a Lie subgroup if H is path connected (by not necessady Cm paths); see Yarnabe, On an arcwire cotrnected sllbg~oupofa L e group, Osaka Math. J. 2 (1950), 13-14. (d) If H c G is a subgroup and an irnrnersed subrnanifold, then H is a Lie subgio up.

19. For a E G, consider the rnap b

M

aba-l = L,R,-'(b).

The rnap

is denoted by Ad(a); usually Ad(a)(X) is denoted sirnply by Ad(a)X. (a) Ad(nb) = Ad(n)oAd(b). Tlius we llave a 1iornoinorphisrn Ad : G + Aut(g), where Au/(n), the autornorphisrn group of g, is the set of al1 non-singular linear transforrnatioi~sof the vector space g onto itself (tllus, isornorphic to GL(n, R) i f n has dirnension n). Tlie map Ad is called the adjoint representation. (b) Show tl~ai exp(Ad(a)X) = n(exp ~)a-' . Hini: T1iis follows irnrnediatcly frorn one of our propositions.

(c) For A

E

GL(n, R) and M

E

gl(n, R) show that

(11suffices to show this for M in a neighborhood of O.) (d) Show that Ad(expíX)Y = Y + í [X, Y] + 0 ( i 2 ) . (e) Since Ad: G + g, we have the map Ad*e : (= Ge)

+

tangent space of A u t ( ~ )at the identity map i of Q to itself.

Tl~istangent space is isornorphic to E n d ( ~ )where , End(9) is the vector space of al1 linear transformations of g into itself: If c is a curve in A u t ( ~with ) c(O) = l,, tllcn to regard c'(0) as an elcment of A u t ( ~ )we , let it operate on Y E Q by

(Compare with the case 9 = Rn, A u t ( ~ )= GL(n, IR), End(g) = 12 x n matrices.) Use (d) to show that Ad*e(X)(Y) = [X, Y]. N

N

(A proof may also be given using the fact that [X, Y] = L F ~ . )The map Y J-+[X, Y] is denoted by ad X E End(g). (f) Conclude that Ad(exp X ) = exp(ad X)

==

lo

+ ad X + (ad2!x ) ~ +

-

e

.

.

(g) Let G be a connected Lie group and H c G a Lie subgroup. Show that H is a iiormal subgroup of G if and only if I) = L ( H ) is an ideal of Q = L ( G ) , tllat is, if and only jf [X, Y] E lj for al1 X E n, Y E I).

20. (a) LRt f : M + V, where V is a finite dimensional vector space, wjth basis v i , . . . ,ud. For Xp E M p , define Xp(f) E V by

xd

where f = I=I j' vi for f i : M + R. Show tliat this definition is indepeiident of the cl~oiceof basis vi, . . . ,vd for V.

(b) If o is a V-valued k-forrn, sliow that d o rnay be defined invariantly by the formula in Theorern 7-13 (using the definition in part (a)). (c) For p: U x V + W. show that p(w A q) rnay be defined invariantly by

Conclude, in particular, that

(d) Deduce tlle structure equations frorn (b) and (c). 21. (a) If o is a U -valued k-forrn and q is a V-valued I-forrn, and p: U x V + W, then d ( p ( o A $1) = p ( d o A 9) (-lIk p ( o A dq).

+

(b) For a pvalued k-foi-rn o and 1-forrn q we have

[ o A q] = (-l)kl+'[qA o]. (c) Moreover, if A is a 9-valued n7-forrn, then

22. Let G c GL(II,IR) be a Lje subgroup. The inclusion rnap G + GL(n, IR) + IRn2 will be denoted l y P (for "point"). Then d P is an IRn2-valued 1-forrn (it corresponds to the ideiltity rnap of the tangent space of G jnto itself). We can also consider d P as a rnatrix of 1 -forrns; it is just tlie rnatrix (dxij), where each dxij is restricted to the tangent bundle of G. \Ve also have the ~ " ~ - v a l u e d 1 -forrn (or rnatrix of 1 -forrns) P-' dP, wliere - denotes rnatrix rnultiplication, and P-' denotes the rnap A I-+A-' on G. (a) (b) (c) (d)

P-' - d P = p(P-] A dP), where p: 8"' x 8" + 8"' is rnatrix rnultiplication.

LA*dP= A dP. (Use f *d = df *.) P-' d P is left invariant; and (dP) P-' is right invariant. P-] d P is the natural Q-valued 1-forrn o on G. (It suffices to check that P-' d P = w at 1.) (e) Using d P = P o , show that O = d P - o P - d o , where the rnatrix of 2-foi-rns P . d o is cornputed by forrnally rnultiplying h e matrices of 1-forrns d P and o. Deduce tllar do+o.o=O. m

+

If w is the matrix of 1-forrns o = (oU), this says that

Check that tliese equations are equjvalent to the equations of structure (use the forrn d o ( X , Y) = -[o(X),o(Y)].)

23. Let G

c

GL(2, R) consjst of al1 matrices

(Ol)w i h

n

# O. ~ o conwr

iiieiice, denote the coordinates xll and x12 on GL(2, IR) by x and 1.. (a) Show that for the natural g-valued forrn o on G we have

so tliat dx/x and d y / s are IeA iilvarjant 1-forrns on G, and a left invariant 2-form is (dx A dy)/x2. (b) Find tlie structure constants for these forms. (c) Show that

and find tlle right iiivariant 2-forrns. 24. (a) Show that tlie natural gI(n, R)-valued 1-forrn o on GL(n, R) is giverl by ..

olJ=

(11)

1

det (xaB) k=i

ddx"'

Sliow that botli tlie leR and nght invariant ,12-forrns are rnultiples of 1

(dxJ1A . . . / \ d x n ' )

A ...A

(dxl" A . . . ~ d ~ " " ) .

(det(xa8))"

25. The special linear group SL(,z, R) deterrninant 1.

c GL(n, R) is the set of all matrices of

(a) Using Problern 15: sliow that its Lie a1gebi.a .r;l(~z, R) consists of al1 matrices with trace = 0.

(b) For the case of SL(2,R), sbow that

P

~

P

( -=uvddxx+- x yd du u

vdy - y dv -udy+xdv

where we use x, y, u, v for x", x12, x2', x ~ Check ~ . tliat the trace is O by djfferentiating the equation xv - yu = 1. (c) Show that a left invariant 3-forrn is

26. For M , N E o ( n ) = L(O(n)) = { M : M = -M'), define (N, M ) = - trace M N t . (a) ( , ) is a positive definite inner p~-oducton u(n). (b) If A E O ( n ) ,tliei~

(Ad(A) is defined in Problern 19.) (c) Tlie left inva-iant rnetric on O(!?) witli value ( , ) at invarian t.

is also right

27. (a) If G is a cornpact Lie group, then exp: g + G is onto. Hint: Use Proposition 21. (b) Let A E SL(2,R). Recall that A satisfies its cl-iaracteristic polyilornjal, so 1 -2. A 2 - (trace A)A + I = O. Conclude that trace (c) Show that the following elernent of SL(2, R) is not A 2 for aiiy A . Conclude that it is ilot iil the irnage of exp.

(d) SL(2, R) does not llave a bi-invariant rnetric.

28. Let x be a cooi-diilate systern around r in a Lie group G, let ~ r: G i xG+G be the projectioiis, a i ~ dlet ( y , z) be tlie coordiiiate systern arouiid (e, r ) given by y ' = x ' o i r i , z i = x i o i r i . Define#': G x G + R by

and let Xi be tlie left iilvariant vector field on G with

(a) Show that

(b) Using &Lb = Lab, s l i ~ wthat

Deduce that

xi (b)(xi

and then tliat

Letting

$ = (3;)

-

0

La) = @: (ab).

.

be tlic inverse rnatnx of @ = (y?:),

we can write

This equation (or arly of nurnerous tl-ijng equivalent to it) is known as Lde'sfirsi fundame~talUieorem. Tlie associativity of G is jrnplicitly contained in it, since we used the fact that L, Lb = Lab. (c) Prwe tlie conuene uf G e S jrsi funda~nenhl Ilreurem, which states the followiiig. Let # = (#', . . .,#") be a differentiable fuilction in a neighborhood of O E IR2" [with standard coordinate system . . , v",zl,. . . , s n ] such that #(a,o) = a

for a E IR".

Suppose thei-e are differciitiable functions @; in a iieighborhood of O standard coordinate system rl,.. .,xn] sucli tliat

E

IRn [with

$j(O) = sj n

(*)

a#' m("7b)=C@:(4(a,b)).$;(b) i= l

Toi. (a, b) in a neighborhood of O E Ik2".

Tlien ( a ,b ) M # ( a , b ) is a local Lie group structure on a neighborhood ofO E IRn (it is associative arid has inverses for points close enough to O, which serves as the ider-itity);the corresponding left invariant \rector fields are

[To prove associativity, note that

arid tlien sliow tliat # ( a ,# ( b , z)) satisfies the sarne equation.]

29. Lid secolrdfundonzenid tlteo~e~n states tl-iat tlie left invariant \rector fields Xi of a Lie group G satisfy n

for certain constoak ~ 6 - i n otlier words, tlie bracket of two left invariant vector fields is left invariant. The airn of this problern is to prove tl-ie ~o?¿uer.eof&3 secondfulzdo)nento2 tlteo~em,w1iicl-i states tl-icfollowing: A Lie algebra of vector fields on a rieigliborhood of O E IRn, which is of dirnension iz over IR and cor-itains a basis for RUo,is tlie set of left invariant \lector fields for sorne local Lie group structure on a rieighborhood of O E IRn. (a) Choose X I ,. . . ,X, in the Lie algebra so that Xi (O) = d/dxl l o arid set

tlien tlie oi are tlie dual rorrns, and coiisequently

(b) Let q : IR' x IRn + IR^ be tlie projections. Then

'

Consequeri tly? tlie ideal 9enerated by tlie forrns d (r ox2)(11.: ox2) xl*oi is rhe same as the ideal 9 eeiierated by the forms rn*oj-xI'oj. Using the faci iliat tlie cjk are coirstants: SIIOMIthat d(9) c 9. Hence W" x W" is foliated b\11-dimensional manifolds oii which tlie forms d ( x i o x2) (@/ O x2) xl*oi a H j1ar-iish . (c) ConcIude, as in tlie prooiof Theorem 17, that for fixed a,there is a iunct~on @, : W" + W" satisfying @,(O) = a and n

a (

n

b )=

~A-J

E@f ( @ o ( b ) )- i q b ) . i= l

Now ser # ( a , b) = @,(b): arid use tlie converse of Lie's first fuiidamental tlieorem .

30. Lie's tliirdjiudnn?cr~i~l 11zeoren~states that tlie C;k satisfy equations (1) aiid (2) on page 396, ;.e., tliat tlie left invariant vector fields form a Lie algebra undes [ , 1. The aim of tliis problem is to prove tlie cu:o,zverseo f h ' s tízirdfundamental :Acooi.ern. \vIricli states that any 17-dimerisional Lie algebra is the Lie a1gebz.a Porsorne l'ocal Lie group in a ~ieigliborhoodof O E IRX. Let C; be constants satisfyiiig equations (1) aiid (2) on page 396. We would like to find vector fields XI, . . . , X , on a neigl~borlioodof O E IR" such rhat [ X i . X,] = E;=, C[ X k . Equivalently, we want to fiiid forms o' witli

Tlierl tlie 1-esultwill follow from tlie conver-se of Lie's secorid fundamental tlieoreni. (a) Lec h4 be functions on W x IR" such that

h,k (O, X) = 0. Tliese ai-e equatioris "deperiding on the paranieters x" (see Problern 5-5 (b)). Notr tliat l$(r, O) = 6:r' so tliai h f ( i,O) = 6J. Let ok be tlie 1-form on W x WN defiried b!.

and wrile

d o k = hk + (di A a k ) .

where hk and a k do not involve d t . Show that

(b) Show that

(c) Let

Show that

+

terrns not involving d f.

Using

and equation (2) on page 396, show that

+

terrns not involving di

FinaUy deduce that

dek = dt

A

- E C; j,i

xjg1 + terrns not involving d t .

(d) MIe can wnie

wliere gk.f J (O.

.Y)

= O (Mrli!;?). Using (c), show ihai

Cor-iclude that 8' = 0. (e) M;e 11ow ha\:(-

CHAPTER 11 EXCURSION IN THE REALM OF ALGEBRAIC TOPOLOGY

T

liis cliapter exl~loresfurther properties of tlie de Rharn cohornolog-y \lector. spaces of a riianifold. Our rnain results will be restaternents, in terrns of thr. de Rharn colrornology of fundamental properties of the ordinary cohornolog? whicli is studied 111 algebraic topologv. Because we deal only with rnanifolds. rnar-iv of the proof3 I~ecornesignificantlv easier. On tlie other hand, we will be usir-ig sorne of tlie riiairi tools of algebraic topology rl-ius retaining rnuch of the flavor of rlia t subject. Along tl-ie wav \ye \vil1 deduce al1 sorts of in teresting consequences? irrcludirig a tl~eor-ernabou t tlie possil3ility of irnbeddirig ir-rnanifolds in ~ ? l + l Lct M be a nianiiold with M = U U 1/ ior open sets U . V C M. Before exarniiiing tlie coliornology of M we will sirnply look at tlie vec.tor s p c e ck( M ) of k-forrns orr M. Ler

l ~ etlie inclusions. T l ~ c nwe have two linear rnaps cu and

/?.

defiried b!

Her-e iw*(o) is just the restr-iction of o to U'. etc. Clearly /? o a = & ln other w ~ ~ 'iriiag~' d ~ I a C ker /?. h4oreover; tlic converse liolds: ker /?C iniage a. Foq if P ( h r , h 2 )= 0: [he11i r= A 2 on U n V. so we can define o on M to be h I on U and h2 on 1/. arld tllen a(w)= (AI,h2). The equation irnage ar = ker /?is expressed by sayirig tliat tlie above dia~i-arnis exact at tlie rniddle vector space. \Ve can esrend thic diasrarn by putting tlie vector-space coritaining orily O at thc

e~ids;rl~eai.r.o\trs at eitlier crid of the following sequerlce are tlie only possilil~linear rnaps. 1. LEh/lh/lA. The sequericc

is exacr at al1 places.

I'ROOF. lt is clear tliat a is one-one. Tliis is equi\~aleritto exactness at ck( M ) . sirice tlie irnage of the firsi rnap is (O) c ck( M ) . Sirnilarly, exactness at C k (U 11) is equi\,alerit to #l be ir^^ onro. To projle tliar /?is onto, let (#u, #VI be a partiiiori ol'uriity suliordiiiate to ( U , V ) . Tlien o E c k ( U n V ) is wl~er-e# v o denotes tlie Iorrn equal to #vo ori U U - ( U n V I . +>

n

1/, arld equal to O o11

By p u t t i r ~ir1~rlie rnaps (1. ~ - c cari expand our diagram as follows:

so rliat rlie

arme all exacr. lt is easy to check rhat rliis diagram cornrnuies. tliat is, any rwo cornpositions Irorn orie vector space to anotlier are equal: I-OM'S

Our first rnain theor~rndeperids onlv on tlie siinple algebraic structure inherent in this d i a ~ ~ - a r nTo . isolate this purelv al9ebr-aic structure, we rnakr* ilie following definitions. A cornplex C is a sequence of wctor spaces C" k = 0,1,2,. . . , togerller with a sequence of linear rnaps

satisfyilig dk+' o d A = 0: or biiefly, d 2 cornplexes is a sequerice of linear rnaps

=

O. A rnap a: CI + C2 briweeii

sucli that tlie followirig diagrarn cornrnu tes for al1 k .

Tlie rnost irnportaiit exarnples of cornplexes air obtained by clioosin~C" C' ( M ) foi. soine nianifold M, with d k tlie opei-atoi- d oii k-forins. Anotlier exarnple, irnplicir in our discussioii, is rlic direct sum C = Cl C2 of tmio coinplexes? defiiied 11).

l'or any cornplex C

IW

caii define tlie cohomology vector spaces of C by k

(')

ker clX dk-1

= image

Katuially, if C = (c'(M):, tlien H'(c) is just H ~ ( M ) .lf a : Ci + C2 is a rnap betwren cornplexes. tl~eriwe have a rnap. also denored by a.

To deíiiie a wc iiotr diat e17ery clerneiit of HQC,) is deterrniiied by sornr .Y E C? i'4tlt d d i k ( , ~ =) O. Cornrniitativity of [he above diagi-arn s1io1z.s tliat k k di"(s) = O, SO a k ( x ) detei-rniiies aii elernent of H ~ ( c ~ ) ) , d2 (a (x)) = a ~1hicl-i~ $ define. ~ e ro b r @([heclass deterinined bv A*). This rnap is ~vell-defined.

for if \ve cllange x to x + dik-' (y) for sorne y E Clk-', then lo ak(x dlk-i(B))= a k ( x ) ak(dik-'(y))

olk ((x)

is clianped

+

+

k

= a (x)

+ d2k - I ( f f k - l

(Y)).

wliich deiel-iniiies tlle sanie elerneiit of H k (4).\Ylien Clk = C k ( ~ )~2~ , = @((N). and a : C k ( M ) + C ~ N is ) f * for f :N + M, then this map is jusi I*: H ~ ( M + ) H~(N). h o w suppose that \\le havc an eiact seqlrence of cornplexes

ivl~iclii.eally ineaiis a vasi coinmutative diagrarn in ~+lliichall rows are exacr.

W i a i does iliis iniply aboui tlic rnaps a : HB(C1) + H P(

c ~alid ) B : H k (c?)3

Hk(C3)? Tlir iiiccsi tliiny iliat could liappeii would be for ihe followiiig diagrarn lo bc exact: a B O + H k ( C 1 )+ Hk(C'2) + HL(C'3) + O . l i s S o e . l:or csanildc, if U and V are oiei-lsppiiig poiíions of s2 IOI wliich ilier-e is a deforrnariori i.erractiori of U flV inio S', then we llave aii exacf sequen cc

but

nol

an exact sequencc

h-e\lertheless, sornetliing very nice is true: ff

B

2. THEOREh4. lf O + CI + C2 + C3 + O is a short exact scqucncc o i cornplexes, tlieii tliere are linear maps

so that the follo\ving infinitely long sequence is cxact (eveqwhere):

PROOE Throughout tlie proof, diagrarn (*) should be kept at hand. Let x E C3'< with d3'astlie same H' as a disjoint union of infinitely many copies of S'; tliis sliows that H 1 ( M )is infinite dimensional (see Roblem 7 for more infoi.niation about tlle cohomology of M). O n the other hand: 20. PROMSITION. If M has fiiiite type, then HP[ M ) and H;[M) are finite dimensional for al1 k .

PROOF. BY iriduction on tlle number- of open sets i. in a ~iicccover; It is clear for r = l . Suppose it is true for a certain r , and consider a nice cover. {U', . . .,U,, U] of M. T l ~ e ntlle tlleor-em is true for V = Ur U . - .U U, and

for U. It is also true for UnV, sii-ice thisl~asthe nicecover { U n U i , . . .,UnU;j. Now consider the Mayer-Vietoris sequence

The map o maps H k ( M ) onto a finite dimensional vector space, and the kernel of o is also finite dimensional. So H k ( M ) must be finite dimensional. Tlie pioof for H: (M) is similar, For ai-iy mariifold M we can define (see Problem 8-31) t l ~ ecup product map w

H ~ ( M x) H / ( M ) -+

H~+I(M)

by We can also define

by tlie same formula, since w A q Iias compact support if q does. Now suppose that M" is connected a r ~ doriented: witli orientation ,u. Tliere is tlien a uriiqur element of H:(M) repi-esented by any q E C , ( M ) witli

It is comler-iient to also use ,u to denote both tliis element of H:(M) and the isomorphism H:(M) + W which takes this element to 1 E R. Now every a E H" M) determines an element of the dual space H:-~ ( M)* by

We denote this clement of H:-'(M)' that we llave a map

by P ! ( o ) , the "Poincaré duai" of o, so

One of tlle fundamental tlreorems of manifold tl~eorystates that P ! is always an isomorpllisrn. We are al1 set up to prove this fact, but we shail restrict tlle tl~eoi-emto manifolds of finite type, in order not to plague ourselves with additional tecl~rlicaldetails. As witl~most big tl-ieorems of algebraic topology, tlle maiil part of the p o o f is called a Lemma, and the theorem itself is a simple corollan:

21. LEMMA. If M = U U V for open sets U and V and PD is an isomorpllism for al1 k on U, V: and U n V: then PD is also an isomorphism for al1 k on M.

PROOF. Let 1 = I r

- k.

Consider the following diagram, in wl~icht l ~ etop row is the Mayer-Vietor-is sequence, and the bottom row is the dual of the MayerVietoris sequence for compact supports.

By assuinption, al1 vertical maps, except possibly tlle middle one, are isomor-

pllisms. It is not hard to check (Problem 8) that every square in this diagram commutes up to sign, so tllat by changing some of tlle vertical isomor.pllisins to tlleir llegatives, we obtain a commutative diagiam. We now forget al1 abour our manifold and use a purely algebraic result . "TH E Fl VE LEh/lMA". Consider tlle followil~gcommutative diagram of vector spaces and linear- i-ilaps. Suppose that the rows are exact, and tllat #i, $2. are isomorpl~isms.Then #3 is also ail isomorpliism. 44,

PROOF. Supposc 4 3 ( ~=)O for some x E V3. Tllcn P3#3 (x) = O, so #4a3(x) = O. Hence a 3 ( x ) = O, sirlce #4 is an isomorpl-iism. By exactness at V3: tllere i5 1. E V2 witl-i A- = a2(y). Thus O = # 3 ( ~ - ) = #3a2 (y) = P2#2 (y). Hencc #2()*) = r91 ( z ) for some 5 E Wl . Moreover, := 41 (w) for some w E VI. Then

wllicll implies that y = al (w). Hence

So $3 is one-onc. Tllc ~ i ' o o tl-iat f #3 is onto is similar, and is left ro the reader. This proves thr, original Lemma.

22. THEOREM (THE POINCARE DUALITY THEOREM). ~f M is a connected or-iented n-manifold of finite type, then the map

PD: H ~ ( M + ) H:-~(M)* is an isomorpllism for ail k.

PROOE By induction on tl-ie number I. of open sets in a nice cover of M. The theorem is clearly true for r = 1. Suppose it ic true for a certain r, and consider a nice cover {Ui,.. .,U,, U) of M. Let V = U1 U . U U,. The theorem is true for U, V, and for U n V (as in the proof of Proposition 19). By the Lemma, it is true for M. Tllis completes the induction step. 4% 23. COROLLARY. If M is a connected oriented n-manifold of finite type, tllen H k ( M ) and N:-'(M) have the same dimension.

PROOE Use tlle Theoren~and fioposition 19, noting that V* is isomorphic to V if V is finite dimensional. 4% Even though tlie Poincaré Duaiity Tlleorem holds for manifolds which are not of finite type, Coi-ollary 23 does not. In fact, Problem 7 sliows that N i (IR2 - N) and H,' (IR2 - N) have dúrerent (infinite) dimensions. 24. COROLLARY. If M is a compact corlr-iected orientable n-manifold, then H k (M) and H " - ~( M ) have the same dimension. 25. COROLLARY. If M is a compact or-ierltable odd-dimensional manifold, then x ( M ) = 0.

PROOF. In the expression for x ( M ) , tlle terms

( - 1 1 ~dim H'(M) aiid

cancel in pairs. ++: A more iinrolved use of Poirlcaré duality will aleritually allow us to say much more about tlle Euler characteristic of any compact connected oriented manifold M''. ]Ve begirl by consideiing a smootll k-dimensional orientable vector bundle 6 = x : E + M over M. Orientatjons p for M and v for 4 give an orientation p v for the (n + k)-mar~ifoldE. since E is locally a product. If {Ui,.. . , U, 3 is a i~icecover of M by geodesically convex sets so small that each bundle cIUi is ti-ivial: thei~a slight modification of the proof for Lemma 19

sliows that {ñ-' ( U ] ) ., . ., ñ l 1(U,)] is a nice cover of E , so E is a manifold of finite type. Notice also that for the maps

ño

s=

s oñ

identity of M

is smootllly homotopic to identity of E,

so ñ*: H ' ( M ) + H ' ( E ) is an isornorphism for al1 I . The hincaré dualih tl~eoremshows that there is a unique class U E H ~ ( Esuch ) that

This class U is called tlle Thom class of t. Our first goal will be to find a simpler property to characterize U. Let Fp = ñ-' ( p )be the fibre of 6 over any point p E M , and let jp : Fp + E be the inclusion map. Silice jp is proper, there is an element jp*U E H:(Fp). 0 1 1 the other hand, the orientation v for determines an orientation vp for Fp: and hence an element vp E H: ( F ~ ) .


. Theorem 7-14 sllows tllat

Now, on tlle one hand we have (Problem 8-1 7)

O n tlie otller llaild, we claim tllat tl-ie last integral in (3) is O. To prove tllis, i i ~.learlysuffices to prove tllat tlle integral is O over A' x Fp for any closed ball A' c A . Since x*p A cih = fd ( x * p A dh).

(5)

J

x*p ~

d =h

A' x Fp

S

d(x*p~h)

A' x Fp

a'pnh

=f

where n*p A A has compactsupporton A' x Fp by (2) bv Stokes' Tl-ieorem

a A ' x Fl,

= o.

becaiisc tlle form x*p A h is cleai-ly O on a A' x Fp (sincc a A' is (n - 1)-dimeiisional). Conlbii~ing(3), (41, (5) we see tllai

This shows that JF jpiois independent of p, for p E A . Using connectedness. P it is easy to see that it is inde~endentof p for al1 p E M, so we will denote it simply by JF j'o. Thus

Con-iparing with equation (l), and utilizing partitions of u n i g we conclude tl-iat

wliich proves t l ~ efirst part of tlie theorem. Now suppose we llave another class U' E H,k (E). Since

it follows tliat U' = cU for some c

E

R. Consequently,

Hence U' l ~ a sthe sarne propcrty as U only if c = 1. 4 3 The Tl-iom class U of $ = n : E + M can now be used to determine an element of Hk(M). Let S : M + E be any section; tliere always is one (namel): i11e O-sectior-i)and ar-iv two are clearly smoothly liomotopic. We define t l ~ Euler e class ~ ( $ E1 H k ( ~of) 6 h\.

R'otice that if 6 has a non-zero section S: M + E, and o E C;(E) repr-esents U, then a suitable multiple c . s of S takes M to the complement of support o. Hence, in tllis case

T l ~ termiiiology e "Euler class" is connected witli the special case of the bundle TM, wliose sections are, of course, vector fields ori M. If X is a vector field on M wl-iich has aii isolaced O at some point p (tl~atis, X ( p ) = 0, but X(q) # O for q # p iri a rieighborl-iood of p), tl~en,quite independently of our- previous considei.atioi~s,\ve can define an "ii~dex" of X at p. Consider first a vectoi-

field X on an open set U c Wn with an isolared zero at O E U. We can define a function fx: U - {O) + S"-' by fx(p) = X(p)/lX(p)l. If i: S"-' + U is i(p) = EP, mapping S"-' into U, then the map fx o i : S"-' + S"-' has a certain degree; it is independent of E, for small E , since the maps i ~ i2 , : S"-' + U corresponding to E] and ~2 will be smoothly homotopic. This degree is called the index of X at O.

index O

index O

index

-1

index 1

index I

index -2

index 2

iildex 1 in R"

index (-1)"

Now consider a diffeom~r~hism 8: U + V h,X is the vector field on V with

in Rn

c Wn with h(0) = O.

Recall thar

Clearly O is also an isolated zero of h , X

27. LEMMA. If h : U + V c R" is a diffeomorpllism with hIO) = 0, and X has ai-i isolated O at O, tllen tlle ii-idex of Ir,X at O equals the index of X at O.

PROOF. Suppose first that h is orientation preserving. Define U : IR" x [0,1]+ Rn

This is a smooth homotopy; to prove that it is smooth at O we use Lemma 3-2 (compare Problem 3-32). Each map U, = A- t+ H I x , t ) is clearly a dfleomorphism, O < r < 1. Note that U i E SO(n), since h is orientation preserving. There is also a smooth homotopy {Ut], 1 _< I < 2 with each Ht E SO(n) and Hz = identity, since SO(n) is connected. So (see koblem 8-25), tlle map h is smoothly llomotopic to the identity, via maps wliich are dfieomorphisms. This sliows that fh,x is smoothly homotopic to fx on a sufficiently small region of IRn - { O ] . Hence the degree of _fh,x o i is the same as the degree of fx o i. To deal witll non-orientation preseniing h, it obviously suffices to check the theorem for h ( x ) = ( x l , .. . ,xn-l, -xn). In this case

which shows that degree

h , i~= O

degree fx

o i.

43

As a consequence of Lemma 27, we can now define tlie iiidex of a vector field on a manifold. If X is a vector field on a manifold M , with an isolated zero at p E M , we clloose a coordinate system ( x ,U ) with x ( p ) = 0 , and define the index of X at p to be the index of x,X at O.

28. THEOREM. Let M be a compact connected manifold with an orientation F , whicll is, by definition, also an orientation for the taiigent bundle 6 = IT: TM + M . Let X : M + T M be a vector field with only a finite iiumber of zeros, and let a be the sum of the indices of X at these zeros. Then

PROOF. k t pl ,.. . , p, be t he zeros of X. Clloose disjoint coordiiiate systems ( U ] x, l ) , . . . ,(Ur,x r ) with xi(pi) = 0, and let

If w E CF(E) is a closed form representiilg the Thom class U of $, then we are trying to prove that f

\Ve can clearly suppose tliat X(q) # Support w for q #

Ui Bi. So

tlius it suffices to prove that X *(u)= index of X at pi.

(*)

It will be conueilient to drop the subscript i from now on. We can assume tliat T M is trivial over 3: so that n f l (B) can be identified with B x M,. Let j p aiid n2 llave tlie same meaning as in the proof of Tlieorem 26. Also clioose a iiorm 11 11 on M p . We can assume tllat uiider tlie identification of n-' (3)~ i t hB x Mp, the su1ipo1-1of wln-' (3)is contained iii {(y,u) : y E A , llvll 5 11. Recall fiom tlle proof of Theorem 26 tliat

Since we can assume tliat X(q)

support h for y

E

8 3 , we llave

O n tlle maiiifold Mp we llave jp* w = dp

p aii

[ I T - 1 )-form oii Mp (witll non-compact support).

If D c Mp is tlic unii disc (witli respect to tlle iiorni 8D c M,, tliei-r

- 1. -

11 11)

and S"-' deiiotes

by Tlieorem 26, and the fact tllat support jP*w C D.

Now, for q

and (31

E

B - {p}, we can define

f :aB + T M is smoothly homotopic to X :

S, x*x2*(jp*u)

=

al3 + T M . So

J, x * x 2 * d p X*x2*p

by Stokes' Theorem

From tlle defiilition of tlle iildex of a vector field, together with equation (2), it follows tllat

( x 2 0 Z * ) p = index of X at p. L B

Equations (1), (31, (4) togetlier imply (*). 43

29. COROLLARJ7. If X aiid Y are two vector fields witli only finitely many zems on a compact orieiitable maiiifold?tllen the sum of the indices of X equals tlie sum oi' tlie indices of Y. At tlie monicilt, we do not even kilow that there is a vector field on M with finitely nianv zeros, iloi- do we know what this constant sum of the indices is (althougli our terniiilology certainly suggests a good guess). To resolve these questions, we consider once again a triangulation of M . MJe can then find a vector field X witll just oile zero in each k-simplex of the triangulation. We begin by drawing tlie integral curves of X along the 1 -simplexes, with a zero at cacli O-simplex and at oiie point in eacll 1-siinplex. \)Ve then extend this picture

to ir~cludetlie integral cunres of X on the 2-simplexes, producing a zero at oní

point in cach of tliem. We tlien continue similarly until tlie n-simplexes are filled.

30. THEOREM (POINCARÉ-HOPQ. Tlie sum of tlie indices of t his vectoi. field (and Iience of aiiy vector field) on M is tlie Euler characteristic X( M ) . Tlius, for $ = n : TM + M we have x($) = x ( M ) p .

PROOF. At each O-simplex of tlie triangulation, the vector field looks like

with index 1. Now coiisider tlle \lector field in a iieigliborhood of tlie place where it is zero on a 1-simplex. The vector field looks like a vector field on W" = W 1 x IR"-' wliicli points djrecOy iiiwairls on W 1 x {O) and directly outwards on {O) x W"-' .

t

(b) j? = 3

For 11 = 2, the index is clearly - l . To compute the index in general, we note that fx takes the "north pole" N = (0, . .. ,O, 1) to itself and no other point qoes to N. By Theorem 8-12 we just have to compute signN fx. NOWat N we can pick projection on Wn-' x (O) as the coordinate system. Along the inversc image of the x'-axis the vector field looks exacily like figure (a) above, where w already know the degree is - 1, so fx. takes the subspace of S"-'N consisting of tangent vectors to this cuive into the same subspace, in an orientation reversing way. Along the inverse image of the x2-, . . . ,xn-'-axes the vector field looks likc L

so fx. iakes the correspoiiding subpaces of S n - I N into themselves in an orientation preservjng way Thus signN fx = -1: which is tlierefore the index of the vector field. In general, aear a ze1.o within a k-simplex, X looks like a vector field on W" = IRk x Wn-hrliicli points directly inlvards on JRk x (0) and directly outwaids on (O) x IRn-'. The same argument shows that the index is ( - 1 )k. Consequently, the sum of the indices is a0

- al

+ a2 -

. = X ( M).

43

\Ve end tliis chapter with one more obsenlation, which we will need in the last chapter of Volume V ! Let 6 = n : E + M be a smooth oriented k-plane bundle over a compact conliected oriented 12-manifold M, and let ( , ) be a Riemannian metric for 6 . Then we can form tlie "associated disc bundle" and "associated sphere bundle"

D

=

(e : (e,e) 5 1 )

S = ( e : (e,e) = 1 ) .

+

It is easy to see tl-iat D is a compact oriented (iz k)-manifold, wjth aD = S; mor.eover, tlie D constructed for any other Rieniannian metric is diffeorn~r~liic to this one. We Iet no ; S + M be 171s.

31. THEOREM. A class a multiple of

E

H k ( ~satisfies ) no*(a)= O if and only if a is a

~(6).

PROOF. Consider the following picture. The top row is the exact sequence

Hk(M) for ( D ,S ) @ven by Theorem 13. The map S : M +- D - S is the O-section: while i: M + D is tlie same O-section. Note that everything commutes.

no*= i* o ( n1 D)* S*

= S* o C)

since no

= ( n I D )o i;

since extending a form to D does not affect its value on S ( M ) ,

and that

S* o ( n l D ) * = identity of H k ( M ) , since [n1 D ) o i is smootlily homotopic to the identity Now lei a E H ~ ( Msatisfy ) no*(a)= O. Then i*(n1 D)*a = O, so (nID)'a E iinage L.. Since D - S is diffeomorpl~icto E, aiid every element of H: ( D - S ) is a niultiple of the TIiom class U of t, we conclude that

Tlie proof of the coiiverst: is similar.

Iixirn:rio?i i?r lhr. Ren hn of. A Lyth?uic 7Új)ologl. C

PROBLEMS 1 . Find Hk(S' x dim H' = (;).]

.

x S') by induction on the number rr offactors. [Answei.:

2. (a) Use the Mayer-Vieioris sequence to determine Hk( M - { p ) )in terms of H k ( ~ )for , a connected manifold M. (b) If M and N are two connected 11-manifolds, let M # N be obtained by joining M and N as shown below. Find the c o h o m o l o ~of M # N in terms of that of M and lb'.

(c) find

x for ihe rt-holed torus.

[Answer: 2 - 21n.J

3. (a) Find ~ ~ ( A 4 o b i strip). us (b) Find Hk(lP2). (c) Iind Hk(P"). (Use Problem 1-15(b); it is necessary to consider wliether a neighborliood of BW-'iii B" is orientable 01. not.) [Answer: dim H k ( P " ) = 1 if k even and 5 11: = O othenuise.] (d) R n d Hk( ~ l e i nbot de). (e) Fiiid tlie c o l i o n i o l o ~of M # (Mobius strip) and M # (Kleiri bottle) if M is the n-l-ioled torus.

4. (a) Tlie fipr+eI~eIowis a triangulation of a rectangle. If we perform the indicated ident&catioris of edges we do ?iol obtain a triangulation of the torus. Why not?

(b) Tlie figure below does give a triangulation of the torus when sides are iden-

tified. Find ao, u ' , u2 for this triangulation; compare with Theorem 5 and Problem l .

5. (a) For any trianplation of a compact 2-manifold M, show that

(b) Sliow that for triangulations of

s2and tlie torus T~ = S'

x S' we have

Find iriangrilatioiis for wlijch iliese incq uaiities are al1 equalities. 6. (a) Tiiid H! (S"x R"') 11y iiiductioii oii n, using tlie Mayer-Vietoris sequenci for compact supports. (h) Use tlie exact sequeiice of ille pair [S" x Rm, { p ) x R m ) to compute the same vector spaces. (c) Compute H ~ S x" snl-' ), using Tlieorem 13.

7. (a) The vector space ~ ' ( - N)3may be described as tlie set of ail scqueiices of real numbers. Using the exact sequeiice of the pair ( R 2 ,N ) , show tliat H!(IR2 - N ) may l ~ ecorisidered as the set of al1 real sequeiices ( a , ) such tliat a, = O for al1 but fiiiitely many 1 1 . (h) Describe tlie map PD: H1(IR2- N ) + H,'(IR2 - N)* in terms of these desci-iptionsof H' (LR2 - N ) and H: ( IR2 - N ) . aiid show thai it is an isomoi-phism. (c) Clearly H:(LR2 - N ) has a countable basis. Show that H1(LR2 - N) does iiot. Hiiil: If vi = { u i j ) E H 1 ( R 2- N ) , choose ( b 1 , b 2 )E R2 linearly iiidependent of ( a 1 1 , a 1 2 )tllen ; choose ( b 3 ,b4,b s ) E R3 linearly independent of boih ( a l 3 a, l 4 ,a l S ) and ( í 1 2 3 , nz4, nz5);etc.

8. Show that the squares in the diagram in tlie proof of Lemma 21 commute, except for the squaw

wliich commutes up to tlie sign ( - I ) ~ . (It will be iiecessary to recal] howvarious maps are defined, which is a good exercise; the only slightly difficult maps are tlie ones involved in the above diagram.)

9. (a) Let M = M i U M2 U M3 U. - be a disjoint union of oriented n-manifolds. H: ( M i ) , this "direct sum" consisting of al1 sequences Show that H! ( M ) ( a l ,a 2 , a 3 , . . . ) with aj E H:(Mi) and al1 but finitely rnany ai = O E H$ ( M i ) . (b) Sliow that H k ( M ) % H k ( M ~ ) this : "direct product" consisting of nll sequences ( a l ,a2, a3, . .. ) with ai E H k ( M i ) . (c) Show that if tlie Poincaré duality theorem holds for each Mi, then it holds for M . (d) The figure below shows a decomposition of a triangulated 2-manifold into tliree open sets U*,U1, and U2.Use an analogous decomposition in n dimensions to prove that Poincarii duality holds for any triangulated manifold.

ni

U. is union of shaded

(-1)

U2is union of shaded

i\ .:.i;i:'

,,;,

10. Let 6 = rr : E + M and 6' = rr' : E' + M be oriented k-plane bundles. over a compact orieiited manifold M , and (f, f ) a bundle map from t' to t \vliicli is an isomorphism on each fibre. (a) If U E H: ( E ) and U' E H: ( E') are the Thom classes, then f * ( ~ = ) U'. (b) . [ * ( x ( c ) )= ~ ( 6 ' )(Using . tlie iiotation ofProbIem 3-23, we Iiave f * I x I E ) ) =

xrf*rt>>.) 11. (a) Let 6 = T : E + M be an oriented k-plar-ie bundle over an oiiented rnalijfold M , witli Tliom class U. Using Poincaré duality, prove tlie Tliom Isomorpliism Theorem: Tlie map H i ( E ) + H:+~( E ) @ven by a t+ a v U is an jsomoi-phism for a11 i . (11) Since we can also coiisider U as being in H k ( E ) , we can form U v U E H : ~ ( E ) .Using anticominutativity of A, sliow that tliis is O for k odd. Concludr tliat U represents O E H ~ E ) so ) , that ~ ( = t 0.) It follows, in particular, tliai ~ ( =t O)wheii 6 = rr : TM + M foi- M of odd dimension, providing anothel proof that x ( M ) = O in tliis case.

12. IS a \rector field X has aii isolated singularity at p E M n , show that thc iiidex of - X at p js (-1)" times tlie index of X at p. This provides ariothelproof that x ( M ) = O for odd n. 13. (a) Let pr, . . .,pr E M. Usir-ig Problem 8-26, sl-iow that there is a subsei D c M diffeomor.pliic to the closed ball, sucli tliat al1 p; E interior D. (h) If M is compact, tlien tliere is a vector field X oli M with only one siiiyularitx (c) is a fact that a Cm niap j.: S"-' + Sn-' of degi-ee O is sinootlily Iioniotopic to a coiistant map. Usii-ig this, show tliat if x ( M ) = O, theii tliere is a lio~rliereO vector field on M . (d) lf M is connected ai-id not compact, then there is a nowl-iere O vector field ciii M . (Begiri wjth a tria1i~gu1atioi-i to obtaiii a vector field with a discrete set oj zeros. join tliese by a ray goirig to infirlity, enclose tliis ray in a cone, and pus11 ever-ything off to irifiriit\:j

(e) If M is a coririected riianjfold-witli-boiriidary.with aM # 0, tlien tliere is a rioivher-e zero vector field on M .

14. This Problem proves de Rliam's Theorem. Basic knowledge of singular col~omoIogyis rcquired. l e wiil denote tlie group of singular k-chains of X by Sk(X). For a manifold M, we let S r ( M ) denote the Cm singular k-chains, aiid let i : S r ( M ) + S k ( M ) be the inclusion. 11is not hard to show that theiac is a cliair-i map r : S k ( M ) + M ) so that r o i = identity of M), wliile i o r is chain homotopic to the identity of Sk(M ) [basically, r is approximatioi~ by a C m cbain]. Tliis mearis that we obtain the correct singular cohomology of M if we consider the complex H o m ( S r (M), R).

Sr(

Sr(

(a) If w is a closed k-fo~-mon M, let Hl(w) E H o m ( S r [ M ) , W) be

Show tliat Rli is a cllain iiiap from ( c ~ ( M ) ) to ( H o m ( S ~ ( M ) , R ) ) .(Hitii: Stokes' Tlieol-em.) It follows t hat t here is an iriduced map Rii from the de Rliam cohomology of M to tlie singular cohomology of M. (b) Sliow tliat Rii is aii isomorphism on a smootlily coritractible manifold (Lemmas 17, 18, and 19 will not be necessary for this.) (c) Imitate the proof of Theorem 21, using the Mayer-Vietoris sequence for singular. col-iomology, to show t hat if Rii is an isoniorphism for U, V , and U n V: tlieti it is an isomorphism for U U V. (d) Coiiclude that Rii is aii isomorpliism if M is of finite type. (Using the method of Problem 9: it follo\vs that Rii is an isomorpl-iism for any trianLgulated manifold.) (e) Check that tlie cup product defined using A corresponds to tlie cup product defined in singular cohomology.

APPENDTX A CHAPTER 1 Following the sugpestions in tl-iis chapcer: we wiII no-\vdefine a manifold to b i a topological space M such that (1) M is Hausdofi

(2) For eacli x E M there is a neighborliood U of x and an integer n such that U is homeomorphic to IR".

>O

Condition (1) is necessary, for there is eveii a l-dimensional "manifold" which is i-iot Hausdorff. l t consists of R U (*) where * # ~ 4 t htlie following topo lo^-\.: A set U is open if and only if

(1) U (2) If

n R is opeii:

* E U. thcn (U n R) U (O: is a neighborliood of O (in R). Thus tlie neigliborlioods of * look just like neigl~borhoodsof O. Tliis space mav also be obtained by idei~tif~ring al1 points except O in one copy of R with thr correspondirig point in anotlier copv of R. Alt l-iougli non-H ausdofl manifold? are important in certain cases, we ~rillnot consider tllem. We llave just seen that the Hausdoflproperty is not a "local property", but local compactiiess is, so every maiiifold is locally compact. hgoreover, a Hausdorfflocally compact space is regular. so every manifold is regular. (By the wa)i this argunient does i ~ owor-k t fol. "infiiiite dimensional" manifolds, which are loc a l ] like ~ Banacli spaces; these iieed iiot be regular even if they are Hausdofl.) O n tlie otlier liaild, there are manifolds whicli are not iiormal (Problem 6). E\:ery mariifold is also clearly locally coniiected, so e v e q component is open, and ihiis a mariifold iiself. Before exhibitjiis noii-rnetrizable manifolds, we first note tl-iat alnlost al1 "nice" propertie.; of a manifold are equivalerit. THEOREhl. The following pr-operties are equivalei-itfor any manifold M: (a) Each cornponent of M is a-compact.

(b) Eacli coinpoiieilt of M is secoild coui-itable (lias a countablc base for t he t o p o l o ~. j (c) M is metrizable. (d) M is paracompact. (In particiilai: a compact mariifold is metriza ble.)

FIRST PROOF. (ai a (b) foIlo\m immediately fi-om tlie simple propositioil tliai a o-compact IocaIlv second countable space is second countable. (11) a (e) follo-\vsfi-om tlie Urysohn metrization tlieorem. (cj 4 (d) because any metric space is paiaacoinpact (Kelley. Ckieraf 7Üpofog.i.. pg. 160). TIie second pr-oof does iiot rely on this dificult theorem. (dj + (a) is a consequeiice of the follouliii?, LEMMA. A coiiiiecied. locally coinpact, ~~ai.;icoiiipact space is o-compaci.

Ao@ T1iei.e is a locally fiiiite covei' of tlie space by open sets with coinpact closure. If U. is oiie ofthese. then U. can interseci oiily a finite numl~ei-L'i, . . . ,L',,, of ilie otbers. Siiiiilai.ly ÜoU Üi U - - U U,I, iiiiersects only Un,+i.. . . ,U,,: and so on. The uiiioii

is clcarl!; opeil. It is also closed, foi- if A- is i i l ilir closure! then x níiisi be i i i the closure of a finiie iiilion of these Ui. becausc s has a neighboi.liood \vliich iiitei.sects oiily fiilitel!~niany. Tlius x is iii the uilioii. Siilce the spacc js coililecied, ii equals this couiitable unioii oi' coiilpact sets. Tliis proIres tlie Leninia aild the Tlieorcm.

SECO.ArR l'R001': (a) d (b) 3 (c) aiid (d) d (a) as befoi-e. (c) a (a) is Theorem 1-2. . ~ Ci is conlpact. Clcai-lv C1 ha' (a) d (d). Le1 M = C1 U C7 U . . \ v l ~ c ieach an opeii iieiglil~orlioodUi \r?iili compaci closure. Tlieii Ül U C2 lias a11ol~eil iiriyliborhood fi wjtll ('OiiilIact closui*e. Coniiniiiiig iii thjs way, we obtaiii open seis Ui witli Ui roiiqlaci aiid G. c Ui+i. whosr uiiioii coniaiiis al1 Ci. aiid lieiicr is M. It ir; easv to sliow li-onl thjs tliat M is pai-acoml~act.43 a

a

It iuriis out thai t l i ~ iaiLr - ~ eveii 1 -mailifolds wliich a1.e noi paracoinpact. Thrcoilst ruction of ihrse exain~~lcis I-equires t11e ordiilal iiumbers, whicli are bricfl~. cxplaiiied herc.. (Oi.diiial iiiiiii11ei.s will iioi 11c needed for a 2-diineiisioiial example io come latei: ORDINAL NUMBERS Recall tliat a n orderine < on a set A is a i~latioilsucli tliai

(1) a < b and h < í-implies a < c for al1 a , b, r, E A (transitivit?.~

(2) For al1 a , b E A : one aild onlv one of the follo1~41ig holds: (i) a = h (ii) a < (iii) b < a (also written cr > h q

(trichotom?~).

An ordered set is iust a pair ( A : i ) wliere < is an orclering;on A . 1-1410 oi-dered sets ( A ,