Differentiable Manifolds: A First Course

C I I I I C B I U\B B Y w w m M-XedTexts BaW LehrtxlcM Lawrence Conlon Differentiable Manifolds A First Course ...

Author: Lawrence Conlon

135 downloads 1461 Views 18MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

C

I

I

I

I

C B I U\B B Y w w m

M-XedTexts BaW LehrtxlcM

Lawrence Conlon Differentiable Manifolds A First Course

Birkhauser Boston Base1 Berlin

Lawrence Conlon Department of Mathematics Washington University St. Louis, MO 631 30-4899 USA

]Library of Congress Cataloging-in-Publiation Data Conlon, Lawrence,1933Differentiable manifolds : a first course / Lawrence Conlon. -- Boston ; Basel ;Berlin : Birkhguser, 1993 (Basler Lehrbticher, a series of advanced textbooks in mathematics; Vol. 5) p. cm. Includes bibliographical references. ISBN 0-8176-3626-9 (hard : acid-free). -- ISBN 3-7643-3626-9 (hard : acid-free) 1. Differentiable manifolds. I. Title. QA614.3.C66 1992 92-31098 CIP 5 16.3'64~20 Printed on acid-free paper O Birkhiuser Boston 1993 Copyright is not claimed for works of U.S. Government employees. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission of the copyright owner. Permission to photocopy for internal or personal use of specific clients is granted by Birkhiuser Boston for libraries and other users registered with the Copyright Clearance Center (CCC), provided that the base fee of $6.00 per copy, plus $0.20 per page is paid directly to CCC, 21 Congress Street, Salem, MA 01970, U.S.A. Special requests should be addressed directly to B W u s e r Boston, 675 Massachusetts Avenue, Cambridge, MA 02139, U.S.A. ISBN 0-8176-3626-9 ISBN 3-7643-3626-9 Typeset by the Author in AMS-uTEX. Printed and bound by Quinn-Woodbine, Woodbine, NJ. Printed in the U.S.A. 9 8 7 6 5 4 3 2 1

This book is dedicated to my wife Jackie, with much love.

TABLE OF CONTENTS

Preface

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

Acknowledgments

xi

... . . . . . . . . . . . . . . . . . . . . . . . xi11

.

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

1

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

1

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

3

Chapter 1 Topological Manifolds 1.1. Locally Euclidean Spaces

1.2. Manifolds

. . . . . . . . . .7

1.3. Quotient Constructions and 2-Manifolds 1.4. Partitions of Unity

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

15

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

18

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

20

1.5. Imbeddings and Immersions 1.6. Manifolds with Boundary

Chapter 2. Local Theory

. . . . . . . . . . . . . . . . . 25

2.1. Differentiability Classes

25

2.2. Tangent Vectors

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

27

. . . . . . . . . .

34

2.4. Diffeomorphisms and Maps of Constant Rank

38

2.5. Smooth Submanifolds of Euclidean Space

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

42

2.3. Smooth Maps and Their Differentials

2.6. Constructions of Smooth Functions 2.7. Smooth Vector Fields 2.8. Local Flows

. . . . . . . . . . . 47

. . . . . . . . . . . . . . . . . 50

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

55

. . . . . . . . . . .

64

2.9. Critical Points and Critical Values

.

Chapter 3 Global Theory

. . . . . . . . . . . . . . . . 67

3.1. Smooth Manifolds and Mappings

3.2. Diffeomorphic Structures 3.3. The Tangent Bundle

. . . . . . . . . . . . 67

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

73

. . . . . . . . . . . . . . . . . . 74

3.4. Cocycles and Geometric Structures

. . . . . . . . . . .

78

CONTENTS

........

3.5. Global Constructions of Smooth Functions

84

3.6. Manifolds with Boundary

87

3.7. Submanifolds

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

91

3.8. Homotopy and Isotopy

93

3.9. Degree Theory

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

95

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

.

Chapter 4 Flows and Foliation

. . . . . 4.2. The Lie Bracket . . . 4.3. Commuting Flows 4.4. Foliations . . . . . . . 4.1. Complete Vector Fields

.

Chapter 5 Lie Groups

. . . .

. . . .

. . . .

. . . . . . . . . . . . 101 . . . . . . . . . . . . 107 . . . . . . . . . . . . 110 . . . . . . . . . . . . 116

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

5.1. Basic Definitions and Facts

101

127

. . . . . . . . . . . . . . 127

. . . . . . . . . . . . . 136 . . . . . . . . . . . . . . . . . . . 139

5.2. Lie Subgroups and Subalgebras 5.3. Closed Subgroups

. . . . . . . . . . . . . . . . . 145 . . . . . . . . . . . . . . . . . . . 150

5.4. Homogeneous Spaces 5.5. Principal Bundles

.

Chapter 6 Covectors and 1-Forms

...........

. . . . . . . . . . . . . 6.2. Line Integrals 6.3. The First Cohomology Space . . 6.4. Some Topological Applications .

. . . .

. 159 . 167 . 173 . 180

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

189

6.1. The Space of 1-Forms

.

Chapter 7 Multilinear Algebra

. . . . . . . 7.2. Exterior Algebra . . . . . . 7.3. Symmetric Algebra . . . . . 7.4. Multilinear Bundle Theory . . 7.5. The Module of Sections . . . 7.1. Tensor Algebra

. . . . .

. . . . . . . . .

. . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

159

. . . . . . . . . .189 . . . . . . . . . . 199 . . . . . . . . . . 207 . . . . . . . . . . 209 . . . . . . . . . . 212

CONTENTS

ix

........

.

Chapter 8 Integration and Cohomology

221

. . . . . . . . . . . . . . . . 221 . . . . . . . . 228 8.2. Stokes' Theorem and Singular Homology 8.3. The Poincar6 Lemma . . . . . . . . . . . . . . . . . 243 8.4. Exact Sequences . . . . . . . . . . . . . . . . . . . 249 8.5. Mayer-Vietoris Sequences . . . . . . . . . . . . . . . 252 8.6. Computations of Cohomology . . . . . . . . . . . . . 257 8.7. Degree Theory . . . . . . . . . . . . . . . . . . . . 260 8.8. Poincar6 Duality . . . . . . . . . . . . . . . . . . .263 8.9. The de Rham Theorem . . . . . . . . . . . . . . . .268 8.1. The Exterior Derivative

.

Chapter 9 Forms and Foliations

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

277

. . . . . . . . . . . 277 9.2. The Normal Bundle and Transversality . . . . . . . . . 281 9.3. Closed, Nonsingular 1-Forms . . . . . . . . . . . . . . 286 9.1. The Frobenius Theorem Revisited

.

Chapter 10 Riemannian Geometry

...........

. . . . . . . . . . . . 10.2. Riemannian Manifolds . . . . . . . . . . . . . . . . 10.3. Gauss Curvature 10.4. Complete Riemannian Manifolds . . . 10.5. Geodesic Convexity . . . . . . . . 10.1. Connections

10.6. The Cartan Structure Equations 10.7. Riemannian Homogeneous Spaces

. . . . .

. . . . .

293

. . . . . . .294 . . . . . . . 302 . . . . . . . 306 . . . . . . . 314 . . . . . . . 327

. . . . . . . . . . . . 331

. . . . . . . . . . . 344

. . . . . . . . . . . . 349 . Appendix B . Inverse Function Theorem . . . . . . . . . . . 353 Appendix C. Ordinary Differential Equations . . . . . . . . 359 Appendix A Vector Fields on Spheres

C.1. Existence and uniqueness of solutions C.2. A digression concerning Banach spaces

. . . . . . . . . . 360

. . . . . . . . . 362

CONTENTS

C.3. Smooth dependence on initial conditions C.4. The Linear Case

. . . . . . . . . . . . . . . . . . .366

. Appendix E. de Ftharn-cech

Appendix D Sard9sTheorem

E.1. ~ e c cohomology h

. . . . . . . . . . . . . . . . 367

. . . . . . . . . . . . 371 . . . . . . . . . . . . . . . . . . . 371 Theorem

E.2. The de ~ h a m x e c hcomplex Bibliography Index

. . . . . . . . 364

. . . . . . . . . . . . . . 375

. . . . . . . . . . . . . . . . . . . . . . . . .383

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 385

PREFACE

This book is based on the full year Ph.D. qualifying course on differentiable manifolds, global calculus, differential geometry, and related topics, given by the author at Washington University several times over a twenty year period. It is addressed primarily to second year graduate students and well prepared first year students. Presupposed is a good grounding in general topology and modern algebra, especially linear algebra and the analogous theory of modules over a commutative, unitary ring. Although billed as a "first course", the book is not intended to be an overly sketchy introduction. Mastery of this material should prepare the student for advanced topics courses and seminars in differential topology and geometry. There are certain basic themes of which the reader should be aware. The first concerns the role of differentiation as a process of linear approximation of nonlinear problems. The well understood methods of linear algebra are then applied to the resulting linear problem and, where possible, the results are reinterpreted in terms of the original nonlinear problem. The process of solving differential equations ( 2 . e., integration) is the reverse of differentiation. It reassembles an infinite array of linear approximations, resulting from differentiation, into the original nonlinear data. This is the principal tool for the reinterpretation of the linear algebra results referred to above. It is expected that the reader has been exposed to the above processes in the setting of Euclidean spaces, at least in low dimensions. This is what we will refer to as local calculus, characterized by explicit computations in a fixed coordinate system. The concept of a "differentiable manifold" provides the setting for global calculus, characterized (where possible) by coordinate-free procedures. Where (as is often the case) coordinate-free procedures are not feasible, we will be forced to use local coordinates which vary from region to region of the manifold. When theorems are proven in this way, it becomes necessaxy to show independence of the choice of coordinates. The way in which these local reference frames fit together globally can be extremely complicated, giving rise to problems of a topological nature. In the global theory, geometric topology becomes an essential feature.

xii

PREFACE

These themes of linearization, (re)integration, and global versus local will be emphasized repeatedly. Although a certain familiarity with the local theory is presupposed, we will try to reformulate that theory in a more o r g a n i d and conceptual way that will make it easier to treat the global theory. Thus, this book includes a modem treatment of the elements of multivariable calculus. Fundamental to the global theory of differentiable manifolds is the concept of a vector bundle. The easiest nontrivial example is the tangent bundle to the n-sphere which we introduce from a purely topological point of view in Section 1.2. The subtleties involved in this bundle are illustrated in the discussion of the vector field problem on spheres in Section 1.2 and Appendix A. As the global theory is developed, the tangent bundle, the cotangent bundle, various tensor bundles, and the associated (principal) frame bundles will play increasingly important roles, as will the related notions of infinitesimal G-structures and integrable G-structures. For conceptual simplicity, all manifolds, functions, bundles, vector fields, Lie groups, homogeneous spaces, etc., will be smooth of class Coo. It is possible to adapt the treatment to smoothness of class Ck, 1 5 k < 00,but the technical problems that arise are distracting and the usefulness of this level of generality is limited. On the other hand, in much of the literature, the study of Lie groups and homogeneous spaces is carried out in the real analytic (CW)category. In these treatments, it is customary to note that C" groups can be proven to be analytic, hence that no generality is lost. It seems to the author, however, that nothing would be gained by this approach and that the ideal of keeping this book as self contained as possible would be compromised. Finally, the proofk of certain theorems have been relegated to appendices. Two of these, the inverse function theorem and the uniqueness, existence, and smoothness of solutions of systems of ordinary differential equations, are fund* mental and will be used repeatedly. Their proofs, however, are relatively long, add little insight, and can be distracting in a limited set of lectures. The remaining appendices establish results that are appealed to in the book, but are somewhat less central. These are Sard's theorem, the de R.ham-&~h cohomology theorem, and the Radon-Hurwitz-Eckmann construction of independent vector fields on spheres.

ACKNOWLEDGMENTS I would like to thank Robby Gardner and his students at Chapel Hill who "beta tested" eight chapters of a preliminary version of this book in an intensive, onesemester graduate course. Their many suggestions were most helpful in the find revisions. Others whose input was helpful include Gary Jensen, Alberto Candel, and Nicola Arcozzi. Finally, I particularly want to thank Filippo De Mari, whose beautiful class notes, written when he was one of my students in an earlier version of the course, were useful in subsequent revisions and first suggested to me the idea of writing this book.

CHAPTER 1 Topological Manifolds

This chapter pertains to the global theory of manifolds. See also [3, Chapter I] and [38, Chapter 11.

1.1. Locally Euclidean Spaces Classical analysis is carried out in Euclidean space, the operations being defined by local formulas. One might hope, therefore, to extend this classical theory to all topological spaces that are locally Euclidean. While this is not generally possible without further restrictions on the spaces, the locally Euclidean condition is fundamental. 1.1.1. A topological space X is locally Euclidean if, for every DEFINITION x E X, 3 n 2 0 (an integer), an open neighborhood U C X of x, an open subset W & Rn and a homeomorphism cp : U + W .

If we can show that n is uniquely determined by x, we will write n = d ( x ) and call this the local dimension of X at x.

EXAMPLE1.1.2. Any open subset X R n is a locally Euclidean space that is also Hausdorff and 2nd countable. We will see that the local dimension is n at every x E X. EXAMPLE 1.1.3. Let X = R LI {*I, where * is a single point and 11 denotes disjoint union. Topologize this set so that a basis of open subsets V c X consists of the following: a If * 4 V , then V is open as a subset of R. If * E V, then 0 4 V and there is an open neighborhood W c R of 0 such that V = (W \ (0)) U {*I. This space is locally Euclidean and znd countable. It is not Hausdorff since every open neighborhood of * meets every open neighborhood of 0. In this case, the local dimension is everywhere 1, even a t * and at 0.

-

EXAMPLE 1.1.4. In [38, Appendix A], there is described a bizarre space called the long line. It is connected, Hausdorff, and locally Euclidean with d(x) 1,

2

1. TOPOLOGICAL MANIFOLDS

but it is not 2nd countable. In fact, the long line contains an uncountable family of disjoint, open intervals. The following difficult result, known as the Brouwer theorem on invariance of domain, is needed in order to show that local dimension is always well defined on locally Euclidean spaces. The proof will not be given. It is best carried out by the methods of algebraic topology [9, p. 3031, [37, p. 1991, [12, p. 1101. For amore classical proof, see (20, pp. 95-96]. In the theory of smooth manifolds, differential calculus reduces the appropriate analogue of this theorem to elementary linear algebra.

THEOREM 1.1.5 (BROUWER).If U & Rn is open and f ous and one to one, then f (U) is open in Rn.

: U -,Rn

is continu-

COROLLARY 1.1.6. If U C Rn and V & Rm are open subsets such that U is homeomorphic to V, then n = m .

# n, say, m < n . Define i : Wm i(xl,. . . ,xm) = (x1 , .. . ,xm,0,. . . ,O).

PROOF.Assume that m

L,

Rn by

v n-m

This map is continuous and one to one and i(Rm) is not open in Rn and does not even contain a subset that is open in Rn. By assumption, there is a h o m m morphism cp : U --, V, so the composition

is continuous and one to one. Also, U Rn is open, but f (U) = i(V) C i(Rm) cannot be open in Rn. This contradicts Theorem 1.1.5. [7 COROLLARY 1.1.7. If X is locally Euclidean, then the local dimension is a well defined, locally constant function d : X -,Z+.

PROOF. Let x E X and suppose that there are open neighborhoods U and V of x in X, together with open subsets @ E Rn, Rm, and homeomorphisms

Since V n U is open in X , it follows that

are inclusions of open subsets and, similarly, that $(V

n U) is open in Rm. But

is a homeomorphism, so m = n by the previous corollary. All assertions follow. I7 COROLLARY 1.1.8. If X is a connected, locally Euclidean space, then the local dimension d : X -,Z+ is a constant called the dimension of X .

1.2. MANIFOLDS

3

EXERCISES (1) Prove that each connected component of a locally Euclidean space X is an open subset of X. (2) Prove that a connected, locally Euclidean space X is path connected. (3) Give an example of a connected, 2nd countable, Hausdorff space that is not path connected. (4) Let X and Y be connected, locally Euclidean spaces of the same dimension. If f : X + Y is bijective and continuous, prove that f is a homeomorphism. (5) RRcall that a regular space X is one in which any proper, closed subset C c X and point x E X \ C can be separated by disjoint, open neighborhoods of each. Prove that a locally compact (in particular, a locally Euclidean), Hausdorff space must be regular.

1.2. Manifolds Some authors call any locally Euclidean space a manifold. It is more common, however, to require more. DEFINITION 1.2.1. A topological space X is a manifold of dimension n (an n-manifold) if (1) X is locally Euclidean and d(x) G n = dim X; (2) X is Hausdorff; (3) X is 2nd countable. Of the three examples in the previous section, only the open subsets of R n were manifolds. LEMMA1.2.2. If X is a compact, connected, rnetrizable space that is locally Euclidean, then X is an n-manifold, for some n E Z+. PROOF. Indeed, X is Hausdorff because it is metrizable. It is 2nd countable because it is locally Euclidean and compact. Being locally Euclidean and connected, X has constant local dimension. Here are some examples of manifolds.

I llvll = 1) is an n-manifold. EXAMPLE 1.2.3. The n-sphere Sn = {v f One way to see that it is locally Euclidean is by stereographic projection. Let p+,p- E Sn be the north and south poles, respectively. Then the stereographic projections (see Figure 1.1)

are homeomorphisms and {Sn \ { p - ),Sn \ {p+ )) is an open cover of Sn. Since Sn is compact and metrizable, it is an n-manifold.


FIGURE1.1. Stereographic projection from p +

EXAMPLE 1.2.4. If N is an n-manifold and M is an m-manifold, then N x M is an (n m)-manifold. Indeed, if (x, y) E N x M, let U be a neighborhood of x in N homeomorphic to an open subset of Rn and V a neighborhood of y in M homeomorphic to an open subset of Rm. Then, the neighborhood U x V N x M of (x, y) is homeomorphic to an open subset of Rn x Rm = Rn+m. Since M and N are Hausdorff and 2nd countable, so is N x M.

+

EXAMPLE 1.2.5. The n-torus 7

n factors

is an n-dimensiond manifold. Indeed, by Example 1.2.3, S1 is a 1-manifold and Example 1.2.4, applied successively, then implies that Tn is an n-manifold.

EXAMPLE 1.2.6. A vector w E Itn+'is defined to be tangent to Sn at v E Sn if w I v. This conforms to naive geometric intuition and can be seen to conform to the general definition of tangent vectors to differentiable manifolds (cf. Exercise (5) on page 38). In order to keep track of the point of tangency, we will denote this tangent vector by (v, w) E RnS1 x Rn+'. Thus, the set of all tangent vectors to Sn is T(Sn) = {(v, w)E Rn+' x Rnfl

I IIvII

= 1,w

Iv } .

This space is topologized as a subspace of Rn+l x Rn+l. We also define the continuous map p : T(Sn) + Sn by p(v, w) = v . Thus, p assigns to each tangent vector its point of tangency. For each vo E Sn, consider

1.2. MANIFOLDS

5

the set of all vectors tangent to Sn at vo. This is an n-dimensional vector space under the operations

This structure, p : T(Sn) -, Sn,is called the tangent bundle of Sn. The space T(Sn) is the total space of the bundle, the space Sn is the base space of the bundle, and p is the bundle projection. By a common abuse of terminology, the total space is often referred to as the tangent bundle itself. In Exercise (3),you are going to prove that T(Sn) is a 2n-manifold. DEFINITION 1.2.7. If U Snis an open subset, then T(Sn)IU = T(U) is the space p-l(U). Then p : T(U) + U, the restriction of p to T(U) T(Sn), is called the tangent bundle of U. DEFINITION 1.2.8. If U E Snis open, a vector field on U is a continuous map s : U -,T(U) such that p o s = idv.

A thorough understanding of the tangent bundle of Sn eluded topologists for several decades. For instance, it was long unknown what is the maximum number r(n) of vector fields s, : Sn -, T(Sn), 1 5 a 5 r(n), which are everywhere linearly independent. That is, we require that, for each u E Sn, the vectors {sl (v), . . . , s ~ (u)) ( ~ be ) linearly independent in T. (Sn) and that no set of r(n) +1 fields has this property. The problem of computing r(n) was known as the "vector field problem for spheres". DEFINITION 1.2.9. The sphere Sn is parallelizable if r(n) = n. This brings us to a striking example of global versus local properties. If Sn is parallelizable, Exercise (3) will imply that T(Sn) 2 Sn x Rn. For general n, this same exercise will show that the tangent bundle T ( S n ) is locally a Cartesian product of an open set U C Sn with Rn, but it is only globally such a product when Sn is parallelizable. Not every sphere is parallelizable. For instance, it has long been known (and will be proven in Section 8.7) that r (2n) = 0. This means that every vector field on S2"is somewhere zero. In the case of S2,this is sometimes stated facetiously as "you can't comb the hair on a billiard ball". It was also known for some time that S1,S3,and S7are parallelizable. The following was finally proven in the late 1950s [4], [27].

THEOREM 1.2.10 (BOTT AND MILNOR,KERVAIRE).The sphere Sn as p a d lelizable if and only zf n = 0,1,3, or 7. The case n = 0 is the trivial fact that the O-sphere So= {kl) c R admits 0 independent fields. There is an interesting relationship between Theorem 1.2.10 and the problem of defining a bilinear multiplication on R n without divisors of zero. Such a multiplication is a bilinear map

6


written p(u, w) = vw, such that uw

=0 H

u = 0 or w = 0.

THEOREM 1.2.11. There is a multiplication on Wn+' without divisors of zero if and only if n = 0,1,3, or 7. Indeed, R1 = R, R2 = C, and W4 = W (the quaternions). The multiplication on W8 is given by the Cayley numbers, a nonassociative division algebra whose elements are ordered pairs (x, y) of quaternions [40, pp. 10&109]. This proves the "if" in Theorem 1.2.11. Conversely, such a multiplication can be used to construct n everywhere independent vector fields on Sn (Exercise (6)), so the "only if" part of the theorem is given by Theorem 1.2.10. The full solution to the vector field problem for spheres was given by Frank Adams in the early 1960s [I], culminating a long history of research on that problem by several algebraic topologists. We state Adam1 result. Define the function p(n), n 2 1, by requiring that Sn-' admit p(n) - 1 everywhere linearly independent vector fields, but not p(n) such fields (thus, r(n) = p(n 1) - 1). Write each natural number n uniquely as

+

where r, c, d are nonnegative integers and c 5 3. This uses the unique factorization theorem and the division algorithm mod 4.

Remark that p(odd) = 1, since c = d = 0. This gives the classical result that every vector field on an even dimensional sphere is somewhere zero. The easier part of Adams' theorem is that Sn-' does admit at least 2C 8d - 1 independent vector fields. The original proof, using only linear algebra, was given by W o n and Hurwitz and, in 1942, an improved proof was given by Eckmann [8]. A detailed outline of a modern proof will be found in Appendix A. The harder part of the theorem, that there are at most 2C 8d - 1 such fields, is too advanced for this book.

+

+

EXERCISES (1) In Example 1.2.3, write down formulas for the stereographic projections .~rhand prove carefully that they are homeomorphisms. (2) Given vo E Sn, show that there is an open neighborhood U C Sn of uo and vector fields s j : U -,T(U), 1 5 i 5 n, such that

is a basis of the vector space Tu(Sn),V v E U.

(3) Let U c Sn be as in the previous exercise. Using that exercise, construct a continuous bijection cp : U x Rn -, T(U) and prove that p is a homeomorphism. (This is not very deep. You do not, for instance, need Theorem 1.1.5.) Using this, prove that T(Sn) is a 2n-manifold. Prove also that, for each u E U, the formula pv(w) = cp(v, w) defines an isomorphism cpu : Rn + Tv(Sn) of vector spaces.

1.3. QUOTIENT CONSTRUCTIONS

7

(4) Using Theorem 1.2.12, show that p(n) = n if and only if n = 1,2,4, or 8. This gives back Theorem 1.2.10. (5) Without appeal to Theorem 1.2.12, show that p(2) = 2. (6) If Rn+I admits a multiplication without divisors of zero, prove that Sn is pardelizable. 1.3. Quotient Constructions and 2-Manifolds

We continue the project of constructing manifolds. In this section, examples will be constructed using the quotient topology. Before giving a careful definition of quotient spaces, we look a t some intuitive examples of 2-manifolds constructed in this way. Consider the square D = [O, 11 x [0, 11. In Figure 1.2, the arrows indicate that the opposite sides of D are to be glued together so that the vertical edges are glued bottom to top and the horizontal edges from left to right. When the first pair of edges are glued, the result is a cylinder. When the second pair are glued, the cylinder becomes the 2-torus T 2= S1 x S1.

FIGURE 1.2. The 2-torus

In Figure 1.3, the vertical edges of D are identified in the same way, but the top and bottom are glued together in opposite senses. If the top and bottom are identified first, the result is a Mobius strip. It is not very easy, then, to picture the rest of the identification. If the left and right are identified first, the result is a cylinder whose boundary circles are then to be identified with an orientation reversing "flip". The resulting 2-manifold is called the "Klein bottle" K and can only be pictured in R3 if one allows the surface to intersect itself. Figure 1.4 is an attempt at such a picture, the self-intersection occurring along a circle. This circle of intersection corresponds to the horizontal line L and the circle c in Figure 1.3. In order to view K 2without self intersection, it is necessary to situate it in a 4dimensional framework. This is not as psychologically hopeless as it might seem.


FIGURE 1.3. The Klein bottle

FIGURE 1.4. K~ with self-intersection One can, for example, color K~ by shades of grey, varying continuously over the Klein bottle, in such a way that the circle c in Figure 1.3 has no points with the same shade as any point in the line 1. By continuously assigning numbers from 0 to 1 (lightest to darkest) t o these shades, one introduces a fourth "dimension" and the shaded Klein bottle has no self intersections. What was the circle of intersection is now two disjoint shaded circles l and c. One can think of the shaded Klein bottle as a topologically imbedded Klein bottle in R4. In Figure 1.5, we identify each pair of opposite sides of D with a reverse of orientation. The resulting 2-manifold is called the projective plane P and, once again, it cannot be imbedded in R3.The topologist (but not the geometer) can view D as the unit disk D 2 = {v E R2 1 llvll 5 1 ) in such a way that the gluing identifies antipodal pairs of boundary points. That is, if J J v J=J 1, then v and -v are identified. Another way to think of P2 is to start with S2 C IR3 and to identify the antipodal pairs { w ,-w). To see that this also yields P2,first carry out the identification for the antipodal pairs not lying on the equatorial circle z = 0. The resulting disk has the equatorial circle as boundary and the remaining identifications give the previous description of P 2. DEFINITION 1.3.1. Let MI and M2 be 2-manifolds and let Di C Mi be imbedded disks, i = 1,2. Let Mi' = Mi \ int(Di), i = 1,2, and glue these together


FIGURE 1.5. The projective plane by a homeomorphism of a M i = aDl to aMi = a D 2 . The resulting 2-manifold Ml # M2 is called the connected sum of M I and M2. When this definition has been put on a rigorous setting, it can be shown that the connected sum is well defined up to homeomorphism. Thus, strictly speaking, the symbols M I , M2, and MI # M2 should denote homeomorphism classes of 2-manifolds. With this understanding, connected sum can be seen to be commutative, Ml#M2 = M2#Ml, associative, (Ml ttM2) #M3 = M l # (M2 # M 3 ) , (hence parentheses can be dropped) and to admit (the homeomorphism class of) S2 as a 2sided identity

In this way, the set of (homeomorphism classes of) surfaces becomes an abelian semigroup. Of particular interest is the subsemigroup of compact, connected surfaces. THEOREM1.3.2. The semigroup of compact, connected 2-manifolds is generated by T~ and P 2 . Indeed, every element other than the identity S 2 can be uniquely written as a connected sum of finitely many copies of T 2 or of finitely many copies of P 2 . This is the classification theorem for compact, connected surfaces. It is a classical result. For a proof of this classification, see Massey [26, Chapter 11. He also shows that

10


Thus, the connected sum operation does not have cancellation. An important tool in the proof of Theorem 1.3.2 is a "triangulation" of a compact surface M. Tkiangulations are useful for many purposes, so we give a brief discussion. Let A c It2 be the convex hull of the points vo = (0, O), vl = (1,0), and v2 = (0,l). That is,

a closed, triangular region in the plane having vertices (vo, vl, v2) and edges eo = {(x, y E A) I x + y = 1) el = ((0, y) E A) e2 = {(x, 0) E A). The triangle A C Kt2 is called the "standard Z-simplex". One decomposes M into pieces Al ,A2, . . . ,A,, together with homeomorphisms

in such a way that any two of the triangles Ai, Aj are either disjoint, have in common just one vertex v i ( V t ) = vj (vk), or have in common just one edge Pi(et) = vj(ek). Such a decomposition is called a triangulation of M. THEOREM1.3.3 ( R A D ~ )Every . compact surface M admits a triangulation. Equivalently, M can be constructed, up to homeomorphism, by taking finitely many copies of the standani 2-simplex A and gluing them together appropriately along edges. This theorem is intuitive but nontrivial [31, pages 58-64]. The first proof was given by T. Rad6 [36]. The standard 2-simplex is "oriented" by a choice of ordering of its three vertices. Two orderings give equivalent orientations if they differ by an even permutation. Thus, up to equivalence, there are two orientations of A. Such an orientation induces a direction along the edges of A, leading to the terminology "clockwise orientation" and "counterclockwise" orientat ion. The standard orientation of A, given by the ordering (vo, vl ,v2), is the counterclockwise orientation. Note that the homeomorphism a : A + A, defined by a(x, y) = (y, x) is orientation reversing. Triangulations give us a way of defining the notion of "orientability" for a compact surface. The idea is that a triangle A, = vi(A) has the standard orientation (9, (vo), vi(vl), 9, ( ~ 2 ) ) .TWOtriangles Ai and A j that have a common edge e are coherently oriented if their standard orientations induce opposite directions along e. A little thought should convince the reader that this is the natural notion if we are to picture the orientations of both A and A j as counterclockwise. If A, and A j have no edge in common, they are also said to be coherently oriented. Given a triangulation of a connected surface M , one can attempt to make all the orientations coherent as follows. Starting with one triangle, say A 1, look at any triangle, say A2, having an edge in common with A1. If they are coherently oriented, well and good. If not, replace 9 2 with pa o a, making them


Vl

FIGURE 1.6. A triangulation of S2 coherently oriented. Continuing in this way either orients the triangulation or leads to conflicting orientations on at least one simplex. In the first case, the triangulation is said to be orientable and, in the second, to be nonorientable. If M has more than one component, each must be treated separately and the triangulation of M is orientable if and only if the triangulation of each component is orientable. It can be proven that some triangulation of M is orientable if and only if every triangulation of M is orientable. DEFINITION 1.3.4. We say that M is orientable if some, hence every, triangulation of M is orientable. EXAMPLE1.3.5. The sphere S2 is orientable. Indeed, define the standard h i m p l e x A3 c W3 to be the convex hull of the set of points

That is, A ~ = { ( X , ~ , IZx), y , z ~ O a n d x + y + z 6 1 ) . Any three of the vertices {vo,vl, v2, v3) correspond to a triangular face of A3. These four triangular faces unite to form a surface homeomorphic to S', defining thereby a triangulation of S2 (Figure 1.6). Orienting these faces by (vl ,v2, us), (vO,v2, vl), (213, 212, vo), and (us, vo, vl ), respectively, gives coherent orientations to all the triangles. An interesting invariant of surfaces is the Euler characteristic. If the compact surface M has a triangulation 7, let V denote the number of points of M that are vertices of the triangulation, E the number of arcs in M that are edges of the triangulation, and F the number of triangular faces.

+

THEOREM1.3.6. The number F - E V depends only on M , not on the choice of triangulation 7. This number is called the Euler characteristic of M and is denoted x(M).


12

This theorem has many proofs. One method, useful only for smoothly imbedded surfaces in R3, is to use the Gauss-Bonnet theorem, which asserts that the integral of the Gauss curvature over M is 27rx(M) ([16, p. 1111, [34, pp. 380-3821, [41, pp. 237-2391). EXAMPLE1.3.7. In the triangulation of S2given in Figure 1.6, V = 4, E = 6, and F = 4, so X(S2)= 2. In Exercise (4), you are invited to show directly that an arbitrary triangulation of S2always has F - E V = 2.

+

Our discussion of the topology of 2-manifolds belongs to the "cut and paste" brand of topology. To put such constructions on a rigorous footing, it is necessary to introduce quotient spaces. Let X be a topological space and let be an equivalence relation on X . That is, xNx,vxEX x-y*y-x x~yandy-Z+X-2. This relation partitions X into a collection {X,),Em of disjoint subspaces, the equivalence classes of -.

-

-

DEFINITION 1.3.8. The set {X,),fm of equivalence classes of X is called the quotient space of X modulo

and is denoted by X/-.

The surjection

is the map that assigns to each x f X its equivalence class ~ ( x E) X/-. A subset U & X/- is said to be open if and only if 7r-'(U) is open in X. It is trivial to check that these open sets constitute a topology on the quotient space X/-. This is the quotient topology, characterized as the largest (i. e., the finest) topology on XI- relative to which the canonical projection

is continuous.

-

EXAMPLE 1.3.9. Let X = W x (0, I), topologized as the disjoint union of two copies of R. Define an equivalence relation on X by setting (x,a) (y,P) if and only if either a = ,d and x = y, or a # P and x = y # 0. Thus {(O,O)) and {(0,1)) each constitute a distinct equivalence class, but all other classes are pairs { (x, 0) , (x, 1)) where x # 0. The quotient space X/- is the non-HausdorfT, locally Euclidean space given in Example 1.1.3. As the above example shows, even if X is Hausdorff, the quotient X/- may fail to be Hausdorff. LEMMA1.3.10. If the space X is compact, so is X/-.

PROOF.The continuous image 7r(X) = X/- of a compact space X is compact. LEMMA1.3.11. If the space X is connected, so is X/-o


13

PROOF. The continuous image 7r(X) = XI- of a connected space X is connected. [3 DEFINITION 1.3.12. A map f : X -* Y respects an equivalence relation X if x y + f (x) = f (y). In this case, the induced map

-

on

is well defined by f(r(x))= f ( 4 .

-

LEMMA1.3.13. Let X and Y be topological spaces, let be an equivalence relation on X, and let f : X -* Y be a map respecting this equivalence relation. Then f is continuous i f and only i f f is continuous.

PROOF.Consider the commutative diagram

Since f = f o 7r, continuity of f implies continuity of f. For the converse, assume that f is continuous, hence that f -'(U) is open in X whenever U is open in Y. But this implies, via the commutative diagram, that

is an open subset of X. By the definition of the quotient topology, F 1 ( U ) is open in X/w, so f is continuous.

EXAMPLE 1.3.14. Let X = [-I, 11 and let Y

= S'

c @.

Define

-

by f ( t )= eint, a continuous surjection. It is not quite one to one since f (1) = f (-1). On X , define the equivalence relation x y by requiring either that x = y or that {x, y) = {f1). Clearly, f respects this relation. Denote X/- by [-I, I]/{& 1) and remark that is bijective. By Lemma 1.3.13, f, is also continuous. But [- 1,1]/{&1) is compact

by Lemma 1.3.10 and S1 is Hausdod. A one to one, continuous map from a compact space onto a Hausdod space is a homeomorphism, so f gives a canonical homeomorphism [-I, I]/{& 1) E S1. Generally, if A X , one can define the equivalence relation N A by writing x - A y if and only if either x = y or x, y E A. The quotient space X/ .v A is thought of as the result of crushing A to a single point in X and will be denoted by X/A. Some care should be taken in using this notation. If X = G is a topological group and H Z G is a subgroup, then G/H denotes the space of left cosets of H, not the space obtained by collapsing H alone to a point.

14


-

The context should make clear which interpretation is intended. Remark that G / H is also a set of equivalence classes, the relation being g 1 92 if and only if g;'g2 E H. Thus, G / H can be given the quotient topology and the continuous map T : G 4 G / H is ~ ( g=)g H .

DEFINITION 1.3.15. If G is a topological group and H E G is a subgroup, the quotient space G / H is called the (left) coset space of G mod H.

EXERCISES (1) Let Dn = {v E W n 1 1 1 ~ 1 1 5 I), the unit n-disk (also called the closed n-ball) with boundary aDn = Sn-I . Prove that Dn/Sn-l is h o m e morphic to Sn. (We proved the case n = 1 in Example 1.3.14.) (2) View W n as an abelian group under vector addition. This is a topological group. The integer lattice

is a (normal) subgroup of W n . Let T n = Rn/Zn be the coset space (actually, a group). Prove that this space is homeomorphic to

our definition of the n-torus in Example 1.2.5. Use this to show that the surface constructed by Figure 1.2 is, indeed, T2. (3) Define an equivalence relation on S n C Etn+l by writing v w if and only if v = f w . The quotient space P n = S n / - is called projective n-space. (This is one of the ways that we defined the projective plane P2.) The canonical projection .rr : Sn + Pn is just ~ ( v = ) {fv). Define Ui c P n , 1 5 i 5 n 1, by setting

-

+

Prove (a) U, is open in P n . (b) {Ul, - . - , Un+1) covers Pn. (c) There is a homeomorphism cp, : Ui + Rn. (d) Pn is compact, connected, and Hausdofl. This proves that P n is an n-manifold. (4) Give an intuitive, but completely convincing, proof that V - E F = 2 for every triangulation of S 2 . (5) Produce triangulations of the surfaces T2,K2, and P 2 . Use these to prove that T 2is orientable and that K 2 and p2 are not. Use T h e rem 1.3.2 to determine all orientable, compact, connected surfaces and all nonorientable ones. (6) For compact, connected surfaces M1 and M2, prove that

+

1.4. PARTITIONS OF UNITY

15

Use this, Theorem 1.3.2 and Exercise (5) to compute the Euler characteristics of all compact, connected surfaces. 1.4. Partitions of Unity

Partitions of unity play a crucial role in manifold theory. Their existence is a consequence of the fact that manifolds are paracompact. In this section, we establish these facts. For further information about paracompact spaces, the reader is referred to [7, pp. 162-1691. DEFINITION 1.4.1. A family C = {C,),Em of subsets of X is locally finite if each x E X admits an open neighborhood W, such that W, n C, # 0 for only finitely many indices cu E ?2l. The proof of the following useful lemma is left as an exercise (Exercise (1)).

LEMMA1.4.2. X , then

If C = {C,),E~ is a locally finite

UaEa C, is a closed subset of X .

family of closed subsets of

DEFINITION 1.4.3. Let U = {Ua},Era and V = {Vp)pEg be open covers of a space X. We say that V is a refinement of U if there is a function i : 93 -+ % such that Vp U,(p), V P E 93. DEFINITION1.4.4. A HausdodT space X is paracompact if it is regular and if every open cover of X admits a locally finite refinement. Actually, it is redundant to require regularity, but it will shorten some arguments. By Exercise (5) on page 3, locally compact Hausclod7 spaces, hence manifolds, are regular. DEFINITION 1.4.5. Let U = (Ua)aEll be an open cover of a space X . A partition of unity, subordinate to U, is a collection A = {A,),,% of continuous functions A, : X + [O,11 such that (1) supp(A,) c U,, Vcu E 2l (where the support supp(A,) is the closure of the subset of X on which A, # 0); (2) for each x E X , there is a neighborhood W, of x such that A, IW, $ 0 for only finitely many indices cy E iZL; (3) the sum C,,, A,, well defined and continuous by the above, is the constant functxon 1.

Our goal is to prove that open covers of manifolds admit subordinate partitions of unity. This will require a proof that manifolds are paracompact, a property that could fail had we not required manifolds to be second countable. LEMMA1.4.6. If X is pamwmpact and l.4 = {U,)aEQ is an open cover of X , there is a locally finite refinement V = {Vp}pEg, i : 23 + %, such that Vp C ui(p), V P E 93-

PROOF.Indeed, paracompact spaces are regular, so it is easy to find a refinement W = {Wr)xsa and associated mapping j : R + %, such that W,C Uj(,), Vrc E A. Passing to a locally finite refinement of W gives a refinement of U with the desired properties.


Recall that a topological space X is said to be normal if, whenever A, B c X B = 0. are closed, disjoint subsets, there is an open set U 3 A such that

an

LEMMA1.4.7. If X is paracompact, it is normal.

PROOF.Let A, B c X be closed, disjoint subsets. The space X being regular, there is a family {U,),Ea of open subsets of X , covering A and such that U, c X \ B,V a E ?2l. The space X being paracompact, the open cover of X , obtained by adjoining X \ A to {U,),Em, has a locally finite refinement. Thus, we lose no generality in assuming that {Ua)aEa is a locally finite family of open sets. If x E M, an open neighborhood W of x meets U, if and only if W meets U,, so {Va},Ea is also locally finite. Set U = U,, U,, an open neighborhood of A. By Lemma 1.4.2, U = U,,% U, and this set does not meet B. In the construction of partitions of unity, we will use the following well known property of normal spaces [7, pp. 146-1471. We do not give the proof here since, ultimately, our interest is in the Cooversion which we will prove later (Corollary 3.5.5). THEOREM1.4.8 (URYSOHN'S LEMMA).If X is a normal space and A, B c X are closed, disjoint subsets, then there is a continuous function f : X + [0,1) such that f lA 1 and supp(f) C X \ B.

THEOREM 1.4.9. Every locally compact, grid countable Hausdorfl space X is paracompact. PROOF. By Exercise (5) on page 3, X is regular. Next, we observe that there is a countable, increasing nest

of compact subsets of X such that

Indeed, let ( W i ) z l be a countable base of the topology of X such that each -

W i is compact. We set K t = and, assuming inductively that K j has been defined, 1 5 j 5 r , we let l denote the least integer such that

and set

e+i

K,+i

=

UITi.

i=l

Let U = {U,),Ea be an open cover of X . We select a refinement as follows. We can choose finitely many Vi = Uai f U, 1 5 i 5 L1, that cover the compact set K l . Extend this by {u,~ ):!-el+1 to an open cover of K 2 . Since X is Hausdorff,

1.4. PARTITIONS OF UNITY

17

<
el. Proceeding inductively, we obtain a refinement V = {Vi)zl of U with the property that K, only meets finitely mimy elements of V, V r 2 1. Given x E X , choose r 2 1 such that x E int(K,), a neighborhood of x which only meets finitely many elements of V. IJ

COROLLARY 1.4.10. Every manifold is paracompact. The existence of partitions of unity can now be established. THEOREM1.4.11. If X is a paracompact space and U = {Ua)aEll is an open cover of X, then there exists a partition of unity subordinate to 24.

PROOF.Choose a locally finite refinement V = {VP)BEg of U, with associated mapping i : B + Q. By Lemma 1.4.6 we can choose a locally finite refinement W = {W,},Er of V, with associated mapping j : R + 93,such that W,c V K E a. For each K E R, use Theorem 1.4.8 to define a continuous function 7, : X [O, 11 such that -+

Given CY E M, let A, = { K E A I a = i ( j ( ~ ) ) } a, possibly empty set of indices. , the union of these sets is closed Since V is locally finite, so is { s ~ p p y , ) , ~ ~, so in X (Lemma 1.4.2), hence is the support of the continuous function

2=

C 7~: X

-+

10,

m).

Of course, if R, = 0,7, = 0. By these choices, it is clear that {?a)rrEm satisfies properties (1) and (2) in Definition 1.4.5. It is also clear that 7 = GaEB 7, < m is continuous and nowhere 0. Therefore, {A, = ?a/7),Emis a partition of unity subordinate to U. El

COROLLARY 1.4.12. Every open cover of a manifold admits a subordinate partition of unity. One use of partitions of unity will be to allow us to define Riemann integration globally on (smooth) manifolds. Another will be to show that every (smooth) manifold has a Riemannian metric. For these and similar applications we will need smooth partitions of unity, a notion that we are not yet ready to define. In the next section, we will use the existence of continuous partitions of unity to prove a topoIogical imbedding theorem for manifolds.

EXERCISES (1) Prove Lemma 1.4.2. (2) Let M be a manifold, U M an open subset, K C U a set that is closed in M, and let f : U -,R be continuous. Prove that the restriction f (K extends to a continuous function f : M + R.

18


(3) Let K C Sn be a closed subset, U > K an open neighborhood of K, v a vector field defined on U. Prove that vl K extends to a vector field on all of sn. 1.5. Imbeddings and Immersions

We will prove that compact manifolds can always be imbedded in Euclidean spaces of suitably large dimensions. DEFINITION 1.5.1. Let N and M be topological manifolds of respective dimensions n 5 m. A topological imbedding of N in M is a continuous map i : N -,M which carries N homeomorphically onto its image i ( N ) . DEFINITION 1.5.2. If N and M are as above, a topological immersion of N into M is a continuous map i : N -* M such that, for each x E N , there is an open neighborhood W of x in N such that i(W : W -* M is a topological imbedding. For example, Figure 1.4 on page 8 depicts an immersion of the Klein bottle K into W3. Every imbedding is also an immersion, but even one to one immersions can fail to be imbeddings. The immersion of W in W2, pictured in Figure 1.7, is one to one, but is not a homeomorphism onto its image.

FIGURE 1.7. The topologist's sine curve

-

DEFINITION 1.5.3. If M is a manifold and X M is a subspace, we say that X is a submanifold if there is a manifold N and an imbedding i : N M such that X = i ( N ) . DEFINITION 1.5.4. The image of a one to one immersion i : N an immersed submanifold of M.

--, M

is called

Some authors use the term "submanifold" to include immersed submanifolds. From the point of view of a topologist, this seems dangerously misleading.

THEOREM 1.5.5. I f M is a compact n-manifold, then there is an integer k > n and an imbedding i : M -+ Ktk.

1.5. IMBEDDINGS AND IMMERSIONS

19

PROOF.Since M is compact, we can find a finite open cover 24 = {Uj};=, of M, together with homeomorphisms 9j : Uj + W j Rn. Let X = {A j } ; = l be a partition of unity subordinate to U.We wiil take k = r ( n 1) and construct an imbedding i : M r Rk. Define z:M-*W~X~.-XR~XRX-.-XIW

+

-r factors

r factors

by i ( ~= ) (Xl(x)(~l(x), , X r ( x ) ~ r ( ~ ) , X l ( .x.. ) , Xr(z)). Here we make the convention that 0 vj (x) = 6 E IRn, even when V j (x) is undefined. Since supp(Xj) c Ujand dom(cpj) = Uj, the expression Xj (x)vj (x) is identically 6 near the set-theoretic boundary of Uj and on all of M \ Uj. This implies that the map i : M + IRk is continuous. Since M is compact and i(M) is Hausdorff, we only need to prove that i is one to one in order to prove that i is a homeomorphism onto its image. Let x, y E M and suppose that i(x) = i(y). Since X is a partition of unity, there is a value of j such that Xj(x) # 0. But the (nr+j)th coordinates of i(x) and i(y) are Xj(x) = Xj(y), so X, Y E s u ~ ~ ( XCj )Uj- Also, Xj(x)vj(x) = Xj(y)vj (y), SO vj (x) = P j (9). Since V j : Uj -* Rn is one to one, it follows that x = y.

+

The imbedding dimension k = r ( n 1) given by this theorem for compact n-manifolds is often much too generous. For example, Exercise (3) on page 14 gives a covering of p2 by r = 3 open sets homeomorphic to R2, so the theorem only guarantees that P2 can be imbedded in R9. In fact, it is possible to imbed P2into It4, as you will show in Exercise (1).

EXERCISES (1) Define f : S2+ R4 by the formula

f (x, y, 2) = ( y t ,xz, xy, x2 + 2y2 + 3z2). Prove that f passes to a well defined, topological imbedding

(It is known that P2 cannot be imbedded in R3.) (2) Let g : S2+ IR3 be defined by Find six points p l , . . . ,p6 E P2such that itself into IR3 is known is a topological immersion. The mapping B of as Steiner's surface. It is simply the imbedding f into R4 followed by projection onto a threedimensional subspace of R4. Prove that 3 does not restrict to an imbedding of any neighborhood of p i , 1 5 i 6. (It

0 with V, c V,, . (Hint: N is 2nd countable.) (b) For {V,,)jEl as in part (a), prove that it is possible to choose the homeomorphisms hk of V,, to open intervals in R so that hk+lJVOlk = hk, Q k 2 1. Show that these assemble to define a homeomorphism h of Ur==, V,, onto an open interval in W. (c) Using the above, show that the partially ordered set Z is inductive, hence contains a maximal element (Zorn's lemma).

CHAPTER 2 The Local Theory of Smooth Functions

In this chapter, we treat the fundamentals of differential calculus in open subsets of Euclidean spaces. Everything will be set up so as to extend naturally to global differential calculus on smooth manifolds.

NOTATION.Elements of Rn, when thought of as vectors, will be written as column n-tuples. When thought of as points, they will be written as rows. 2.1. Differentiability Classes

c

Rn be an open subset. Let x = ( x l , . . . ,xn) denote the general Let U (variable) point of U and let p = . . ,pn) be a fixed but arbitrary point of U . Let f : U + W be a function and let Lp : U + W be an affine (i.e., inhomogeneous linear) map

such that LP (P) = f (P).

DEFINITION 2.1.1. If f and Lp are as above and if lim

f (4- LP(4

= 0) llx - ~ l l Then L, is called a derivative of f at p. If f admits a derivative at p, then f is said to be differentiable at p. x+p

We think of a derivative Lp as a linear approximation of f near p. By the definition, the error involved in replacing f (x) by Lp(x) is negligible compared to the distance of x from p, provided that this distance is sufficiently small. It follows from the definition that an affine map is a derivative of itself. The above definition of "derivative" as a linear approximation embodies the real philosophy of differential calculus. As it stands, however, this definition is a bit unsatisfying. The use of the indefinite article ( a derivative) raises the issue

2. LOCAL THEORY

26

of uniqueness, while the relationship of the notion of a derivative to the familiar operation of differentiation is also unclear. The following lemma, whose proof will be Exercise (I), resolves these doubts. LEMMA2.1.2. If Lp(x) = C +

x7=lbizi is a derivative o f f at p, then

1 5 i 5 n. In particular, i f f is diflerentiable at p, these partial derivatives exist and the derivative L p is unique. Having seen that derivatives are given by partial derivatives, we center our attention on these more familiar operators. DEFINITION 2.1.3. The class of continuous functions f : U -r R is denoted CO(U).If r 2 1, the class Cr (U) of functions f : U -* R that are smooth of order r is specified inductively by requiring that af /axa exist and belong to Cr-I (u), 1 5 i 5 n. The functions that are smooth of order r are also called Crsmooth. One has a decreasing nest

and examples show that these inclusions are all proper. DEFINITION 2.1.4. The set of infinitely smooth functions on U is

It is usual simply to call Cw functions "smooth". We will be concerned primarily with such functions. Remark that the coordinates in U are themselves smooth functions xi : U -, R. Thus, q E U has coordinates xt(q), 1 5 i 5 n, and we can write q = ( x ' ( ~ ) ., .. ,xn(q)).

EXERCISES (1) Prove Lemma 2.1.2 (2) If dimU = 1, prove that the derivative Lp exists if and only if f'b) exists. But if dim U = 2, p = (0,O), find a function f : U -+ R such that both partial derivatives exist at every point of U, but such that the derivative L(olo)does not exist. (3) Let D(U) = { f : U -* R I f is differentiable at x, Vx E U). Show that CO(U) 2 D(U) 2 C1(U). Produce examples to show that both of these inclusions are proper. (Hint: First do this for dim U = 1 and then extend to arbitrary dimensions.) (4) Let U C Wn be open, let f E Cr(U), where 1 5 r 5 oo,and let g : R -* R be Crsmooth also. Prove that the composition g o f belongs to Cr(U).

2.2. TANGENT VECTORS

2.2. Tangent Vectors

We continue to let U C Rn be a fixed but arbitrary open set. We fix p E U and describe the tangent space Tp(U)of U at p. In calculus, it is customary to translate a tangent vector a' at p to the origin 0 E R n , thereby identifying a' canonically with an element of Rn. That is, we set Tp(U) = Rn. This will not do for our purposes since we are trying to set up a local calculus that will make sense on manifolds where, generally, there will be no preferred coordinate system. In standard calculus, the vector

defines a directional derivative Dz at p by the formula

Da(f )

=

lim f(p+h;)-ff(p) h

h+O

=

af C" a'-@), axi i=l

where f is an arbitrary smooth function defined on an open neighborhood of p. Applying this operator to the coordinate functions xi gives so the vector a' is uniquely determined by its associated directional derivative. Another way to obtain this directional derivative is to consider a curve

(where E, 6 > 0 ) , written such that each xi(t) is of class at least C1 and

That is, 5 ' (0)

= pa,

S' (0)= a',

for 1 5 i

5 n. By standard calculus, D a ( f ) = lim f (s(h)) - f @)

h In other words, although the directional derivative was defined as differentiation at t = 0 along a straight line curve with constant velocity a', it could just as well have been defined via any C1 curve out of p with initial velocity a'. While the notion of "straight line" will not have meaning on general manifolds, the notion of "C1 curve" will. h-+O

2. LOCAL THEORY

28

DEFINITION 2.2.1. Given p E U, Coo(U,p) will denote the set of smooth, real valued functions f with dom(f) an open subset of U and p E dom(f).

c1curves s : (-S,e) + U (where S on s) such that s(0) = p is denoted S(U, p).

DEFINITION 2.2.2. The set of all 0 and

E

> 0 depend

>

DEFINITION 2.2.3. If sl ,s2 E S(U, p), we say that sl and sn are infinitesimally equivalent at p and we write sl x, s 2 if and only if

for all f E Cm(U,p). It is easy to check that x, is an equivalence relation on the set S(U,p).

DEFINITION 2.2.4. The infinitesimal equivalence class of s in S(U,p) is denoted (s), and is called an infinitesimal curve at p. An infinitesimal curve at p is also called a tangent vector t o U at p and the set T,(U) = S(U,p)/=, of all tangent vectors at p is called the tangent space to U at p. The following is immediate by the definition of infinitesimal equivalence.

LEMMA 2.2.5. For each (s),

E

T,(U), the operator

is well defined by choosing any representative s E (s), and setting

for all f E Cm(U,p). Conversely, (s), is uniquely determined by the operator

D(9)pBy the definition of infinitesimal curve, there is a natural one t o one correspondence between T,(U) and what, in calculus, is usually called the space of vectors at p, but this notion of tangent vector will make sense on manifolds. Since the whole reason for introducing tangent vectors is to produce linear approximations to nonlinear problems, it will be necessary t o exhibit a natural vector space structure on T,(U), not dependent on the coordinates of Rn.The key lemma for this follows.

LEMMA2.2.6. Let (sl), , (s2), E T,(U) and a, b E B. Then there is a unique infinitesimal cume (s), such that the associated derivatives on Cm(U,p) satisfy


29

PROOF.We are allowed to use the coordinates of R n to prove this assertion. The important point is that the assertion itself is coordinate free. It is clear, then, that s(t) = asl(t) bs2(t) - ( a b - l)p, defined by coordinatewise operations for all values of t sufficiently near 0, is a C curve in U with s(0) = p, and that this curve represents the desired (s) E Tp(U). By the definition of infinitesimal equivalence, (s), is completely determined by D P

+

+

,

DEFINITION 2.2.7. Let (sl),, (s2), E T,(U) and a, b E R. Then is defined to be the element (s), given by Lemma 2.2.6.

LEMMA2.2.8. The operation of linear combination, given by Definition 2.2.7, makes T,(U) into an n-dimensional vector space over R. The zero vector is the infinitesimal curve represented by the constant p. If (s), E T,(U), then - (s), = (s-), where s-(t) = s(-t), defined for all suficiently small values oft. The proof of this lemma is left as an exercise. The operator Db)p,described above, does not "see" all of f E C * (U,p) , only the behavior of f in arbitrarily small neighborhoods of p. The proper way to say this is that D(,!p(f) depends only on the ''germ" of f at p. There is some usefulness in formalizing this.

DEFINITION 2.2.9. We say that the elements f ,g E Cw(U,p) are germinally g if there is an open neighborhood W of p in equivalent (at p) and write f U such that W 5 dom(f) n dom(g) and f (W = g(W.

-,

It is clear that germinal equivalence is an equivalence relation on C * (U,p).

DEFINITION 2.2.10. The germinal equivalence class [f], of f E C w (U, p) is called the germ of f at p. The set CM(U,p)/-,

of germs at p is denoted 8,.

DEFINITION 2.2.11. For each s E S(U,p) the operator is defined by De,,

d

[flp = zf

(s(t))lt=o*

REMARK.The discussion so far would have worked equally well if we had fixed an integer k 2 1, replaced Cw(U,p) with the set C' (u,p) of C k functions defined in neighborhoods of p, and taken 6, = e): to be the germs of these functions. This remark is crucial if one wants to formulate the theory of C manifolds. There is a purely algebraic characterization of T,(U) which, though adrnittedly more formalistic, has its charms. This definition of the tangent space is only valid for the C w case (the default). First we define algebraic operations on germs as follows.

2. LOCAL THEORY

Scalar multiplication: t[f], = [tfIp, V t E R, V [flp E 8,. Addition: [fIp [gIp = [flw glW]p, V[f]p, [glp E @pr where W is an open neighborhood of p in dom(f) n dom(g). Multiplication: [f],[g], = [(flW)(glW)]p, v[f]p,[glp E @pr where W is again as above.

+

+

LEMMA2.2.12. The above operations are well defined and make 8, a commutative and associative algebra over R with unity. The elementary proof of Lemma 2.2.12 is left as an exercise. DEFINITION 2.2.13. The evaluation map e, : 8, + JR is defined by e,[fl, = f @). The following is immediate. LEMMA2.2.14. The evaluation map e, : 8, phism of algebras.

+R

is a well defined hornomor-

DEFINITION2.2.15. A derivative operator (or, simply, a derivative) on 6, is an R-linear map D : 8, + R such that

for all a, b E 8,. A derivative on 6, is also called a tangent vector to U at p. The set of all derivatives on 8, is denoted T, or T,(U) and is called the tangent space to U at p. We define algebraic operations on T,(U). scalar multiplication: (tD)( a ) = t(D(a)), V t E R, V D E T,, V a E 6,; addition: (Dl &)(a) = D l (a) D2(a), V Dl, D2 E T,, Va E 8,.

+

+

LEMMA2.2.16. The tangent space Tp(U) is a vector space over JR under the above operations. Again, the proof will be an exercise.

2.2.17. Define Di,, : 8, EXAMPLE

+R

by

This is a well defined, R-linear map. Furthermore, by the Leibnitz rule for partial derivatives,

Thus DilPis a derivative, 1 5 i 5 n.


EXAMPLE 2.2.18. If ( s ) is ~ an infinitesimal curve, then derivative. Indeed,

31

D(3.p

: Bp + IR is a

where

so the assertion follows from the previous example. It is obvious that

defines a linear map from the space of infinitesimal curves into the space of derivatives of Q3p. It is also clear that this linear map is injective. The fact that it is an isomorphism of vector spaces (Corollary 2.2.22) requires proof. LEMMA2.2.19. If c is a constant function on U and D E Tp(U), then

PROOF. Consider first the case c = 1. Then

from which it follows that D[1Ip = 0. For an arbitrary constant c, D[c], = cD[l], = 0, by linearity. El In order to get more information on Tp(U), we need a technical lemma. Let x = (xl,. .. ,xn) and p = (xl(p), . . . ,xn(p)). LEMMA2.2.20. Let f CE Cm(U,p). Then there exist functions g,, . . . ,g, E C""(U,p) and a neighborhood W C dom(f ) n dom(gi ) f~- - - n do+,) of p such that

PROOF.Define

2. LOCAL THEORY

32

This is clearly a smooth function defined at dl points x sufficiently near p. In order to prove ( 2 ) , consider

In order to prove ( I ) , consider

THEOREM 2.2.21. The set of tangent vectors { D l l p l .. .,Dn,,) is a basis of the vector space T,(U).

PROOF.Suppose that n

For the coordinate functions xj, 1 5 j

5 n,

the Kronecker delta. Thus,

for 1 5 j 5 n. This proves that { D l , , , . . . , D,,,) is a linearly independent subset of T,. We must prove that it is also a spanning set. Let D E T,. Set a' = D[xi],, 1 5 i n. Given an arbitrary I f ] , E 6,,write

1.

(One says that h is Cw-flat at t = 0.)

The graph of h is depicted in Figure 2.1.

FIGURE 2.1. The graph of h The functions h* : R + [O,1 ) are defined by

These are smooth and Coo-flat at t = 0 by exactly the same reasons that h is. The graphs are depicted in Figures 2.2 and 2.3 respectively.

2. LOCAL THEORY

FIGURE 2.2. The graph of h+

FIGURE 2.3. The graph of h , These functions are then combined to produce a C w function k : R

-+

[O, I),

k ( t ) = h-(t - b)h+(t - a ) , where a < b. This function is Coo.It is positive exactly for a < t < b (see Figure 2.4).

FIGURE 2.4. The graph of k LEMMA2.6.3. Let A = ( a l ,b l ) x - .- x (a,, b,) c Rn be a n open, bounded, n-dimensional interval. T h e n there i s a smooth function g : R n -+ [O, 1) such that g > 0 o n A and gl(Rn \ A) = 0.

PROOF.The definition of k gives functions ki,by taking a = ai and b = bi in that definition, 1 5 i n. Then

0. In the definition of C : R -+ [0,11, take a = 0 and b = 6. Then f = e o G : R n -+ [O, 11 is smooth, supp(f) c U, and f l K = 1.

EXERCISES (1) Prove Lemma 2.6.2. (2) Let U Rn be open, f : U -+ R smooth, and p E U. Prove that there is a Cm function f : Rn -+ R such that [f], = [f], in 8,. Conclude that, for arbitrary p E Rn,8, can be identified canonically with the set of germs at p of globally defined, smooth, real valued functions on R n. (3) Let C c Rn be a closed subset, U C Rn an open neighborhood of C . Show that there is a smooth, nonnegative function f : R n -+ R such that f lC > 0 and supp(f) C U.

2. LOCAL THEORY

50

2.7. Smooth Vector Fields

Let U C Rn be an open subset. In particular, this is a smooth submanifold and we consider the set X(U) of smooth vector fields on U (Definition 2.5.12). Remark that X(U) is a vector space over R under the pointwise operations. Throughout this section, the preferred way to think of tangent vectors will be as derivatives of the algebra of germs (Definition 2.2.15). Correspondingly, we will be able to view vector fields on U as first order partial differential operators on the algebra of smooth functions. The vector fields Di E X(U), 1 5 i 5 n, assign to each x E U the partial derivative operator Di,, (Example 2.2.17). The canonical identification T (U) = U x Rn is accomplished by the correspondence

From this, it is clear that the general element X E X(U) can be written

where f Z : U -+ R is smooth, 1 5 i 5 n. Remark that Cm(U) can be viewed as an associative algebra over R under the operations of pointwise addition, pointwise multiplication, and the usual multiplication of functions by scalars. This algebra has as unity the constant function 1. From the above observations, it is clear that X(U) is a module over the algebra Coo(U) . In fact, X(U) is a free Cm(U)-module with canonical basis

{Dl, - . ,Dn). We are going to give a deeper algebraic interpretation of X(U). For this, we need some definitions. Let F be an (associative) R-algebra with unity.

DEFINITION 2.7.1. A derivation A of the R-algebra F is a linear map

such that A(fg) = A(f)g be denoted V(F).

+ fA(g), V f , g E F. The set of derivations of F will

LEMMA2.7.2. The set of derivations V(F) is a vector space over R under the linear operations

Va, b E R, VA,, Az E V(F), V f E F. The proof of this is completely elementary and is left to the reader

2.7. SMOOTH VECTOR FIELDS

DEFINITION 2.7.3. If A l , A2 E V ( F ) ,then the Lie bracket

is the operator defined by

V f E F . This is also called the commutator of A 1 and A2.

LEMMA2.7.4. The Lie bracket satisfies the following properties:

v

(1) [A,,A21 E V ( F ) , Al, A2 E W ) ;

( 2 ) the operution , - 1 : V ( F )x V ( F ) + V ( F ) is R- bilinear; ( 3 ) [ A l ,A2]= - [A2, All, V Al ,Az E V ( F ) (anticommutativity); ( 4 ) [A,,[A,,A311 = [ [ A lA,21, A31 + [A,, D l , A311, v A1 A2, A3 E D ( F ) (the Jacobi identity). [

a

The proof is left as an exercise. Thus, we can think of the operation

as a bilinear multiplication making V ( F ) into a kind of R-algebra. This algebra is nonassociative, however, the Jacobi identity replacing the associative law. The algebra is also anticommutative and does not have a unity.

REMARK.One way to remember the Jacobi identity is to notice that, by this identity, the operator [A,- 1 : V ( F ) -4 V ( F ) is a derivation of the (nonassociative) algebra V ( F ) ,V A E V ( F ) . DEFINITION 2.7.5. A nonassociative algebra having the properties listed in Lemma 2.7.4 is called a Lie algebra. LEMMA2.7.6. If

L

E F is the unity and c E R, then A(cL)= 0 , V A E V ( F ) .

The proof is exactly like that of Lemma 2.2.19. Suppose that the algebra F is commutative and define an operation of "scalar" multiplication

F x D ( F ) -+ V ( F ) ,

LEMMA2.7.7. If F is commutative, the scalar product fA is an element of V ( F ) , V f E F , V A E V ( F ) . This makes V ( F ) a module over the algebra F . The elementary proof will be an exercise. LEMMA2.7.8. The space X ( U ) is a Coo(U)-submodule of V ( C o o ( U ) ) .

2. LOCAL

THEORY

PROOF.Indeed, given X E X(U), write it as

where f i E CM(U), 1 5 i 5 n. The Leibnitz rule for each partial derivative Di implies that X(fg) = X(f)g + fX(917 v ~ , ~ E c ~ ( u a) . We are going to prove the reverse inclusion

hence that X(U) is the full Lie algebra and Cm(U)-module V(CM(U))of derivations of the function algebra CM(U). LEMMA2.7.9 (KEY LEMMA).Let A E V(Cm(U)), f E Cm(U), and suppose that V E U is an open set such that f lV = 0. Then A(f)lV -= 0.

PROOF.Let x

E V.

By Theorem 2.6.1, we find cp E Coo(U)such that

Indeed, since {x) is compact, we find .II, E C" (U) such that @(x) = I and supp(.II,)c V. Then p = 1- $ is as desired. Since f lV 0, we see that cpf = f . Thus, A ( f ) = A(Pf = A(cp)f + vA(f 1, hence A ( f )(XI = A(v)(x)f (4+ v(x)A(f )(x). But f (x) = 0 = cp(x), so A( f)(x) = 0. Since x E V is arbitrary, A(f) lV 0.

>

-

COROLLARY 2.7.10. Let A E V(CM(U)), f E CM(U), and x E U. Then A(f)(x) depends only on A and the germ [f], E 8,. PROOF.Let f , g E CM(U) have the same germ [f], = [g],. Choose an open neighborhood W E U of x such that f 1 W = g ) W.By the Key Lemma 2.7.9,

Given w E @,, x E U, there exists f E CM(U)such that w = [f],. This is by Exercise (2) on page 49. This allows us to make the following definition.

DEFINITION 2.7.11. Given A by

V w = If],

E V(Cm(U)) and x E U, Ax : @, -,R is given

Ax (4= A(f >(x), E 8,, where f E CM(U).

By the above discussion, it is clear that Ax is well defined, Vx E U, V A E N C " (41.

2.7. SMOOTH

VECTOR FIELDS

53

PROPOSITION 2.7.12. If A E V(Cm(U)) and x E U, then Ax E T,(U). PROOF. Let [f],, [g], E 6,, where f , g E Cm(U). Then,

It isclear that A x :6, - + R is linear, so Ax E Tx(U). Given A E V(Cm(U)), define i: U -+ T(U) by i ( x ) = Ax E Tx(U). This is a section of the tangent bundle. If this section is smooth, then i E X(U) . Write

remarking that the smoothness of

is equivalent to

PROPOSITION 2.7.13. If A E V(Cm(U)), then

f' E Cm(u),1 5 i 5 n. E X(U).

PROOF. Consider the coordinate functions x i E U, 1 5 i 5 n. Let

Then

We summarize this discussion as follows.

-

THEOREM 2.7.14. Under the correspondence A L, V(Cm(U)) is canonically identified with X(U) as a Cm(U)-module, thus defining a Lie algebra structure on X(U) . From now on, we use either point of view freely, but we denote this Lie algebra and Cm(U)-module only by X(U). EXAMPLE2.7.15. One is often interested in certain Lie subalgebras of X(M). We give here an example on the group manifold Gl(n) to which we will be returning later. Let g[(n) c X(Gl(n)) denote the subspace of left invariant vector fields. That is, X E gI(n) if and only if Lp,(X) = X, V P E Gl(n). We can identify gI(n) with the space m ( n ) of n x n real matrices. Indeed, for each A E m ( n ) , view the right translation map R A : Gl(n) + !Dl(n) as a vector field on Gf(n). The value of this vector field a t Q E Gl(n) is the matrix QA E 1)31(n) = TQ(Gl(n)). Evidently, Lp,(QA) = PQA, the value of the field RA a t P Q = Lp(Q). That is, Lp+(RA)= R A , proving that RA E g[(n). Of course, the value of a left invariant field on Gl(n) at one point, say the identity I, determines the whole field, so

54

2.

LOCAL THEORY

RA is the unique left invariant field whose value at I is A E T r ( G l ( n ) )= !?R(n). It is evident that this one to one correspondence between g((n)and !?R(n)is an isomorphism of vector spaces. In Dt(n),define the bracket to be the usual commutator of matrices [A, B]= A B - BA, Q A, B E %R(n). The proof of the following lemma will be left as an exercise.

LEMMA 2.7.16. The commutator operation makes m(n)into a Lie algebra, and the canonical identification DZ(n) = g[(n)turns the commutator into the Lie Q A, B E !XQ(n),so bracket of vector fields. More precisely, [RA,RBI = g l ( n ) is a Lie subalgebra of X(Gl(n)) isomorphic t o the Lie algebra m(n)of n x n matrices under the commutator bracket.

NOTATIONAL CONVENTIONS. When v ( C M ( U ) )is thought of as the space X ( U ) of smooth vector fields, it is customary to denote its elements by capital letters from the end of the alphabet. The value of a field X E X ( U ) at a point q E U is usually denoted by X , and, if f E C m ( U ) , one writes Xq(f ) for X q [f],.

As noted earlier, a diffeomorphism z : U -+ V onto an open subset V R n can be thought of as a change of coordinates. The new coordinate functions are

Similarly, x = z-l gives the coordinates xi as functions of (zl,. . . ,zn). Recall that such a change of coordinates translates formulas in differential calculus to new formulas relative to the new coordinates. In the present context, vector fields X E X ( U ), viewed as first order differential operators, are carried to vector fields z , ( X ) E X ( V ) . One can think of this as a change of formula for the operator X in terms of the new coordinate system (V, z l , . . . , zn). In order to define z , ( X ) from this point of view, we need some definitions.

DEFINITION 2 . 7 . 1 7 . If z : U -+ V is a smooth map between open subsets of Euclidean spaces, then z* : C m ( V ) -t C M ( U )is defined by setting

for all f E C m ( V ) .

LEMMA2.7.18. If z : U 4 V is a diffeomorphism between open subsets of Rn, then r* : C M ( V )4 C m ( U ) is an isomorphism of algebras. This is completely straightforward to check. DEFINITION 2.7.19. If z : U 4 V is a diffeomorphisrn between open subsets of Rn, then t , : X ( U ) -t X ( V ) is defined by setting

for all X E X ( U ) and all f E C m ( V ) .

FLOWS

2.8. LOCAL

55

The fact that z,(X) E J ( V ) is elementary, as is the following. These are left as an exercise for the reader. LEMMA2.7.20. If z : U -+ V is a difleomorphisrn between open subsets of then z, : X(U) -+ X(V) is an isomorphism of Lie algebras.

Rn,

EXERCISES (1) Prove Lemma 2.7.4. (2) Prove Lemma 2.7.7. (3) Prove Lemma 2.7.16. (4) Prove Lemma 2.7.20 and the preceding statement. (5) Show that, if X E E(U) is viewed as a smooth section of the tangent bundle .rr : T(U) -+ U, then the section Z = z, ( X ) of the tangent bundle .rr : T(V) + V is given by ZC= d ~ , - l ( ~ ) ( X , - l ( ~VC ) ) ,E V. 2.8. Local Flows

We begin this section by showing how a vector field X E X(U) can be viewed as an (autonomous) system of ordinary differential equations (O.D.E.) on U. DEFINITION 2.8.1. Let s : (a, b) to E (a, b) is

-+ U

be smooth. The velocity vector of s at

REMARK.If T = t + t o , a - to < t < b - to, then a ( t ) = S(T) has velocity b(t) = d(r) = d(t to). In particular, i ( t o ) = b(0) = (a),(,o).We will write this infinitesimal curve as (s) ,(to).

+

DEFINITION 2.8.2. The map i : (a, b) the smooth curve s : (a, b) -+ U.

-+

Remark that p o s = s, where p : T(U) remark that s is smooth.

T(U) is called the velocity field of

-+

U is the bundle projection. Also

DEFINITION2.8.3. Let X E J ( U ) and xo E U. An integral curve to X through xo is a smooth curve s : (-6, E) -+ U, defined for suitable 6, E > 0, such that s(0) = xo and i ( t ) = Xs(,), -6 < t < e. Suppose that s is an integral curve to X E J ( U ) through xo E U. Write

and

2. LOCAL THEORY

56

At each t E (-6, E), the Jacobian matrix of s is

The vector

$1,

E Tt(-6, e ) = W 1 is the canonical basis element 1 E W 1 , so

;[ ] d"'( t )

ds.(:I,,

=

.Rn

dt (t)

But

Thus, s is an integral curve to X if and only if

dx' dt

-( t )= f " x l ( t ) ,. . . zn( t ) ) , )

-6 < t < E, 1 < i 5 n. This is an (autonomous) system of O.D.E. with solution s subject to the initial condition s(0) = xo. The existence and uniqueness of integral curves is guaranteed by the following theorem.

THEOREM 2.8.4. Let V G Rr and U Rn be open subsets, let c > 0, let f a E CDO((-c, c) x V x U ) , I _< i 5 n , and consider the system (nonautonomous with parameters b = ( b l , . . . , 6') E V ) of O.D.E.

Let a = (a1,.. . , an) E U. Then there are smooth functions x i ( t ,b), 1 _< i _< n , defined on some nondegenemte interval [-6, E] about 0, which satisfy the system (*) and the initial condition

firtherrnore, zf the functions Z i ( t , b), 1 5 i < n, gzve another solution, defined on [-6,4 and satisfying the same initial condztion, then these solutions agree on [-6, e] n [-8, q. Finally, if we w i t e these solutions as xi = xs'(t,b, a ) (in order to emphasize the dependence on the initial condition), there is a neighborhood W of a in U,a neighborhood B of 6 in V , and a choice of e > 0 such that the solutions xi(t, z , x) are defined and smooth on the open set (-e, E) x B x W IW"+'+~.

2.8. LOCAL FLOWS

57

This is the well known theorem giving the existence, uniqueness, and smooth dependence on initial conditions and parameters of solutions of systems of ordinary differential equations. The proof is given in Appendix C (where the smooth dependence on initial conditions will require some elementary facts from calculus in Banach spaces). DEFINITION2.8.5. The system (*) is autonomous if the functions f do not depend on t , 1 5 a 5 n. The problem of finding integral curves to a vector field is the problem of solving an autonomous system of O.D.E. without parameters. The proof of the existence, uniqueness, and smooth dependence of these curves on the initial conditions, however, involves an inductive step that requires the general formulation of Theorem 2.8.4. From now on, we consider only autonomous systems without parameters. COROLLARY 2.8.6. If X E X(U) and x E U, then there is an integral curve to X through x. Any two such curves agree on their common domain. DEFINITION2.8.7. A local flow @ around xo E U is a smooth map

(written @(t,x ) = a t ( x ) ) , where W is a suitable open neighborhood of x o in U, such that (1) a0: W --+ U is the inclusion W L) U; (2) @t,+t,(x) = @ t , ( @ t 2 ( x ) )whenever both sides of this equation are defined.

If r E U, the flow line through z is the curve a ( t ) = Qt(z), -E

< t < E.

THEOREM 2.8.8. Let X E X(U)andxo E U . Then there is a EocalfEow around xo such that the flow lines are integral curves to X . Two such local flows agree on their common domain.

PROOF. By Theorem 2.8.4, we find an open neighborhood W of xo in U and a number E > 0 such that the integral curve s,(t) through z is defined for - E < t < E and for all r E W. By the smooth dependence on initial conditions, we define a smooth map @ : (-E, E) X W -+ U

Theorem 2.8.4 also assures us that, if 6 is another local flow around xo with flow lines integral to X, then @ and 6 agree on their common domain. We must show that @ is a local flow. Since a. (r) = S , (0) = r, it is clear that Go is the inclusion map W U. Fix to E (-E, E) and r E W and define the curves

-

s(t) = sz(t

+ t o ) = Gt+to(z)

2. LOCAL THEORY

58

and ~ ( t=) S s z ( t o ) ( t ) = Qt(sz(t0))= %(@to (z)). These are defined for small values of t and are both integral to X. They satisfy s(0) = s z ( t o )and ~ ( 0=) s ~ , ( ~ ~ ) (=O s) z ( t o ) ,so s ( t ) = u ( t ) whenever both sides are defined. That is, @ t + t o ( r = ) a t ( G t 0 ( z ) whenever ) both sides are defined. El The locd flow @ associated to X E X(U) as in Theorem 2.8.8 is said to be generated by X. Also, the vector field X is called the infinitesimal generator of the local flow @. DEFINITION2.8.9.

FIGURE 2.6.

EXAMPLE 2.8.10. Let U

Flow lines for X = x l D l

+ x2D2

+

= R2 and X = X ' D ~ x2D2. The integral curves

s ( t ) = ( x l ( t ) ,x2( t ) ) will satisfy

The solution curve with initial condition s(0) = (a l , a 2 ) is

(see Figure 2.6). All of these solutions are defined for -oo < t < cm. Remark that @ t , + r 1, ( a2 , ~ ) =1 ( tl+t2 a e ,a2et1+t2)=~t,(@s(a1,a2)). This curve is stationary for ( a 1 ,a 2 ) = (0,O) and in all other cases follows a radial trajectory out of the origin, but is not parametrized linearly. The "speed" of the trajectory increases proportionally with the distance from the origin.

2.8. LOCAL

FLOWS

FIGURE 2.7. Flow lines for X EXAMPLE 2.8.11. Let U = It2 and X s(t) = (xl (t),x2(t)) will satisfy

=

= x l D l - x2D2

x1Dl - x2D2. The integral curves

The solution curve with initial condition s(0) = (a1, a 2 ) is

(see Figure 2.7). All of these solutions are defined for -w < t

< oo.

Again

This curve is stationary for (a1, a2) = (0,O). Tkajectories of points on the x2-axis (other than the origin) stay on the x2-axis and head toward the origin. Trajectories of points on the xl-axis (other than the origin) stay on that axis and head away from the origin. The remaining trajectories follow hyperbolic paths asymptotic to the coordinate axes. LEMMA2.8.12. Let X E X(U) generate a local flow @ : (-E,E) x W --+ U. Then, for each q E W, there is a neighborhood V of q in W and a number 6 > 0 such that (1) @,(V) W, -6 < t < 6; (2) at : V + W is an imbedding, -6 < t < 6; (3) (@t)*p(Xq)= Xar(p)7 -6 < t < 6. The proof of Lemma 2.8.12 is left as an exercise. So far, we have used vector fields to differentiate functions. Now we will show how a vector field can be used to differentiate another vector field. Let X E X ( U ) , q E U ,and let @ : (-E, e ) x W --+ U be a local flow about q generated

2. LOCAL

60

THEORY

by X. Let Y E X(U). By Lemma 2.8.12, there is a value 6 > 0 such that the vector (@-t),*,(,)(Ya,(,)) E T,(U) is defined, -6 < t < 6. Generally, this vector differs from Y,, although they are equal if Y = X (Lemma 2.8.12 again). The difference quotient

can be thought of as the average rate of change of Y near q along the integral curve to X through q. This lives in the vector space Tq(U) = Rn,so lirnt,0 Z,(t) makes sense. If this limit exists V q E U, we obtain a (conceivably not smooth) vector field @-t*(Y) - y Lx(Y) = lim t -40 t DEFINITION 2.8.13. If Lx (Y) exists on W, it is called the Lie derivative (on W ) of the field Y by the field X.

REMARK. One can also define the Lie derivative of a function by the formula Lx(f) = lim @t*(f)- f t-0 t The use of at instead of is due to the fact that @,' pulls functions back, while at, pushes vector fields forward. The reader should have no trouble seeing that Lx(f) = X(f).

THEOREM 2.8.14. If X, Y

E

X(U), the Lie derivative of Y by X is defined

and smooth throughout U and

The proof requires a couple of lemmas. We will fix q E U and a neighborhood : ( - E , E ) x W -+ U is defined. Write

W of q so that

so limt+o Zq(t) exists if and only if limt,o Ci(t) exists, 1 5 i

< n.

LEMMA2.8.15. The limit A, = limt,o Zq(t) exists if and only iA for each f E Coo (U) , lirnt+0 Zq(t)( f ) exists, in which case this limit is A , ( f ). PROOF. Assume that the limit A, = ~ : = , a 2 D i ,exists. , That is,

a' = lim

t-40

1 i: i

< n.

exists.

ci(t),

Then, for arbitrary f E Cm(U), it is clear that

2.8. LOCAL FLOWS

For the converse, suppose that, for each f E Cm(U), n

lirn Zq(t)(f) = lirn t+O

t+O

af 1cl(t)-(q) axi i=l

exists. Choose f = x j and define a3 = lirn 2, (t)(x3) = lirn cj (t), t-+O t+O 1 5 j 5 n. Thus, limt,o Zq(t) = A, exists and has coordinates a j , 1 5 j 5 n. For arbitrary f E Cm(U) it is also clear that

LEMMA2.8.16. Given f E Coo(U), there is a function g such that

E

Cm((-E, E) x W)

PROOF.Define h(t,x>= f ( @ - t ( 4 ) - f ( 4 , so h E Cm((-E,E) x W) and h(0,x) = 0. To simplify notation, we denote the partial of h with respect to t by h(t, x). Define g E Cm((-e, e) x W) by

Then

giving the first assertion. For the second, consider g(0, x) = lirn g(t, x) t+O

= lirn

f (@-t(x)>- f (5)

-t - f (4 = lirn f (@t(x>> t+O

t-ro

= X d f 1.

t

2. LOCAL THEORY

62

PROOFOF THEOREM 2.8.14. Let f E C" ( U ) and let g be as in the preceding lemma. Let gt : W --, U be given by g t ( x ) = g(t, x ) . Thus, X ( f ) = go = limt+o gt , so limY@,(q)(gt)= Y , ( X ( f ) ) .

t+O

We now compute

lim Z, ( t )( f ) = lim

t+O

t--0

( a - t t qq

) - Yq

t

-Ydf) t Y@t(q) ( f - t9t) - Y , ( f ) = lirn t+O t Y @ t ( q ) (-f )Y , ( f ) = lirn - t+O lim Y@, ( 9 )(gt) t+O t ) Y ( f) ( q ) = lirn Y ( f) ( @ t ( q ) - t+O lim YQ~(,) (gt) t+O t = X , ( Y ( f )) - Y , ( X ( f 1) = [x,YI,(f ). = lirn

Y@t(q)(f O a-t)

t+O

Since f is arbitrary, Lemma 2.8.15 gives the desired conclusion. Let X, Y E X ( U ) and let a, Q denote the respective local flows generated by these fields about some point q E U . We can choose 6, > 0 so that Qt\ES(q)and Q S a t ( q )are defined, -6, < s, t < 6,. As q E U varies, these bounds 6, will also vary. The local flows vary too, but they agree on overlaps by Theorem 2.8.4.

DEFINITION 2.8.17. The local flows of X and Y commute on U if @t!Ps(q)= @,at(q), -6, < s, t < 6,) V q E U . The vector fields themselves commute on U if [X,Y ]_= 0 on U . THEOREM 2.8.18. The vector fields X and Y commute on U zf and only if their local flows commute on U. Suppose that the local flows commute on U . Then, for -6, < t < 6,) < s < 6,) of \E onto another flow line of 3P. By taking the infinitesimal curve point of view, we see immediately that (at),,(Y,) = Y@,(,),-6, < t < S,, V x E U . That is, PROOF.

at carries any flow line {\Es(x) 1 -6,

[ X ,Y ]= lirn t+O

a-t*(Y)- Y Y-Y - lirn =0 t+o t t

-

throughout U . For the converse, we assume that [ X , Y ] 0 on U and deduce that the local flows commute. Let q E U , fix s E (-6,, 6,), and let q' = \E,(q). Define v : (-6,)6,)--, T,, ( U ) by the formula v(t)= (Y@,(,,)).(Were and elsewhere,

2.8. LOCAL FLOWS

63

K an open neighborhood of K, v E X(U).Prove that vlK extends to a smooth vector field on all of M . (4) Let U = {U,),Em be an open cover of the smooth manifold M. For each a, E 2L, let cp, : M + R have a constant value c, > 0 on U, and be identically 0 on the complement M \ U,. Set

and construct a smooth function f everywhere on M .

:

M

+

R such that 0 < f < cp

3.6. Manifolds with Boundary

Manifolds with boundary were introduced in Section 1.6 from the purely topological point of view. Here we introduce the smooth version. Since Euclidean half space Wn is a subset of the smooth manifold Rn, Definition 3.5.7 allows us to talk about smooth maps and diffeomorphisms between open subsets of Wn. Thus, if M is a topological manifold with boundary, it makes sense to talk about two Wn*harts on M being Coo-related, so we can define a differentiable structure on M to be a maximal Wn-atlas A of Cm-related charts. As in the topological case, we define

88

3. GLOBAL THEORY

The pair (M, A) is a (smooth) n-manifold with boundary dM. Of course, all smooth n-manifolds without boundary are special cases, as is Wn itself. The notion of smooth maps between manifolds with boundary is defined exactly as in the boundaryless case. The following is an immediate corollary of Theorem 3.5.8.

PROPOSITION 3.6.1. If U C_ Wn \ dWn is open, then U is not difleomorphic to an open subset V 5 Wn such that V dWn # 8.

COROLLARY 3.6.2. If M is a manifold with boundary, then M d M = int(M). PROOF.Indeed, the proposition shows that d M = {x E M A, x E Ua , qa (Ua) E int(Wn))-

COROLLARY 3.6.3. If cp : M

4

(Ua,cpa) E

N is a difleomorphism, then

COROLLARY 3.6.4. A smooth n-manifold M with boundary is also a smooth n-manifold e M = int(M) e d M = 0. COROLLARY 3.6.5. If M is a smooth n-manifold with boundary, then a M is a smooth ( n - 1)-manifold and int(M) is a smooth n-manifold. For x E int(M), the n-dimensional tangent space T,(M) is defined as before. If x E d M , T, (d M) is only (n - l)-dimensional, but we are going to define an n-dimensional tangent space T, (M) having T, (dM) as a subspace. The more intuitively appealing approach is that of infinitesimal curves. Indeed, the notion of smooth curve

makes good sense and we can define the corresponding infinitesimal curve ( s )., This can be thought of as a tangent vector to M at x. In this way, one obtains all the vectors tangent t o the boundary and all the vectors that point "into" M. Unfortunately, the negatives of the inward vectors are not so obtained and the resulting set of tangent vectors is not a vector space. Alternatively, the definition of tangent vectors as derivatives of the algebra of germs of C O0 functions at x, while less intuitive, yields a vector space exactly as before. However, in order to show that this vector space is n-dimensional, we will find it convenient to turn to infinitesimal curves. If x E M, whether or not x E d M , the set Coo (M, x) of smooth, real valued functions defined in neighborhoods of x is defined and there is no problem defining germinal equivalence on this set. Thus, we obtain the IR-algebra @ ,of germs. We define Tx(M) to be the vector space of derivatives D : 6, + R. If x E int(M), this agrees with our usual definition and is n-dimensional. If f : M -t N is a smooth map between manifolds with boundary, the different i d s dfz = f*x : Tx(M) Tf(,)(N) -+

3.6. MANIFOLDS

WITH BOUNDARY

are defined by the usual formula

d f z ( D ) [ g l j ( z= ) D[g O f l z and are linear. The proof of the global chain rule (Lemma 3.1.21) goes through equally well in our present context. LEMMA3.6.6. If f : M M N and g : N 4 P are smooth maps between manifolds with boundary, then g o f is smooth and, for each x E M ,

COROLLARY 3.6.7. I f f : M + N is a diffeomorphism between manifolds with boundary, then d f , : T,(M) M T f ( , ) ( N ) is a linear isomorphism, V x E M . We have reached the point where a little work has to be done. We must show that Tx(Wn) is n-dimensional, even when x E dWn. The above considerations will then extend this property to arbitrary manifolds with boundary. When x E dWn c Rn, we will use the notation @,(Rn) for the algebra of germs of Coo ( R n ,x ) and @, (Wn) for the germ algebra of C" (Wn,x ) .

LEMMA3.6.8. Let x E dWn and let p : @,(Rn) p [ f I x = [ f 1 (Wn r 7dom(f )I,. Then p is a surjection.

+

@,(Wn) be defined b y

PROOF. Let U Wn be an open neighborhood of x. If g : U M R is smooth, there is a neighborhood V of x in R n and a smooth extension ij : V 4 R of g1 (V n U n Wn). Then [ j ] ,E @, ( R n ) and p [ j ] , = [g],. For x E dWn, define

p* : T,(Wn)

4

Tz(Rn)

by setting

~ * ( D ) [ f= l . D(p[fl,). It is elementary that this is linear.

L E M M A3.6.9. p* is bijective. PROOF. We prove that p* is one to one. If p f ( D 1 )= p*(D2),then D l ( p [ f ] , ) = Dz(p[f V [ f 1, E 6,( R n ) . Since p is surjective, it follows that & [g], = D2 [ g ] ,, V [ g ] ,E @,(Wn), so Dl = D2. We prove that p* is onto. Let v E T x ( R n ) = Rn. As an infinitesimal curve, this vector is represented by s ( t ) = x tv. As an operator on germs, v = D (,)=. Either v points into Wn (we intend this to include the case that v is tangent to dWn) or v points out of Wn, in which case -v points into Wn. If v points into Wn, then s ( t ) E Wn, V t 0. Define D : @,(Wn) M R by

lz),

+

>

D [ g ] ,= lim g ( s ( t ) )- ilk4 t+o+

t

It is elementary that D E T,(Wn) and that p*(D) = D(,)= = v. If v points out of Wn, then s ( t ) E Wn, V t 5 0. Define D : @,(Wn) M R by

D [g],= lim g ( s ( t ) )- g(z) t+O-

t

3. GLOBAL THEORY

90

Again, D E Tx(Hn) and p* (D) = u.

COROLLARY 3.6.10. The vector space Tx(Wn) is n-dimensional, Vx E dWn. COROLLARY 3.6.11. Let M be a smooth n-manifold with boundary, x E a M . Then the vector space Tx(M) is n-dimensional. PROOF. Let

(U, cp) be an Wn-coordinate chart about x. Then (p*x : TX(U)

+

T,(X)(cp(U))

is an isomorphism. But Tx(U) = Tx(M) and T,(x)(~(u))= Tp(x)(Wn). This latter is n-dimensional. At this point, one can define the tangent bundle .rr : T(M) 4 M , this being an n-plane bundle over M. The total space T(M) is a 2n-manifold with boundary, W ( M ) = .rr-'(d~). In Exercise (4), you will be asked to check this carefully. Vector fields on a manifold M with boundary are smooth sections of T(M). The smooth Urysohn lemma and its consequences extend to this context. In particular, vector fields are derivations of the function algebra Coo(M) and open covers always admit smooth, subordinate partitions of unity. In general, the classification of manifolds, with or without boundary, is impossible. In dimensions one and two, however, the classification has been achieved and there is considerable effort being expended in an effort to classify compact 3-manifolds. The task becomes impossible in dimensions four and greater, where it can be reduced to the problem of deciding, via a finite algorithm, whether or not given finite presentations of two groups (that admit such presentations) force the groups to be isomorphic. This problem, often called the "word problem", has been shown by logicians to be unsolvable.

EXERCISES Let M be a manifold with nonempty boundary. Show that there is a smooth function f : M + R+ such that d M = f-l(0). Let M and N be smooth manifolds of dimension m and n respectively. If d M # 8 = d N , show how to use the differentiable structures on these two manifolds to give M x N the structure of a smooth (m+n)-manifold with nonempty boundary. (3) If, in the previous exercise, both M and N had nonempty boundary, show that M x N is a topological manifold with boundary. Discuss the problem you encounter in trying to give M x N a natural smooth structure of manifold with boundary. This is, in fact, an example of what is called a smooth manifold with corners. (4) For an n-manifold M with boundary, mimic the construction of the tangent bundle ?r : T(M) --t M, showing that one obtains a smooth 2n-manifold with boundary, the projection .rr being identified locally with the canonical projection

3.7. SUBMANIFOLDS

91

3.7. Submanifolds We give a definition of "submanifold" that applies to manifolds with boundary. DEFINITION 3.7.1. Let M be an m-manifold, possibly with boundary. A subset X c M is a properly imbedded submanifold of dimension n if and only if, V p E X , there is an Wm-coordinate chart (U,cp) about p in M in which (p(UnX) = cp(U) n Wn, where Wn c Wm is the (image of the) standard inclusion. Remark that, in the above definition, (U t l X, cpl (U r l X)) can be viewed as an Wn-coordinate chart on X and that the collection of all such charts makes X a smooth n-manifold with boundary d X = X n dM. Thus, if d M = 0, then dX = 0 also. In this case, we generally drop the qualifier "properly". Note also that X cannot be tangent to a M at any point of dX. EXAMPLE 3.7.2. The image of the standard inclusion W imbedded submanifold.

n

L,

Wm is a properly

We will need the following globalization of the submersion theorem. THEOREM 3.7.3 (GLOBALSUBMERSION THEOREM).Let f : M --, N be a smooth map between manifolds of respective dimensions m and n. Assume that ON = 0 and that m > n. If y E N is a regular value simultaneously for f and for d f = f ldM, then f -'(y) is a properly imbedded submanifold of dimension m - n.

PROOF.Let p E f -'(y), and find a suitable coordinate chart about p in M. There are two cases. Case 1. Suppose p E int(M). Choose a coordinate neighborhood (U, x) about p such that U E int(M). Then y is also a regular value of f (U,so Theorem 2.9.6 implies that f -'(y) n U is a smooth submanifold of U of dimension m - n. Case 2. Suppose p E OM and let (U, x l , x2, . . . ,xm) be an MIm-chart about p in M. Assume that f (U) c W, where (W, yl, . . . , y n , is a coordinate chart about y in which y = 0. Let d f 1 (U n d M ) be denoted by cp(x 2 , . . . ,xm) with component functions cpl, . . . , cpn relative to the coordinates of W. Since p is a regular point for cp, U can be chosen so small that the matrix

has constant rank n on UndM. By a permutation of the coordinates x 2 , . . . ,xm, it can be assumed that the last n x n block

3. GLOBAL THEORY

92

is nonsingular on U n dM. Choosing U even smaller, if necessary, the corresponding n x n block in the matrix

is also nonsingular. We then resort to the trick of recoordinatizing U near p by setting zi = xi, 1 5 i 5 m - n, and zm-n+j = f j , 1 5 j 5 n. The inverse function theorem shows, by the above remarks, that this will define an W n-chart on a small enough neighborhood (again called U) of p. But, relative to these coordinates, f(z',z2, .. . , z m ) = (2m-n+l , . . . ,zm). Then f (y) n U is the set of points with coordinates (z , . . . , z " y 0,. . . ,0). That is,

'

-'

f

EXAMPLE 3.7.4. Let f : Wn+'

nU +R

=

w ~ n U. - ~

be given by

Then 1 E R is a regular value both for f and for d f . The hemisphere f -'(l) is the intersection SnnIHn+land is an n-manifold with boundary f -l(l)ndWn+l = sn-1

The following lemma shows that there are plenty of regular values as in Theorem 3.7.3.

LEMMA3.7.5. If d M = 8 and f : N + M is smooth, then the set of points in M that are simultaneous regular values for f and d f is dense in M. PROOF. Clearly, if p E a N is a regular point for d f , it is also a regular point for f . Thus, y E M is a regular value both of f and df precisely when it is a regular value both of f 1 int(N) and of d f . Use countable coordinate coverings {Ui)iEI of int(N), (%IjEJ of d N , and {Wk)kEK of M . For each k E K , consider the countable family of smooth maps

obtained by restrictions. By Corollary 2.9.5, a.e. y E W kis a common regular value of all these maps. Doing this for each k E K, we complete the proof. For simplicity, the following discussion will be carried out only for the case of manifolds with empty boundary. DEFINITION 3.7.6. A smooth map f : N -+ M of an n-manifold into an m-manifold is an immersion if f has constant rank n.

3.8.

HOMOTOPY AND ISOTOPY

93

DEFINITION 3.7.7. If i : N + M is a one to one immersion, i ( N ) is called an immersed submanifold of M. If i is also a homeomorphism onto i(N), this is called an imbedded submanifold. The reader should be warned that many authors call an immersed submanifold i(N) c M simply a submanifold. This is misleading because the relative topology inherited from M may not be the manifold topology of N.

EXERCISES (1) For manifolds without boundary, we now have two definitions of imbedded submanifold. Prove that these definitions are equivalent. (2) Let i : N 4 M be a one to one immersion, let X be a manifold, and let f : X + M be a smooth map with f ( X ) E i(N). (a) Show by an example that i-l o f : X + N may fail to be continuous. (b) If i-l o f is continuous, prove that it is smooth. 3.8. Homotopy and Isotopy

Let M and N be manifolds and assume that d M = 0. We will study smooth maps f : M + N. The set of all such maps will be denoted Coo(M,N). If M and N are diffeomorphic, the set of all diffeomorphisms from M to N will be denoted Diff(M, N) C Cm(M,N). When N = M, Diff(M,M) is a group and will be denoted Diff (M). Since d M = 0, one easily verifies that M x [O,1] is a manifold with boundary. DEFINITION 3.8.1. Elements fo, fl E Cm(M,N) are said to be (smoothly) homotopic if there is a smooth map H : M x [O,1] + N such that (1) f o b ) = H(x, O), v x E M; (2) fl(x) = H(x,l), Vx E M. We write fo fl. The map H is called a (smooth) homotopy between f o and

flOne should think of a homotopy as a smooth deformation of one smooth map to another through smooth maps. It is a "smooth curve" in C "(M, N) connecting fo to fi. We write ft(x) = H(x, t ) , 0 t 1. Similarly, smooth curves in Diff (M) will be smooth deformations, called isotopies, of one diffeomorphism to another through diffeomorphisms.

<
0 and d M = 8 = dN. The manifold M will be compact and N will be connected. If f E Coo(M,N), choose a regular value y E N of f . By Theorem 3.7.3, f-'(y) is a O-dimensional submanifold, hence a set of isolated points. Being a closed subset of a compact space, it must therefore be a finite set. Let k = If (y)I denote the cardinality of this set, an integer 0.

>

THEOREM 3.9.1 (STACKOF RECORDS THEOREM).Under the above hypotheses, let f-'(y) = {PI,.* . , ~ k h Then, if k > 0, there exists an open neighborhood U of y in N such that

a disjoint union, where U, is an open neighborhood of pi in M that is carried by f dzfleomorphically onto U, 1 5 i 5 k.

PROOF.By the inverse function theorem, choose an open neighborhood Wi of pi in M that is carried by f diffeomorphically onto an open neighborhood f (W;) of y in N , 1 5 i 5 k. Since M is Hausdorff, the neighborhoods Wi can be k assumed to be pairwise disjoint. Since M is compact, so is X = f ( M \ U,=, Wi), and this set does not contain y. Let U = f(Wl) n . - -n f(Wk) \ X , an open neighborhood of y in N. Let Ui = f - ' ( ~ )r l W,, 1 5 i 5 k. It is obvious that f carries each U, diffeomorphically onto U and that Ui C f -'(U). But, if k x E f-'(U)\Ui=,Ui, then f(x) ~X,contradictingthefactt h a t X n U =a.

u;,

COROLLARY 3.9.2. The set R of regular values of f is an open, dense subset of N . The function At : R + Z+, defined by Xt(y) = If-'(y)l, is constant on each connected component of R. Indeed, we already know that R is dense and Theorem 3.9.1 shows that it is open. The theorem also shows that X is locally constant, hence constant on each component. DEFINITION 3.9.3. If y E N is a regular value of f , then deg2(f , y) E the mod 2 residue class of Xf (y) .

Z2is

3. GLOBAL

96

THEORY

LEMMA3.9.4 (HOMOTOPY LEMMA). If f , g E C m ( M ,N ) are smoothly homotopic and if y E N is a regular value for both f and g, then deg2(f,y) = deg2 (g, Y). PROOF. Let H be a smooth homotopy of f to g. We consider two cases. Case 1. The point y is also a regular value of H : M x [ O , l ] + N. By Theorem 3.7.3, H (y) is a properly imbedded 1-manifold in M x [0,I]. This submanifold is compact (as a closed subset of M x (0, I]) and

-'

a ~ - (y) ' = H-'(p)

n ( M x {0) U M

x (1)) = f

-'(y) x (0) u g-'

(y) x {I).

+

It follows, by Corollary 1.6.10, that X (y) A,( y) is an even integer, hence that deg2(f, Y) = deg2(g, Y). Case 2. The point y is not a regular value of H. It is, however, a regular value for both f and g, so Corollary 3.9.2 implies that there is an open neighborhood W of y in N on which both Xf and A, are defined and constant. The set of regular values of H is dense, so we choose such a regular value z E W. By Case 1, we get deg2(f, z) = deg2(g, z) , but Xf (9) = Xf (z) and X,(y) = A, (z), so deg,(f,y) =deg,(g,y). Let z E N . By Corollary 3.8.8, choose f f for which z is a regular value. Then deg2(f ,z) is independent of this choice, so we set N

unambiguously, obtaining a function

By Theorem 3.9.1, this function is locally constant. By the connectivity of N, deg, (f) is constant. D E F ~ N I T I3.9.5. ~ N The element deg2(f ) E Z2 is called the degree (mod 2) of f E Cm(M,N). COROLLARY 3.9.6. Iff,g E Coo(M,N ) are homotopic, then

LEMMA3.9.7. If f : M

+N

is not surjective, then deg2(f ) = 0.

PROOF.Any z E M which is not a value of f is a regular value, so deg (f) is the residue class mod 2 of Xf(z) = 101 = 0. COROLLARY 3.9.8. If N is not compact, deg2(f ) = 0. DEFINITION 3.9.9. A map f E C m ( M ,N ) is essential if it is not homotopic to a constant map. A manifold M is contractible if idM is not essential. COROLLARY 3.9.10. If degz(f )

# 0, then f

is essential.

COROLLARY 3.9.11. If M is compact and connected with empty boundary, then M is not contractible.

3.9. DEGREE THEORY

97

PROOF. Indeed, deg2(idM)= 1, so id^ is essential.

THEOREM 3.9.12 (BOUNDARY THEOREM).Suppose that M = dW for a compact manifold W and let g E Cm(M,N). If g extends to a smooth map

then deg2(g) = 0. PROOF. Let y E N be regular, both for G and g = dG. Then G l l ( y ) is a compact, onedimensional manifold with dG-'(y) = G-' (y) r l dW = g-' (y). As usual, this set has an even number of elements, so deg2(g) = 0. EXAMPLE 3.9.13. Let f : C -, C be smooth and let W c C be a compact region bounded by smooth, closed curves. A basic question is whether or not f has a zero in W, assuming that f has no zeros on d W. Define g : dW + S by

'

a smooth function between compact 1-manifolds, S being connected. If f (z) # 0, V z E W, then g extends smoothly to G : W + S by

'

hence deg2(g) = 0. This will be enough to prove "half" of the fundamental theorem of algebra. 3.9.14. I f f : C THEOREM a zero in C .

+C

is a polynomial of odd degree m, then f has

PROOF. No generality is lost in assuming that f has leading coefficient 1. Write f (z) = zm alzrn-I . . . am,

+

+ +

and define a homotopy by H(z, t) = f t ( t ) = tf (z)

+ (1 - t)zm = zm + t(alzrn-' + - .. + a,).

Then, fO(z)= zm and fl(z) = f(z). Suppose that W, C C is a closed disk, centered at 0 and of radius r > 0. We claim that, for r sufficiently large, f t has no zeros on dWrj 0 5 t 5 1. Indeed,

and the term in the parenthesis converges to 0 as z Thus, for r > 0 sufficiently large, we define

+ oo.

G : aw, x [o, 11+ s1

3. GLOBAL THEORY

98

This is a homotopy between

and

re re'') = ei n 0 , so deg2(Go) = deg2(G1). But G;' (TJ) contains exactly m points, V y E S1, SO deg2(GI) = 1 since m is odd. It follows, by Example 3.9.13, that f has a zero in w,. There is an integer-valued degree for maps f f C (M, N) when M and N are both orientable. This can be used to give a proof of the full fundamental theorem of algebra. We will take this up in Chapter 8 when we study differential forms and de Rham cohomology.

DEFINITION 3.9.15. Let W be a compact manifold, possibly with boundary, and let X & W. A retraction of W to X is a smooth map f : W -,X such that f lX = idx. THEOREM 3.9.16. If W is a compact manifold with dW # 0 connected, then there is no retraction f : W + dW. PROOF.Indeed, deg2(fldW) = 1, since fldW = idaw, and this contradicts the existence of the extension f : W + d W by Theorem 3.9.12.

is smooth, there is a point x E D n such that f (x) = x.

PROOF.Suppose f has no fixed point. Define g : D n + dDn = Sn-l as follows. For each x E Dn, construct the ray R, starting at f (x) and passing through x # f (x). Let g(x) be the unique point R , n S1. If x E dDn, it is clear that g(x) = x, so if g is smooth, we have contradicted Theorem 3.9.16. The smoothness of g is left for the reader to check (Exercise (1)). Another famous theorem that can be proven using mod 2 degree is the JordanBrouwer separation theorem (smooth version). We introduce the key idea, that of "winding number", and then, in the exercises, lead you through a proof of the separation theorem in the plane (the smooth version of the Jordan curve theorem).

DEFINITION 3.9.18. Let f : S1 4 R2 be a smooth map and let p E R 2 \ f (sl). Define jp: s1+ s1

by the formula

3.9. DEGREE THEORY

99

where 11. I( denotes the usual Euclidean norm. Then the (mod 2) winding number of the closed curve f around p is

Remark that the winding number is defined for an arbitrary smooth closed curve f . It is not required that f be an imbedding or even an immersion. In the case that f is a diffeomorphic imbedding (a smooth Jordan curve), you will show in the exercises that the open set IR2 \ f (sl) has exactly two components, distinguished from one mot her by the fact that w (f , p) = 0, for every point p in one component, and w2(f ,p) = 1, for every point p in the other. The component in which w2(f , p) = 0 is unbounded (and called the "outside" of f ( S I)), while the other component is bounded (the "inside"). A key lemma, to be proven in Exercise (3), is the following.

LEMMA3.9.19. Let f be a smooth Jordan curve, let U be a connected component of R2 \ f(S1),and let p,q E U. Then wz(f,p) = w ~ ( f , q ) . DEFINITION 3.9.20. If p E R2, the ray in R2 out of p and having direction given by the unit vector v E S1 will be denoted Rp(v). The second key lemma, to be proven in Exercise (4), is the following.

LEMMA3.9.21. I f f : S1+ R2 is a smooth Jordan curve and p E R2 \ f (S1), then v E S1 is a critical value of f p : S1 + S1 if and only if the ray RP(v) is somewhere tangent to the Jordan curve f . EXERCISES (1) In the proof of Corollary 3.9.17, prove that the map g is smooth. (2) Prove that Theorem 3.9.17 holds when f is only assumed to be continuous. For this, use the Weierstrass approximation theorem to construct, for each E > 0, a Coofunction f, : D n+ D nsuch that 11 f (x)- f,(x)(l 5 e, Vx E Dn.(The Weierstrass theorem asserts that, on a compact set such as Dn c Rn,any continuous function f : D n + Rn can be uniformly a p proximated to within any S > 0 by a polynomial function P : D + Rn,) (3) Prove Lemma 3.9.19. (Hint: The mod 2 degree is a homotopy invariant .) (4) Prove Lemma 3.9.21. (5) Prove the smooth Jordan curve theorem: I f f is a smooth Jordan curve, then R2 \ f(S1) has exactly two components, one of which (called the inside off) is bounded (i.e., has compact closure) and the other of which (the outside o f f ) is unbounded. For every point p in the outside o f f , w2(f,p) = 0, and for every p on the inside, w2(f , p) = 1. Finally, f (S1) is the set-theoretic boundary of each of these components. Proceed as follows. (a) If p E Kt2 \ f (S1), prove that w2(f, p) is the number of points f (S1),for v E S1 any regular value of f p . (Hint: mod 2 in Rp(v) 17 Use Lemma 3.9.21.)

100

3. GLOBAL THEORY

(b) Use (a) to prove that there are points p, q E R2 \ j(S1) such that ~ ~ ( f# ,w~2 ( )f , q ) By Lemma 3.9.19, conclude that R2 \ f (S1) has at least two components. Also remark that the winding number is 0 about points in at least one of these components and is 1 about points in at least one of the other components. (c) Using the fact that f is a smooth imbedding, choose a coordinate chart (U, u, w) about any point j ( z ) in which and f (S1) n U is just a horizontal line segment w r 0. Show that every point of R2 \ f (sl) can be connected by a continuous path in R2 \ f (S1) either to the point (0,l) E U or (0, -1) E U. This proves that R2 \ f (S1)has at most two connected components. (d) Prove that one of these components is bounded, that the other is not, and that the winding number of f is 0 about the points in the unbounded component. (e) Show that each point of f (S1) lies on an arbitrarily short arc that meets both components. Conclude that f (S1) is the common settheoretic boundary of these components.

CHAPTER 4 Flows and Foliations

In this chapter, we investigate the global theory of ordinary differential equations (flows), referred to as O.D.E., and the Frobenius integrability condition for kplane distributions (foliations). Although this latter topic concerns global partial differential equations (P.D.E.) , our approach will be largely qualitative, with very few explicit partial differential equations in evidence. Unless otherwise indicated, all manifolds will have empty boundary. 4.1. Complete Vector Fields The space X(M) of smooth sections of the tangent bundle is a module over the algebra Coo(M) and a vector space over R. Viewed as the space of derivatives of C m ( M ) , X(M) is a Lie algebra over R. By the local theory of O.D.E., for each q E M , a vector field X E X(M) generates a local flow @ : (-e, E)x V 4 U ,where (U, xl, . . . ,xn) is a local coordinate chart about q, V is an open neighborhood of q with compact closure in U, and E > 0 is sufficiently small. Any two local flows generated by X agree wherever both are defined. The notion of a local flow about a point makes sense even when V and U are not coordinate neighborhoods. A system of suitably coherent local flows covering a manifold M will be called a local flow on M. Here is the precise definition. DEFINITION4.1.1. A local flow

on M is a family of smooth maps

written P ( t , x) = @r(x),such that (1) V, C U, C M are open sets and {Va)aEa covers M; (2) aa and @P agree on ((-c,, c,) rl (-fp, ea)) x (V, rl VP), V a,,B E a; (3) : V, + U, is the inclusion map, V cu E IZl; = atq o wherever both sides are defined, V a E 94. (4)

DEFINITION 4.1.2. If @ is a local flow on M and q E M, curves of the form (q), -E, < t < e,, where q E V,, are called flow lines of through q. s t ( t )=

102

4. FLOWS AND

FOLIATIONS

r

By the definition of local ffow, any two flow lines s and s t through q agree on their common domain (-fa, E m ) f l ( - E @ , E ~ ) .Thus, the velocity vector

is well defined, V q E M , and this yields a vector field X E X ( M ) . The definition also implies that X.: = ir ( t ), V t E (-E, , e,). DEFINITION 4.1.3. The vector field X obtained from the local flow @ as above is called the infinitesimal generator of @. The proof of the following lemma will be Exercise (1). LEMMA4.1.4. Every vector field X E X ( M ) the infinitesimal generator of a local flow @ on M . If two local flows @ and \E have the same infinitesimal generator X , then @ U Q is a local flow with the same infinitesimal generator X . Consequently, by partially ordering local flows by inclusion we see that, given a local flow @ on M , there is one and only one maximal local flow on M containing @. COROLLARY 4.1.5. Them is a one to one comspondence between maximal local flows @ on M and vector fields X E X ( M ) given by letting X be the infinitesimal generator of @. Working with local flows can be somewhat uncomfortable. Happily, there are natural situations in which the maximal local flow of a vector field contains an honest (i.e., global) flow. DEFINITION4.1.6. A (global) flow on M is a smooth map @ : R x M written a t ( x ) = @(t, x), such that

--+

M,

(1) = idM; (2) @ t 1 + t 2 = atl 0 ata, v t l ,t 2 E El. If the maximal local flow of a vector field X E X ( M ) contains a global flow, we say that X is a complete vector field. In this case, X is also called the infinitesimal generator of the global flow. EXAMPLE 4.1.7. We define a global ffow on T~ = S1x S1. Fix p E R and, for each t = (eZRia,elnib) E T' and t E R, define

It is clear that this defines a global flow @ p on T ~Of. some interest is the way in which the qualitative behavior of this flow depends on the value of the constant p. If p is rational, there is a least positive integer k such that pk is also an integer. Thus, (e2nia, elnib) = qj;(e2nia , elnib). One concludes rather easily that each flow line @ p ( R x (2)) is an imbedded circle, the flow being periodic of period k (cj. Exercise ( 5 ) ) . By contrast, if p is irrational, each flow line @p(R x (2)) is a one to one immersed copy of R that is everywhere dense in T ~ This . is the two dimensional version of a theorem of Kronecker that we will prove in Chapter 5 (Example 5.3.8). The two

4.1. COMPLETE VECTOR FIELDS

103

dimensional version will also follow from a result to be proven in Section 4.3 (cf. Corollary 4.3.10 and Exercise (2) on page 116). Remark that the infinitesimal generator X of the flow @ p "lifts" to a well defined vector field on R2 relative to the canonical projection

More precisely, the constant vector field

2 E X(R2), defined by

-

satisfies

P*(",Y)X(",V)= XP(",V)? for all (x, y) E IR2. The reader can check this easily. It is noteworthy that the vector field 2 on lit2 is also complete, generating the translation flow

This lifted flow is quite tame, regardless of whether p is rational or irrational. Not every vector field is complete. For example, Exercise (2) on page 63 showed that et $ E X(R) is not complete.

LEMMA4.1.8. If the maximal local flow of X E X ( M ) contains an element of the form @ : (-€,

with

6

> 0, then X

PROOF.Let t

€1 x

M

+M

is complete.

E R. Then one can find k E Z and

t = r + k 4 2 . Given x E M, define

r E ( - ~ / 2 ,c/2) such that

If @t(x)is well defined by this formula, -oo < t < oo, then it will be an integral curve to X . To see this, remark that (x), - $ < r < ~ / 2 ,is integral to X and use the fact that (X) = X (Lemma 2.8.12). We show that at(x) is well defined. For simplicity, let t > 0. Obvious modifications of the argument give the general case. Suppose that

where s,r E ( - ~ / 2 ,~ / 2 )and k, q E Z. It follows that r - s E (-E, E), hence that q - k = 0,1, or - 1. If q - k = 0, then r - s = 0 and we are done. Assume,

4. FLOWS AND FOLIATIONS

104

therefore, that q - k = f1. Without loss of generality, take q - k = 1. Then, r - S = €/2, SO

Thus, Qt(x) is well defined, -00 < t < oo, Vx f M , and is an integral curve to X. We must show that @ : R x M + M is smooth. Let (to,xo) E R x M. For smallenoughq > 0, we fix ko E Z such that t = r+ko.e/2, V t E (to - q , t o + q ) and suitable r E (-~/2,€/2). Then, (to - q, to q) x M is an open neighborhood of (to,xo) in R x M on which

+

This is a smooth function of (r,x) = (t - koc/2, x), hence a smooth function of (t, 2). THEOREM 4.1.9. If X E X ( M ) has compact support, then X is complete. PROOF.Since supp(X) is compact, cover it with open subsets U1,. . . , Ur C_ M such that the local flow of X contains elements

1 5 i 5 r. Let Uo = M

\

supp(X), an open set with X(UoE 0. Define

by a:(+) = I , Vx E UO,Vt E IR. Since {Ui)i=o covers M and the 0' agree on - - E i and overlaps, we have a local flow on M generated by X. Let e = min l 0 and a smooth imbedding

such that @t(4

= cp(e2 r i t / c o

1 7

-00

< t < 00.

In this case, the flow line Rx is the imbedded circle c p ( ~ land ) we say that x is a periodic point of the flow of period co. (6) Let 9 be a flow on M. A subset C M is said to be a minimal set of @ if (a)

C # 0;

(b) C is closed in M; (c) C is @-invariant; (d) C contains no proper subset with all of these properties. For example, if x is a periodic point (Exercise (5)), the flow line R x is a minimal set. Prove that, if M is compact, every closed, nonempty, @invariant subset of M contains at least one minimal set. (In particular, by Exercise (4), every flow line approaches at least one minimal set.) Show by an example that M itself may be a minimal set. (7) Let @ be a flow on M. One defines the a-limit set and the w-limit set of a flow line Rx as follows:

a ( x ) = (y E M 1 3 t k 1-00 such that lim @t,(x) = y) k--roo

w(x) = { y E M 1 3 t k f

00

such that lim k+oo

at,(x) = y).

If M is compact, prove that each of these limit sets is a compact, nonempty, @-invariant set. Show by examples that a(x) and w(x) may or may not be equal and may or may not be minimal. (Remark: The a- and w-limit set terminology is standard and seems to have its origin in a biblical quotation (Revelation 1:8).)

4.2. LIE BRACKET

107

4.2. The Lie Bracket

As we have already remarked, X(M) is a Lie algebra over R under the Lie bracket. Vector fields D, when viewed as derivations of the function algebra C" (M), localize to open subsets U c M. This localization is equivalent to the restriction DIU of D as a smooth section of the tangent bundle, so it is elementary to check the following. LEMMA4.2.1. If X, Y E X(M) and if U

M is open, then

In particular, properties of the bracket that were proven with coordinates can be extended to global properties on M. We apply this remark to Lie derivatives. Since every vector field X E X(M) generates a local flow on M, the definition of the Lie derivative Lx(Y) = lim t+o

a-t*(Y) t

-

y

makes sense pointwise on M and defines a new field L x ( Y ) E X(M). The proof that Lx (Y) = [X,Y] (Theorem 2.8.14) that was given in R can be carried out in local coordinate charts hence, by Lemma 4.2.1, globalizes. THEOREM 4.2.2. If X, Y E X(M), the Lie derivative of Y b y X is defined and smooth throughout M and

Similarly, Theorem 2.8.18 globalizes. Here, commutativity of local flows @ = and !I' = {!I'P}pEB means that !I'f o 8: = o !I'f wherever both sides are defined. Commutativity of vector fields X, Y E X ( M) means that [X,Y] = 0 on M. THEOREM 4.2.3. Vector fields X , Y on M commute if and only zf the local flows they generate on M commute. COROLLARY 4.2.4. Complete vector fields X, Y E X(M) commute if and only if the flows that they generate commute. We are going to be interested in Lie subalgebras of X ( M ) . If F C_ T ( M ) is a k-plane subbundle (also called a k-plane distribution on M), r ( F ) C X(M) is a Coo(M)-submodule and a real vector subspace. It is not generally a Lie subalgebra but, when it is, there are important geometric consequences.

DEFINITION 4.2.5. The k-plane distribution F

C_ T(M) is a F'robenius distri-

bution if r ( F ) is a Lie subalgebra of X(M). REMARK.If f , g E C w ( M ) and X, Y E X(M), it is easy to verify the identity Consequently, if F S T (M) is a k-plane distribution and if X 1, X2, . . . ,Xr E r ( F ) span r ( F ) over C m ( M ) , then F wilI be a F'robenius distribution if and only if [X,, Xj] E I'(F), 1 5 i, j 5 r.

108


EXAMPLE 4.2.6. On the group manifold Gl(n), we will define a couple of interesting Frobenius distributions. First recall (Example 2.7.15) that the space of left invariant vector fields ~ [ ( nc) X(Gl(n)) is a finite dimensional Lie subalgebra, canonically identified with the Lie algebra im(n) of n x n real matrices under the commutator bracket. Let s[(n) c g[(n) be the set of matrices of trace 0 and let o(n) c g[(n) be the subset of skew symmetric matrices. These are clearly vector subspaces and the reader should have little difficulty in computing dim s[(n) = n 2

-

1

n(n - 1) 2 Since tr(AB) = tr(BA), we see that tr[A, B] = 0,hence s[(n) is a Lie subalgebra of gI(n). If A, B E o(n), then dim o (n) =

so o(n) C gl(n) is also a Lie subalgebra. If {S1,.. . ,S n 2 - l ) is a basis of sI(n), extend it to a basis of g[(n) by adjoining a suitable vector S n 2 and view { s ~ )as $ a~set of left invariant vector fields on Gl(n). It is clear that, at each point P E Gl(n), these fields give a basis {PSI,.. ., PSn2)of Tp(G1(n)). Let Sp c Tp(Gl(n)) be the (n2- l)-dimensional subspace spanned by { P S ~ ) ~and ; ' remark that this subspace does not depend on the choice of basis of s[(n), Let

This is an (n2- 1)-plane distribution on Gl(n). Indeed, the vector fields {s~):!~ define an explicit trivialization of T(Gl(n)) r Gl(n) x ~ [ ( n relative ) to which S becomes the trivial subbundle Gl(n) x s[(n). (Warning: We are not using the standard trivialization of T(Gl(n)) = Gl(n) x im(n). This would give an imbedding Gl(n) x sI(n) c T(Gl(n)) different from the one we have defined and not very interesting.) Similarly, extending a basis of o(n) to one of gl(n) and viewing these as left invariant vector fields on Gl(n), we obtain a distribution 0 c T(Gl(n)) of fiber dimension n ( n - 1)/2 and independent of the choices. By the remark preceding this example, the fact that d ( n ) and o(n) are closed under the bracket implies that S and 0 are Frobenius distributions on Gl(n). Also, by our construction, if P E Gl(n), then

Recall that the special linear group is the subgroup Sl(n) c Gl(n) consisting of the matrices of determinant 1 (Example 2.5.5). In Example 2.5.8, we showed that TI(Sl(n)) is the subspace of Tr(Gl(n)) = m ( n ) consisting of the matrices of trace 0. For each Q E Sl(n), LQ carries Sl(n) onto itself, so it follows from our construction that TQ(Sl(n)) = SQ. Also, for arbitrary P E Gl(n),

4.2. LIE BRACKET

109

We say that the left cosets P - Sl(n) are integral submanifolds to the distribution S. In exactly the same way, using Exercise (3) on page 46, we see that the left cosets P - O(n) of the orthogonal group are integral submanifolds to the distribution 0 C T(Gl(n)). These are the first examples of foliations in this course. The left cosets of these groups are the leaves of the foliation integral to the respective distributions. A distribution F on M with (one to one immersed) integral submanifolds through each point of M is said to be integrable. We will see that the Frobenius property is precisely the integrability condition (cf. Example 3.4.16). This is the theorem of Clebsch, Deahna, and Frobenius, commonly called the Frobenius theorem. Generally speaking, a smooth map f : M -+ N does not push a vector field X E X(M) forward to a vector field f, (X) E X(N). There are two problems. If f is not surjective, there would be points of N where f ,(X) would not even be defined. If f is not injective, there could be points of N where f , (X) would be multiply defined. Nevertheless, there are situations in which f fails to be bijective, but the following concept makes sense.

DEFINITION 4.2.7. If f : M -+ N is a smooth map between manifolds (possibly with boundary), vector fields X E X(M) and Y E X(N) are said to be f-related if, for each q E M , f,,(X,) = Yf (,). EXAMPLE 4.2.8. It is possible that X E X(M) is not f-related to any Y E X(N). For example, let f : R -+ S1 be the map f (t) = eZTit.Then

There is clearly no vector field on S1 satisfying this. EXAMPLE 4.2.9. It is possible that X E X(M) may be f-related to many vector fields in X(N). For example, let f : S1 -+ (C be the inclusion map. Let U c @ be an open neighborhood of S1 such that U # @. Let X : @ -t [ O , l ] be smooth such that supp(X) C U and XIS1 1. Define the vector field Y, = iz on C. Since z I iz, we obtain X E X(S1) by setting X, = iz, Vz E S1. Clearly, X is f -related to both Y and XY and these are distinct fields on @

-

--

PROPOSITION 4.2.10. Let f : M -+ N be smooth and let X , Y E X(M) - -be f -related to X , Y E X(N), respectively. Then [X,Y] is f -related to [ X ,Y]. Equivalently, Lx (Y) is f -related to La(?). PROOF.For each h E C w ( N ) and for each q E M , Y(h

f )(q) = Yq(h 0 f ) = f*q(Yq)(h)= Yf (*) (h) = ( F ( h ) 0 f)(q).

That is, Y(ho f ) = F(h) 0 f , Vh E C w ( N ) .


110

There is a similar relation between X and

2.Thus,

V h E Ca(N). That is,

This is the assertion of the proposition.

EXERCISES (1) Let f : M -+ N be a one to one immersion. Let Y E X(N). Prove that there exists a field X E X(M) that is f-related to Y if and only if Yf( q ) E f.,(T,(M)), V q E M. In this case, prove that X is unique (we call X the restriction of Y to the immersed submanifold). (2) Let f : S2n-1L) R2n be the inclusion (an imbedding, hence a one to one immersion). Find Y E X(R2n)that restricts, as above, to a nowhere zero vector field X E X(S2"-I). (By Theorem 1.2.12, this is false for the inclusion f : S2n R2n+1. This fact is more elementary than Theorem 1.2.12, however, and will be proven in Chapter 8, Theorem 8.7.5.)

-

4.3. Commuting Flows

An Rn-chart (U, xl, . . . ,xn) about q E M determines n commuting vector fields

The corresponding local flows about q are of the form

Conversely, the following implies that commuting, linearly independent vector fields correspond to a coordinate chart.

THEOREM 4.3.1. Let M be an n-manifold without boundary, let q E M, let

U be an open neighborhood of q, and let X I , . . . ,Xk E X(U). If these vector fields commute and if {x:, . . . ,x;} is linearly independent, then there is a local coodinate chart (W,(P) about q such that

4.3. COMMUTING FLOWS

111

PROOF.Making U smaller, if necessary, we assume that it is a coordinate chart, hence view it as an open subset of Wn. We can do this so that q becomes the origin 0 and so that the vectors

form a basis of To(U). Let

be a local flow about q = 0 generated by X', 1 5 i 5 k. We can choose of the form (-E, E )with ~ E > 0 so small that the formula 8(x1,. . . , x") = a:, is defined, V (zl,. . . ,xn) E

0

.

0

% to be

02, (0,. . . ,o, x k+l , . . . ,xn)

F.This defines a smooth map

Since [X" X j ] = 0 on U, 1 5 i, j 5 k, we have

for each permutation a of {1,2,.. . ,k). Also,

-

For r = (rl, ... , r n ) E W and 1 < i 5 k,

Here, the notation @fi is a common device for indicating omission of the term . In particular, B,O : Wn + Wn is nonsingular, so we can assume (choosing r > 0 smaller, if necessary) that 0 : f? -+ U carries % diffeomorphically onto an open neighborhood W of 0. By the above,

The desired coordinate chart (W,cp) is obtained by setting cp = 8 -' .

112

4. FLOWS AND

FOLIATIONS

In particular, if X is a nonzero vector field defined in a neighborhood of q E M , then there is a coordinate chart (U, x l , . . . ,xn) about q such that

DEFINITION 4.3.2. A k-flow on M is a smooth map 9 : R x M 9(v, x) = 9,(x), such that (1) 90= idM (2) QVfW= 9, 0 Qw, v v , w E Rk.

-+

M , written

It follows that 9-, = \EL', so 9, E Diff(M), V v E IRk. The map Rk -+ Diff (M), defined by v I+ 9,, is a homomorphism of the additive group JRk into the group Diff (M). We think of 9 as a smooth action of the group R on M. Given q E M define 9 4 : IRk 3 M by setting 'Q4(v) = Qv(q). In particular, \kg (0) = q. DEFINITION 4.3.3. Given a k-flow 9, the S-orbit of q E M is 99(IRk). When a choice of 9 is fixed, the Q-orbit is also called the Rk-orbit of q.

REMARK. Points p, q E M are said to be equivalent under the k-flow if and only if there exists v E Rk such that 9,(p) = q. The fact that this is an equivalence relation is a trivial consequence of Definition 4.3.2. The R k-orbits are the equivalence classes. DEFINITION 4.3.4. The k-flow Q is nonsingular if

is one toone, Qv E Rk, Vq E M. LEMMA4.3.5. The k-flow Q is nonsingular if and only if

is one to one, V q E M .

PROOF. For fixed v E Itk, let

T,

: IRk

-+Rk denote translation byv. Then,

Vv E Rk, Vq E M. By the chain rule,

But (T-~),, = idRk and 9:o is one to one.

is bijective, so \k$, is one to one if and only if

4.3. COMMUTING FLOWS

113

PROPOSITION 4.3.6. The set of k-tuples ( X l , . . . ,x') of complete, commuting vector fields on M is in natural, one to one correspondence with the set of k-Bows Q o n M . The fields X I , . . . , xk are pointwise linearly independent i f and only i f Q is nonsingular.

PROOF. Given the k-tuple (X1,. . . , x k ) of complete, commuting fields, let a', . . . ,@ be the corresponding flows. Since these flows commute, the formula defines a k-flow on M. Conversely, given the k-flow Q, let { e 1, . . . ,e k} be the standard basis of IRk and set

af = Q t e ; , 1 5 i 5 k. This defines k commuting flows and their corresponding infinitesimal generators, X I , . . . are complete, commuting vector fields on M. Finally, these fields are linearly independent at q E M if and only if Q:, is one to one. Thus, the previous lemma gives the final assertion.

,xk

THEOREM 4.3.7. Let M be a connected n-manifold. If there exists a nonsinfor some integer gular n-flow on M , then M is difleomorphic to T~ x k = 0 , 1 , ..., n. Modulo one technical point, the proof of Theorem 4.3.7 is quite straightforward. Nonsingularity of the n-flow implies that, for each q E M , the smooth map 9 9 : Rn -+ M has constant rank n. Consequently, each Rn-orbit is an open subset of M. Being equivalence classes, distinct orbits are disjoint, so the connectivity of M implies that there is only one Rn-orbit and Qq is a surjection. Fix q E M. If v , w E Rn, then

It is clear that G = (Qg)-l(q) is an additive subgroup of R n , so Qq passes to a well defined homeomorphism

If G were the subgroup 2Zk x {O} c C kx

we would have a natural structure of smooth n-manifold on Rn/G = T~ x IRn-k and, 9 9 being locally a diffeomorphism, the homeomorphism

would be a diffeomorphism. Therefore, it will be enough to find a linear automorphism A : Rn + Rn such that A(G) = 2Zk x (0). This is the technical point mentioned above.

4. FLOWS AND

114

FOLIATIONS

DEFINITION 4.3.8. An additive subgroup G C Rn is a kclimensional lattice if it is generated by a linearly independent subset { v . . . ,v k ) C Rn . If G is a k-dimensional lattice for some k = 0,1, . . . , n, then G is called a lattice subgroup.

0, one can find m i E Z such that 11 fi-migoll < C , V i -> 1. By Lemma 4.3.12, { fi-migo)El c G contains only finitely many distinct elements. That is, Ti = fi (GnV) = (fi - migo) -( G n V) assumes only finitely many distinct values as i --t oo. Since limi+oo f i = 0, we conclude that Ti = 0 for large enough values of i. Therefore, there is a neighborhood U' C Rn+'/V of 0 such that U' n (G/(G n V)) = (0). By the inductive hypothesis, G/(G n V) is a lattice generated by a linearly independent set {gi (G n v))L, C Rn+'/V. Given g E G, write

+

+

+

+

That is, there is ro E Z such that

Thus, {go,91, . . . ,ge) generates G and this set is clearly linearly independent in IWn+l.

REMARK.Following Milnor [30],one defines the rank of an n-manifold M to be the maximum number of everywhere linearly independent, commuting vector fields that the manifold admits. By Proposition 4.3.6, the rank of a compact n-manifold M is the largest integer r 5 n for which there exists a nonsingular

4. FLOWS AND

116

FOLIATIONS

r-flow on M. It is a celebrated theorem of E. Lima 1251 that the rank of S3is 1. Since S3 is parallelizable, it admits a nowhere 0 vector field, hence a nonsingular 1-flow. On the other hand, Theorem 4.3.7 implies that the only compact 3manifold of rank 3 is T3. The hard part of Lima's theorem is to show that S3 does not have rank 2.

EXERCISES (1) Let M be an n-manifold, aM = 0. Prove that M is integrably parallelizable (Definition 3.4.8) if and only if there exist pointwise linearly independent, commuting vector fields X l, . . . , Xn (not necessarily complete). Use this t o prove that a compact n-manifold is integrably parallelizable if and only if each component is diffeomorphic to T n. (2) Use Corollary 4.3.10 or Corollary 4.3.11 to give a proof of the assertion in Example 4.1.7 that a line l c R2 of irrational slope projects one to one onto an everywhere dense curve in T2. 4.4. Foliations

Let F T ( M ) be a k-plane distribution on M . For simplicity, we consider only the case in which dM = 8. DEFINITION 4.4.1. An integral manifold of F through q E M is a one to one immersion i : N -+ M of a k-manifold such that q E i ( N ) and

We generally identify N and i(N). This is similar to the customary identification of a curve s : [a,b] + M with its image. The correct topology on the subset i ( N ) of M is the manifold topology of N , not the relative topology. DEFINITION 4.4.2. A k-plane distribution F on M is integrable if, through each q E M , there passes an integral manifold of F.

EXAMPLE 4.4.3. Take M

= R3 and let

F be the 2-plane distribution spanned by the pointwise linearly independent vector fields

Let

7r : IR3 -+

R2 be the projection ~ ( xy,, z ) = (x, 9). Since

4.4. FOLIATIONS

117

an integral manifold of F through q will be carried by 7r locally diffeomorphically onto an open subset of W2. That is, an integral manifold of F is locally a graph = f (x,9) and

That is, f (x,y ) solves the system

This overdetermined system of P.D.E. implies that

That is, a necessary condition for F to be integrable is that -a g= - ah

ay

ax'

It turns out, as we will see, that this is also a sufficient condition for integrability. This integrability condition can also be written in terms of brackets. Indeed,

so the integrability condition becomes

[X, Y J= 0. By our theory of commuting vector fields, this condition implies that there is a local coordinate chart (U, u, v, W) about q in which

118

4. FLOWS AND

FOLIATIONS

and, in this coordinate system, the integral manifolds are given by the equations w = const. Finally, in this coordinate neighborhood, arbitrary fields Z 1, Zz E I'(FJU) are function linear combinations of X, Y, hence integrability implies that [Z1,Zz] E r(F1U). This is true in suitable coordinate charts about each point of W3, hence r ( F ) C X(R3) is a Lie subalgebra and F is a Frobenius 2-plane distribution on R3. This exemplifies the following theorem, due to Clebsch, Deahna, and Frobenius, but generally credited only to the last of this trio.

THEOREM 4.4.4 (THE FROBENIUS THEOREM). If F & T(M) is a k-plane distribution on M , the following are equivalent. (1) F is integrable. (2) F is a Fkobenius distribution. (3) About each q E M there is a coordinate chart (W, x l , . . . ,xn) such that

The proof of the Frobenius theorem is the primary goal of this section. Before giving a proof, however, we discuss some of the consequences. A coordinate chart as in (3) of Theorem 4.4.4 will be called a h b e n i u s chart. Theorem 4.4.4 allows us to find a Cm atlas {(U,, cp,)),En on M such that the associated family of local trivializations

is contained in the Gl(k, n - k)-reduction of T ( M ) corresponding to F. This means that the associated Jacobian cocycle satisfies

Jg,p : U,

n Up

-t

Gl(k,n - k),

V a , p E a. That is, the infinitesimal Gl(k,n - k)-structure determined by F is integrable in the sense of Definition 3.4.11. Fix a coordinate cover {Vx,x i , . . . , satisfying (3) in Theorem 4.4.4 and such that (1) the index set C is at most countably infinite and {VA)XEL: is a locally finite cover of M; (2) x i ranges over the open interval (-2,2), V X E C, 1 5 i 5 n; (3) if Wx C VA is defined by the inequalities -1 < x i < 1, 1 i 5 n, then {Wx)xEc is an open cover of M . This is possible since M is 2nd countable and paracompact. We are primarily interested in the coordinate neighborhoods WA,with the VA's playing an auxiliary role.

0 such that ( l / r ) X E Uo. Thus, cp(expG(l/r)X) E Ve and there is a unique Y,. E Vo such that expH Y, = cp(expG(l/r)X). There is also a unique 2, E (Uo x Vo) n L ( r ) such that Since expG, is one to one on Uo x Vo, this implies that

Take Y =rYT, obtaining ( X , Y ) =rZ, E L(r). COROLLAFW 5.3.10. Let cp : G + H be a continuous homomorphism of Lie groups and let K = ker(cp). Then K is a properly imbedded, normal Lie subgroup of G7 G / K is canonically a Lie group, and the induced map : G / K + H is a one to one immersion of this Lie group as a Lie subgroup of H.

PROOF.Indeed, K is a normal subgroup by standard group theory and K = cp-'(e) is a closed subset of G, so Theorem 5.3.2 guarantees that K is a properly imbedded Lie subgroup of G. By Theorem 5.3.9, cp is a smooth h e momorphism, so Exercise (3) on page 138 guarantees that q ( G ) is a Lie subgroup of H. Obviously, p is an isomorphism of the group G/K onto q(G) , so p can be used to transfer the Lie structure of cp(G) back to G/K.

EXERCISES (1) Let G be a topological group whose underlying space is a topological manifold. Prove that there is at most one differentiable structure on the topological manifold G making G into a Lie group. (The positive solution to Hilbert's fifth problem guarantees that a topological g r o u p manifold does have a smooth (in fact, analytic) structure making it into a Lie group.) (2) Prove the assertion in Example 5.3.8 that the vector v E R n , with rationally independent coefficients, cannot be expressed as a real linear combination of fewer than n elements of the integer lattice Zn. (3) Let v = ( a 1 ,. . . ,a n ) E R" be a point such that 11, a', . . . ,a n ) is linearly independent over Q. Prove that the subgroup A c T n , generated by a = p(v), is everywhere dense. (Hint: Every point in the coset qv Z n has rationally independent coefficients, V q E Z. Prove this and use it

+

5 . LIE GROUPS

144

to show that every 1-parameter subgroup of Tn meeting a nontrivial element of A is dense in Tn.) (4) Show that every finite dimensional Lie algebra contains a nontrivial abelian subalgebra which is not itself contained properly in another such subalgebra. Similarly, show that every compact Lie group G contains a maximal subgroup that is Lie isomorphic to T ~some , k > 1. This is called a maximal t o m of G. (a) When G is compact, prove that the correspondence between Lie subalgebras and connected Lie subgroups sets up a one to one correspondence between the maximal abelian subalgebras of L(G) and the maximal tori in G. (b) Let G be compact and let K E G be a maximal abelian subgroup. Although K need not be connected, you are to use Exercise (3) to prove that there is an element x E K such that the group { x ~ ) , ~ ~ is everywhere dense in K. (c) Let G be compact and connected, K C_ G a maximal abelian subgroup. You are to prove that K is connected, hence that the maximal abelian subgroups of G are exactly the maximal tori. You may use the nontrivial fact that each element of G lies on a t least one 1-parameter subgroup (Corollary 10.7.12 and T h e e rem 10.7.15). (5) Let G be an n-dimensional Lie group and g = L(G) its Lie algebra. Let Gl(g) denote the group of nonsingular linear transformations of the vector space g and let Aut(g) c Gl(g) be the subgroup of Lie algebra automorphisms of 8, Prove the following. (a) Gl(g) has a canonical Lie group structure under which it is (non canonically) isomorphic to Gl(n). Also, for use in Exercise (6), show that L(G1(g)) is canonically the space End(g) of linear endomorphisms of the vector space g, the bracket in End(g) being the commutator product of endomorphisms. (b) Aut(g) is a closed subgroup of Gl(g), hence a properly imbedded Lie subgroup. (c) Assume that G is connected and let C C G denote the center of G, clearly a closed subgroup. Each element a E G determines an inner automorphism of G, denoted by Ad(a) and defined by

Prove that { A ~ ( U ) ) is, ~canonically ~ a Lie subgroup of Aut (g), isomorphic as a group to G/C. This subgroup is denoted by Ad(G) and called the adjoint group of G. (6) Let G be a Lie group and again denote its Lie algebra by g. A derivation D : g + g is a linear transformation such that D [ x , y] = [DX,Y ]

VX, Y

+ [X,D Y ] ,

E g. Let D(g) be the space of derivations of g.

(a) Prove that, under the commutator product, D(g) is naturally identified as a Lie subalgebra of L(Gl(g)) (cf. Exercise (5), part (a)).

5.4. HOMOGENEOUS SPACES

(b) For each X E 0, define ad(X) : g + g by ad(X)Y = [X,Y], V Y E g and prove that ad(X) E D (0). (c) Prove that ad : g + L(Gl(g)) is a homomorphism of Lie algebras. Thus, ad(g) L(Gl(g)) is a Lie subalgebra. (d) Assume that G is connected and prove that the connected Lie subgroup of Gl(g) corresponding to the Lie subalgebra ad(g) is exactly the adjoint group Ad(G). (Hint: Prove that Ad(exp(tX)) = exp(t ad(X)), VX E 0.) 5.4. Homogeneous Spaces Lie groups arise in many natural ways as transformation groups of manifolds. When the group action is transitive, the manifold is called a homogeneous space and one has considerable control over its structure. DEFINITION 5.4.1. Let M be a smooth manifold and G a Lie group. A smooth map p:GxM+M, written p(g, x) = gx, is said to be an action of G (from the left) on M, and G is called a Lie transformation group on M , if (1) g i ( g 2 ~= ) ( 9 1 9 2 ) ~Vgl, ~ 92 E G and Vx E M; (2) ex = x, Vx E M.

REMARK.One can also define a right action

by making the obvious changes in the above definition. DEFINITION 5.4.2. An orbit of the action

is a set of points of the form {gxo I g E G), where xo E M . The action is transitive if M itself is an orbit, in which case M is said to be a homogeneous space of G.

REMARK. It is elementary that the orbits of a group action are equivalence classes, two points x , y E M being equivalent under the action if 3 9 E G such that gx = y. EXAMPLE5.4.3. The orthogonal group O(n) acts on R n in the usual way, leaving invariant the unit sphere Sn-l. Note that, if e l E Sn is the column vector with first entry 1 and remaining entries 0,then Ae 1 is the first column of A E O(n). Every unit vector appears as the first column of suitable orthogonal matrices, so the action 0 ( n ) x sn-'+ sn-'

5 . LIE GROUPS

146

is transitive and Sn-' is a homogeneous space of O ( n ) . In a completely similar way, there is a transitive action

where

s ~c Cn ~ is the - unit ~ sphere in the standard Hermitian metric.

DEFINITION 5.4.4. Let M be a homogeneous space of G and let xo E M. The isotropy group of xo is the set G,, = {g E G I gxo = xo). LEMMA5.4.5. The isotropy group G,, subgroup of G.

as above is a properly imbedded Lie

PROOF.It is obvious that Gxois an abstract subgroup of G. If {gn)F==,c G,, converges to g E G, then, by the continuity of the group action, gxo = lim gnxo = lim xo = xo. n --roo

ndoo

Thus, G,, is a closed subset of G. By Theorem 5.3.2, G,, is a properly imbedded Lie subgroup. EXAMPLE5.4.6. If el E Sn-l is as in Example 5.4.3, the isotropy group

where A E O(n - 1). Similarly, for e 1 E S2n-1,U(n),l is the set

where A E U(n - 1). We are going to show how to put a smooth structure on the quotient space G/G,, and prove that this manifold is diffeomorphic to the homogeneous space M. Under the identification M = G/G,, , the G-action on M becomes the act ion G x G/Gx, GIG,,, g(hG,,) = (gh)G,,. In what follows, we consider an arbitrary properly imbedded Lie subgroup H C_ G, put the quotient topology on G/H, and construct a natural smooth structure on this space. Throughout this discussion, we set tj = L ( H ) . Decompose L(G) = m @ b, where m is any fixed choice of complementary subspace. Let $:m@b--+G be the map $(A, B)= exp(A) exp(B) and choose a neighborhood V of 0 in b and a neighborhood W of 0 in m such that .11, sends W x V diffeomorphically onto a neighborhood U of e in G. Choose a compact neighborhood C c W of 0 with the property that -C = C and exp(C) exp(C) c U. We can assume that coordinates x l , . . . , xk in tj define V by the inequalities - 1 < xi < 1, 1 5 i 5 k. Similarly, coordinates y l , . . . , y4 for m define C by -+

148

5. LIE GROUPS

LEMMA5.4.10. With the above Cooatlas, G / H is a smooth manifold, the projection n : G + G / H is a submersion, and the action

defined b y p(a, bH) = abH is smooth.

COROLLARY 5.4.11. If G is a Lie group and H G is a closed, normal subgroup, then the group G / H has a smooth structure in which it is a Lie group. We return to the smooth transitive action

Let xo E M and let H = G,, be the isotropy group. Define the map by B(aH) = ax0 . This is induced by the smooth map 8 : G -+ M, #(a) = ax0 , so 19 is continuous. Since aH = bH if and only if a-'b E H , we see that xo = a-'bxo, hence ax0 = bxo, if and only if aH = bH. That is, 9 is well defined, one to one, and continuous. Since the action of G is transitive, 13 is a surjection. The following diagram is commutative:

G x M

-

M

P

Thus, if we prove that 0 is a diffeomorphism, 9 will be a canonical identification of G / H with M as a homogeneous space of G.

PROPOSITION 5.4.12. The map I3 : G / H

+M

is a difleomorphism.

PROOF.Let La denote left translation by a E G on both G / H and M. That is,

La(bH) = abH, L ~ ( x=) a x ,

VbH E G / H , V X EM .

Then

8=Lao80La-I, VaEG. Since La : M + M and L a - I : G / H -+ G / H are diffeomorphisrns, it follows that I3 will be smooth at aH if and only if it is smooth at eH. Furthermore, if smoothness has been established, then

will be an isomorphism of T a H( G / H ) onto T a z O ( Mif) and only if

is an isomorphism. We show smoothness at eH. Consider the commutative diagram

5.4. HOMOGENEOUS SPACES

The map

81 exp(Co) : exp(C0) + M is smooth, and the map

is a diffeomorphism onto the coordinate neighborhood U,,so 13 is smooth in a neighborhood of e H . We show that 13*eH : TeH(G/H) -4 Tz,(M) is an isomorphism. Again, this translates, via the commutative triangle, to showing that 8,, is an isomorphism of Te(exp(Co)) onto T,, (M). Let v E Te(exp(Co)) = m and consider the curve s(t) = exp(tv)xo. Since O,,(v) = s(0), we only need prove that i(0) = 0 implies that v = 0. If a = exp(tov), then L,(s(t)) = s(t to), so L,,(s(O)) = i(t0) and i(0) = 0 implies that ~ ( t = ) 0, Vt E R. That is, exp(tv)xo = xo, Vt E R, implying that v E fi m = (0).

+

EXAMPLE 5.4.13. Thus, as a homogeneous space of O(n), where O(n - 1) is properly imbedded as a Lie subgroup of O(n) as in Exam-

ple 5.4.6. Similarly,

~ 2 n - 1- U(n)/

U(n - 1).

EXERCISES (1) Prove Lemma 5.4.8. (2) Prove Lemma 5.4.10. (3) Let Gn,k denote the set of k-dimensional vector subspaces of R n . (a) Show how to make G n l kinto a compact manifold which is a homogeneous space of O(n). This is called the (real) Grassmann manifold of k-planes in n-space. (b) If xo E Gn,k is the standard Rk Rn, identify the isotropy group O(nL0. (c) Show that projective space P"-' is the Grassmann manifold Gntl and identify the standard two-to-one map Sn-I -+ Pn-' as a map O(n)/ O(n - 1) + O(n)/ O(n),, (Remark: Using Cn instead of Rn,one defines in a similar way the complex Grassmann manifolds Gnjk(@)as homogeneous spaces of U(n). Complex projective space is defined to be P n ( @ ) = GnI1(@).The real and complex Grassmann manifolds play an important role in differential geometry and topology.)

150

5 . LIE GROUPS

(4) An orthonormal k-frame in R n is a k-tuple (vl , . . . ,uk) of ort honormal vectors in Rn. Let Vn,k denote the set of d l orthonormal k-frames in Rn, identify this as a homogeneous space of O(n) (called the Stiefel manifold of k-frames in n s p a c e ) . Remark that VnV1= Sn-' and that this case gives back Example 5.4.3. Using C n and the standard positive definite Hermitian inner product on C n , one obtains the complex Stiefel manifolds Vnta(C) and Vn,1(C) = s2"-' . 5.5. Principal Bundles

A good command of the theory of principal bundles is essential for mastery of modern differential geometry (cf. [22], [23]). In recent years, principal bundles have also become central to key advances in mathematical physics (Yang-Mills theory) which have in turn generated exciting new mat hematics ( e.g., Donaldson's work on differentiable 4-manifolds 161, [lo]). In this book, however, principal bundles will only be needed in Section 10.6, where the principal frame bundle will be used to globalize the Cartan structure equations. Let V be a real vector space of dimension n. An n-frame in V is an ordered basis D = (vl,. . . ,v,). If V = Rn, each vi is a column vector and the frame is a nonsingular matrix. That is, the set of all n-frames in R n is naturally identified with the manifold Gl(n). Generally, we denote the set of all n-frames in V by F ( V ) and topologize this set as a subset of Vn = V x . . . x V. Let cp : V -+ R n be an isomorphism of vector spaces and define a diffeomorphism p : Vn -+m ( n ) by ~ ( v l* ., ,vn) = ( ~ ( v l ). ., - , ~ ( v n ) ) . Clearly p ( F ( V ) ) = Gl(n), hence F(V) c Vn is an open subset, diffeomorphic via p t o Gl(n). DEFINITION5.5.1. The smooth manifold F ( V ) is called the frame manifold of V . One might try t o make F(V) into a Lie group via p, but this is a bad idea since, if cpl and cp2 are two isomorphisms of V to R n , it is not generally true that the diffeomorphism pl o (p2)-l : GI(n) -+ Gl(n) is a group isomorphism. That is, there is no canonical way t o make F ( V ) into a Lie group. There is, however, a natural right action

defined by formal matrix multiplication

This is a smooth, transitive action with trivial isotropy group (such an action is said to be simply transitive). Remark that the right action of Gl(n) on itself, defined by the group multiplication, is also smooth and simply transitive and that p : F(V) -+ Gl(n) respects these actions in the following sense.

5.5. PRINCIPAL BUNDLES

LEMMA5.5.2. If cp : V A E Gl(n), then

-+ Rn

151

is a linear isomorphism and if o E F(V),

p(a A) = p(a)A. One says that p is Gl(n)-equivariant. There is a natural map F(V) x R n -+ V defined by formal matrix multiplicat ion

For each fixed a E F ( V ) , this defines a linear isomorphism every linear isomorphism is of this form. We summarize.

P :

Rn

-+

V, and

LEMMA5.5.3. The map defined by (5.2) sets up a canonical identification of F(V) with the set of a11 linear isomorphisms a : R n -+ V. One can canonically reconstruct the vector space V from the frame manifold F(V) as follows. Define the left action of Gl(n) on the space F ( V ) x R n to be the "diagonal action" (5.3)

A . (P,8 ) = (a . A-', AZ),

where A ranges over Gl(n) and (P, a') ranges over F ( V ) x Rn. It is elementary that this is a smooth, left action of the group Gl(n), so there is an associated equivalence relation on F(V) x Rn with the Gl(n)-rbits as equivalence classes. We denote the quotient set by F(V) x ~ l ( Rn. ~ ) The equivalence class of ( P , a') will be denoted by [o, a']. Remark that

We put a vector space structure on the quotient set F(V) x ~ a frame o E F(V) and defining

[o,Zl]

[P, a'z] = [ O , a ' l C[D,

l ( Rn ~ )by

fixing

+Z2]

a'] = [O, &I.

To see that these definitions do not depend on the choice of frame, let u be another choice, let A E Gl(n) be the unique matrix such that u = a A, and use the relation (5.4) to obtain

-. + [u,

[u,all

821 = [O,

= [ol

AZl] + [o,AZ2 ]

+

+ li2)]

= [~,a'l 521 C[U,

a] = c[o,AZ] = [0, A&]

= [.,&].

The map defined by (5.2) passes to a well defined linear map

5 . LIE GROUPS

152

Fixing a frame u E F ( V ) , we define

by

j ( ~Z ) = [o,Z]. It is clear that .1C, and j are mutual inverses, hence that j does not really depend on the choice of frame b. We summarize these remarks in the following.

L E M M A5.5.4. The vector space operations on F ( V ) x ~ l ( Rn ~ ) are well defined, independently of the choice of the frame o, and the map 1C, is a canonical isomorphism of vector spaces, the inverse j being well defined independently of the choice of frame. These remarks generalize, fiber by fiber, to vector bundles. Let 7r : E -+ M be an n-plane bundle over an m-manifold. Each fiber E x = 7rV'(x) is an ndimensional real vector space, so we form the associated frame manifold F ( E x ) and the disjoint union

We also define p : F ( E ) + M by p(F(E,)) = x , V X E M . If

U

-

U

id

is a local trivialization of E, then cpx : E x inducing a diffeomorphism

-+

Rn is a linear isomorphism, V x

E U,

This determines a commutative diagram

where p is bijective, hence defines a topology and smooth structure on p-'(U). It is straightforward to check that, on overlaps p - l ( U ) f7pp1(V),corresponding to two local trivializations of E , the two topologies and smooth structures coincide and t hat this makes F ( E ) into a smooth manifold of dimension m n 2 . Furthermore, p : F ( E ) + M is a smooth submersion and the inclusion map, i x : F ( E x ) F ( E ) smoothly imbeds the fiber p-'(x) as a proper submanifold,

Vx

E

M.

-

+


153

We have defined a new kind of bundle over M, called the frame bundle of E. The fibers F ( E x ) are not vector spaces. Instead, they are manifolds diffeomorphic to Gl(n) which admit a canonical right action of Gl(n). This defines a right action F ( E ) x Gl(n) -+ F(E) which is simply transitive on each fiber. By Lemma 5.5.2, the local trivializations turn this action into the action

defined by ( x , B ) . A = (x,BA). This is clearly a smooth action, so

is locally, hence globally, a smooth action. By Lemma 5.5.3, the fiber F(Ex)over x E M of the frame bundle of E is exactly the set of linear isomorphisms of the "standard fiber" R n to the specific fiber Ex. The construction for recovering a vector space V from its frame manifold F ( V ) globalizes in a natural fashion to give a canonical way of recovering the vector bundle E from its associated frame bundle F(E). The diagonal action of Gl(n) given by (5.3) extends by the same formula to a left action

and we let F(E) x ~ l ( , )Rn denote the corresponding quotient space (with the quotient topology). The bundle projection p : F(E) + M passes t o a well defined map : F(E) x ~ 1 ( , )

Rn

+M

and the map

F(E) x Rn

+ E,

again defined as in (5.2), passes to a well defined map

These maps are continuous and the diagram

commutes. It has also been arranged, by the very definitions, that $ restricts t o $, on the vector space jT1(x) = F(Ex)x ~ l ( , )Rn c F(E) X G ~ ( Rn, ~ ) carrying it isomorphically onto the vector space Ex, Vx E M. The inverse jx of $,,

5 . LIE GROUPS

154

although defined by a choice of frame tt E F(E,), is independent of that choice (Lemma 5.5.4), V x E M , and these fit together to give a global inverse

In a local trivialization of El one can choose n linearly independent smooth sections, using these t o define j. It follows that j is locally, hence globally, smooth and is a diffeomorphism. We summarize this discussion in a theorem.

+

-

THEOREM 5.5.5. The stmcture jj : F ( E ) x ~ l ( , )Rn + M is a vector bundle, canonically isomorphic t o x : E M . The association E F ( E ) is a one to one correspondence between the set of vector bundles over M and the set of associated frame bundles. The frame bundle F ( E ) is an example of a principal bundle. We give the general definition. -+

DEFINITION 5.5.6. Let M and P be smooth manifolds, G a Lie group, and p :P M a smooth map. Suppose that there is an open cover (Ua)aEaof M and, V a E 24,a commutative diagram -+

where $, is a diffeomorphism. Suppose also that there is a smooth right action P x G -, P, simply transitive on each fiber p - l ( x ) . Finally, assume that is G-equivariant with respect to this action and the standard right G-action on U, x G, Vcu E %. Then, p : P -+ M, together with this G-action, is called a principal G-bundle over M.

+,

EXAMPLE 5.5.7. Let p : P -, M be a principal G-bundle and let a : M --t P be a smooth map such that p o a = idM. As in the case of vector bundles, a is called a section of the principal bundle. While a vector bundle always admits sections (e.g., the 0 section) this fails for principal bundles. Indeed, if a is a section, define cp:MxG+P by 4 x 7 g ) = a ( x ) g7 obtaining a diffeomorphism such that the diagram M x G (P P

commutes. As in the case of vector bundles, cp is called a trivialization of the principal bundle P. Thus, the existence of a section is equivalent t o triviality of a principal G-bundle.

5 . 5 . PRINCIPAL BUNDLES

155

EXAMPLE5.5.8. Suppose that the n-plane bundle .rr : E + M is given an explicit O(n)-reduction. Equivalently, there is a positive definite inner product (. , .), on Ex which varies smoothly with 2 E M. That is, (sl(x), s2(x)), is a smooth function of x, Vsl, s2 E r(E) (cf. Lemma 3.4.14 for the case E = T ( M ) ) . In fact, this smoothly varying inner product can be thought of as a smooth "field" of inner products (cf. Exercise (3)), often called a Riemannian metric on the bundle E. One can then define O(E) C F ( E ) to be the subset of frames that are orthonormal with respect to this smooth field of inner products and restrict p to a projection p : O(E) -+ M. If U M is an open, locally trivializing neighborhood for E, let ( s l , . . . , s,) be a smooth frame field on U (i.e., a smooth section of F(E1U)). By an application of the Gram-Schmidt process to this frame field, we can assume that it is everywhere orthonormal, hence is a smooth section a of O(E1U). All of O(EIU) can be swept out by applying right actions of O(n) to a and one obtains a local trivialization

by setting O(x,A) = a(x) A, Vx E U, VA E O(n). Thus, O(E) c F(E) can be thought of as a subbundle which is invariant under the right action of O(n) c Gl(n). In fact, O(n) is simply transitive on the fibers and O ( E ) is a principal O(n)-bundle. The standard application of partitions of unity shows that there are infinitely many choices of Riemannian metrics on E and corresponding orthonormal frame bundles. EXAMPLE5.5.9. The n-plane bundle .rr : E + M always admits O(n)reductions, but we know that it may or may not admit a Gl(k, n - k)-reduction. In fact, such a reduction is equivalently a k-plane subbundle Ek C E (Exarnple 3.4.16). Given such a subbundle, let Fk(E) F ( E ) consist of the frames of E whose first k entries form a frame of Ek. As in the previous example, this forms a locally trivial subbundle of F ( E ) which is invariant under the right action of Gl(k,n - k), a simply transitive action on each fiber. This is a principal Gl(k, n - k)-bundle.

EXAMPLE 5.5.10. An O(n, n - k)-reduction of E corresponds to an indefinite inner product , on Ez (cf. Example 3.4.15) which varies smoothly with x E M. Such a reduction may or may not exist but, if it does, one obtains a principal O(k, n - k)-bundle of frames that are "orthonormal" with respect to this indefinite metric. The reader can supply the details. (a

a),

EXAMPLE5.5.11. Let G be a Lie group, H G a closed subgroup. Then the quotient projection p : G -+ G / H is a principal H-bundle over the homogeneous space G/H. Indeed, the local triviality is by Lemma 5.4.8 and the required property of the right H-action is obvious. EXAMPLE5.5.12. The case of a principal bundle with d2screte structure group G is noteworthy. A discrete group is simply an abstract group with the discrete

156

5 . LIE GROUPS

topology (each point is an open set). This is a topological group and, if G is at most countably infinite, it can also be viewed as a Lie group of dimension 0. A principal G-bundle p : P + M is called a reguiar covering space if G is a discrete group. It is evident that dim P = dim M and that p is locally a diffeomorphism. In fact, the local triviality of the bundle guarantees that each point x E M has a neighborhood U C M such that p-l(U) Z U x G is a disjoint union of copies of U ,one for each point of G, each of which is carried by p diffeomorphically onto U (compare Theorem 3.9.1). I t can be seen that the group of homeomorphisms g : P -t P such that p o g = p is exactly the group G, acting (from the right) on the total space P of the bundle (Exercise (5)). This is called the group of covering transformations. For instance, the standard projection p : Rn -+ T n is a regular covering with Zn as the group of covering transformations. The projection p : S n -+ Pn is another example, the covering group being Z2.

EXERCISES (1) Let p : P -, M be a principal G-bundle and let F be a manifold. If p : G x F -+ F is a smooth action, define the "diagonal" action of G on P x F in analogy with (5.3) and denote the quotient space by P x , F. Show that the bundle projection p passes to a well defined map p : P x , F -+ M, and that this is a smooth, locally trivial bundle with fiber F (define carefully what this means). It will be helpful to consider first the case in which M reduces to a single point (i.e., prove that G x , F = F canonically), then the case in which P = M x G is a trivial bundle, and finally the general case. (2) Let p : Gl(n) x R n -,Rn be the smooth action defined by

Given an n-plane bundle E over M , define E * = F ( E ) x,Rn as in Exercise (1) and prove that E* is an n-plane bundle over M with each fiber El canonically isomorphic t o the dual space of E x . Again, it will be help ful to consider first the case F ( V ) x ,Rn = V*, then the case of a product bundle, and finally the general case. (The bundle T * ( M )= (T(M))* is called the "cotangent bundle" of M. Its sections are "covector fields" or differential 1-forms, the topic of Chapter 6.) (3) Define a smooth action p : Gl(n) x '33Z(n) -+ m ( n ) by

Given an n-plane bundle E over M , show that S ( E ) = F ( E ) x '33Z(n) is an n2-plane bundle over M with each fiber S(E), canonically isomorphic to the space of symmetric bilinear forms on E x . (A section of the bundle S(T(M)) is called a symmetric 2-tensor on M . Riemannian metrics and indefinite metrics are examples of such tensors.) (4) Let p : P -+ M be a principal G-bundle, H & G a closed subgroup, and Q C_ M a properly imbedded submanifold such that p(Q) = M . Suppose


157

that Q is invariant under the right action of H (the restriction to H of the G-action on P ) and that p-I (x) n Q is an H-rbit, Vx E M . (a) Prove that p : Q -+ M is a principal H-bundle. This is called an H-reduction of the principal G-bundle p : P -+ M . (b) If X : G x F F is a smooth action, denote the restriction of this action to H by X also and prove that the inclusion -+

induces a commutative diagram

M

-

M id

in which T is a diffeomorphism. (c) Specialize to the case in which F = G / H and X : G x G / H -+ G/ H is the standard homogeneous action. Prove that the principal Gbundle p : P -+ M admits an H-reduction if and only if the associated bundle p : P xx G / H -+ M admits a global smooth section s : M -+ P xx G / H . (d) If ?r : E -+ M is an n-plane bundle, p : F(E) -+ M its associated frame bundle, and if H Gl(n) is a properly imbedded Lie subgroup, prove that 7r : E 4 M admits an H-reduction if and only if p : F ( E ) -+ M admits an H-reduction. Thus, part (c) gives a new existence criterion for infinitesimal H-structures. (5) Let p : P -+ M be a smooth map. A point x E M is said to be "evenly covered" by p if there is a neighborhood U of x such that p-'(U) falls into a disjoint union of open sets, each carried by p diffeomorphically onto U. If every x E M is evenly covered by p, then p : P M is called a covering space. The group G of diffeomorphisms g : P -+ P such that p o g = p is called the group of covering transformations. If this group permutes the set pP1(x) transitively, V x E M, we say that p : P -+ M is a regular covering space. Prove that this definition is equivalent to the one given in Example 5.5.12. -+

CHAPTER 6 Covectors and 1-Forms

An important analytic tool in our study of manifolds M has been the Lie algebra X(M) of smooth vector fields. In this chapter, we begin the study of the dual o b ject, the space A'(M) of differential 1-forms on M. One would expect this space of "covector fields" to be neither more nor less useful than X(M), but for many purposes it is much more powerful. One reason for this is the "functoriality" of A1(M), as will be explained presently. Another is exterior multzplication, an operation which produces higher order objects, called q-forrns. These q-forms can be integrated over suitable q-dimensional domains and differentiated. A version of the fundamental theorem of calculus, called Stokes' theorem, relates these operations and, in the global setting, leads to a remarkable tool (de Rham cohomology) for analyzing the topology of M . This, in fact, is the beginning of a major mathematical discipline called algebraic topology. Finally, theorems stated in terms of differential forms sometimes provide interesting and useful alternatives to equivalent vector field versions. An example of this will be a differential forms version of the Frobenius theorem. These topics will require the next several chapters to do them justice. Here we deal only with 1-forms. 6.1. The space of 1-forms We introduced the cotangent bundle T * ( M ) in Exercise (2) on page 156 as a bundle associated to the principal frame bundle F(T(M)). Here we give a more elementary construction of T * (M), analogous to our construction of the tangent bundle in Section 3.3. Throughout this discussion, we allow the possibility that a M # 8.

DEFINITION 6.1.1. Let M be a differentiable manifold, x E M. The dual space T,*(M) = Homw(T,(M), R) is called the cotangent space of M at x. Each element a E T,'(M) is called a cotangent vector to M at x. A typical cotangent vector is the differential of a map. Let U M be open, x E U ,and let f E Cm(U). Since Tf(,)(R) = R canonically, we obtain a linear

6. COVECTORS AND 1-FORMS

160

functional dfx : Tx(M) R, so df, E T: (M). It is evident that df, depends only on the germ [f 1, E we obtain an R-linear map +

a,, so

LEMMA 6.1.2. For each X , E Tx(M), dfx(Xx) = X x ( f ) . PROOF.Let (U,x l , . . . ,xn) be a coordinate chart about x. Then

-(a.)I

f

[ af

dfx = Jfx = =(x)

, . . . ' ax1

-

If

then dfX(XX) = Jfx . an

i=l

COROLLARY 6.1.3. Relative to local coordinates x ', . . . , xn about x E M, the covectors dx:, . . . , dx; form a basis of T,'(M). PROOF.Since dimT:(M) = n, it will be enough to show that this set of covectors is linearly independent. By the lemma,

Thus

0 COROLLARY 6.1.4. The linear map d : 8,

-+ T,*(M) is

surjective.

DEFINITION 6.1.5. If cp : M + N is a smooth map between manifolds, if x E M , and if u E T&,) ( N ) , then cp: ( a )E T: ( M ) is defined by

Evidently, this defines a linear map cp; : T;(x) (N)

-+

T '(M).

DEFINITION 6.1.6. The linear map c p z is called the adjoint of cp,,.

6.1. SPACE OF 1-FORMS

161

This is just a special case of a standard construction in linear algebra. If Vl, V2 are vector spaces (of respective dimensions m and n) over a field IF and if L : Vl -+ V2 is linear, then the adjoint is a linear map L * : V2* -+ V;C defined exactly as above. In terms of bases { e l , . . . , e m ) of Vl and ( € 1 , .. . , en) of V2,L is represented by an n x m matrix A over IF. The dual basis {e,t)E1 of V;C is defined by

and {E;);=~is defined similarly. Relative to the dual bases, L * is represented by an m x n matrix A*. The following is standard.

LEMMA6.1.7. I f A and A* are as above, then A* = A ~ . We return to

If x l , . . . , x m are coordinates about x and y l , . . . ,yn coordinates about cp(x), we have bases

and the dual bases

We define the adjoint Jacobian by

the matrix representing cp: relative t o these coordinates. The following three lemmas can be seen directly from the definitions or via adjoint Jacobians.

LEMMA6.1.8. If cp : M -+N is smooth, if x E M , and if cp,, is an isomorphism, then cp: is also an isomorphism and (cp:)-l is the adjoint of (cp,,)-l. LEMMA6.1.9. If M = N and cp = idM, then cpz = idT,'(M) LEMMA6.1.10. If M % N

11,

--+

P are smooth maps and x E M , then

162

6. COVECTORS AND 1LFORMS

Lemma 6.1.10 is a version of the chain rule. Let cp : U -+ V be a diffeomorphism between open subsets of Rn. By Lemmas 6.1.9 and 6.1.10 (applied to (p o cp-' and cp-' o cp),

For each x E U , define cp; = (9;I-l : T W )

+

TG(X)(V)

and the associated n x n matrix

Again, we have chain rules

The definition of the cotangent bundle can now be given and agrees, practically word for word, with that of the tangent bundle. Let

be the maximal smooth atlas on M. As a set,

and we topologize each T * (U,) via the bijection

The structure cocycle for M yields the Gl(n)-cocycle {J'g,p)o,pEa for T*( M ) and the local trivializations are

One can show (Exercise (1)) that the tangent and cotangent bundles are isomorphic, but not canonically. We do not identify these bundles. DEFINITION 6.1.11. The Coo(M)-module of covector fields on M is

Covector fields are also (and more commonly) called 1-forms on M. If w E A1(M), then w : M + T * ( M ) is written x I+ w, f T,*(M), V x E M.


163

If w E A1(M) and (U, xl, . . . , x" ) is a coordinate chart on M , then the restriction of w to U can be written

for unique fi E Cm(Ui), 1 5 i 5 n.

REMARKS. Let 7r : E + M be a k-plane bundle with associated Gl(k)+ocycle {yap}a,PE'd. The inverse adjoint defines m o t her Gl(k)-cocycle { ~ h ~ } , ,and p~~ this, in turn, defines a dual bundle 7r : E* + M whose fibers E: are canonically identified with (E,)* = Homw(Ex,R), V x E M. The double dual of an n-plane bundle 7r : E + M is canonically isomorphic to the original bundle. That is, (E*)* = E. Indeed, it is immediate that (?Lo)' =

Denote by (X(M), Cm(M)) the Cm(M)-module of d l Cm(M)linear maps X(M) + C m ( M ) . If w E A1(M) and X E X(M), we obtain w(X) E C m ( M ) by setting

It is clear that w(fX) = fw(X), V so we can view this 1-form as an element

f

E COO(M),

This defines an injective homomorphism

of Cm(M)-modules. We will now show that this is also a surjection, proving that these Cm(M)-modules are canonically isomorphic. Let a E H O ~ ~ ~ ( ~ ) ( X CbD(M)) ( M ) , and let U C M be an open subset.

LEMMA 6.1.12. If X E X(M) and XIU = 0 , then a(X)IU

-

0.

PROOF.Let x E U and choose f E C m ( M ) , vanishing at x and identically equal to 1 on M \ U. Then f X = X and This shows that a ( X ) ( x ) = f(x)a(X)(x) = 0. Since this is true for arbitrary x E U, it follows that a(X)IU

LEMMA 6.1.13. There is a unique

such that Z(XlU) = a(X)IU, VX E X(M).

-

0. El


164

PROOF.Uniqueness is clear. If Y E X(U), define G(Y) E Coo(U) as follows. For arbitrary y E U, choose f E C m ( M ) such that f = 1 on some open neighborhood V c U of y and f ( ( M\ U) r 0. Then we can interpret f Y as a field defined on all of M (= 0 outside of U) and fYlV = YIV. Define &(Y)(y)= a ( f Y ) ( y ) . If f and are different choices, Lemma 6.1.12 implies that the two definitions of Z(Y) agree at y (and, indeed, on the neighborhood V n of y in U). It is clear (X(U), Cm(U)) and that G(XIU) = a(X)IU, that this defines Z E VXEX(M).

P

P

By this lemma, we can define a l U = C, calling this the restriction of a to U.

COROLLARY 6.1.14. If cr: E H O ~ ~ ~ ~ ~ ~ ( X ( M ) , Cthen "(M a(X ) ) ( x ) depends on X, but not otherwise on X , Vx E M , VX E X(M). PROOF.Let x E M . Choose a neighborhood U of x in M over which T ( M ) is trivial and let Y1, . . . , Yn E X(U) give a basis of of the tangent space at each point of U. Then arbitrary X f X(M) can be written on U as

and

n

4x1(5) = ( ~ l u ) ( X l U 2) ) ( = C fi(x)(alu)(yi)(x). i=l

On the right hand side of this equation, the only dependence on X is in the values fi(x), 1 5 z 5 n. The property of a in the above corollary is called the tensor property.

LEMMA 6.1.15. If r] : X(M) --t Coo(M) is an R-linear map, then 7 has the tensor property if and only if r] E Horncm (X(M), C m ( M ) ) . PROOF.We have proven the "if" part. For the converse, assume that 7 has the tensor property and let f E Cm(M),X E X(M). For each x E M , r](fX)(x) depends only on f (x)X,, hence

by R-linearity. Since x E M is arbitrary, q ( f X ) = f q ( X ) . This equivalence between Cm(M)-linearity and the tensor property will recur in the broader context of Cm(M)-multilinearity in Chapter 7. For x E M and a E Homcm(Ml (X(M), Cm(M)), we can define a, E T: (M) as follows. Given u E T,(M), let X E X(M) be any vector field such that X, = u. Define a,(v) = a(X)(x). By the above, this depends only on v, not on the choice of extension X, so we get a, E T,+(M),Vx E M . Using local coordinates, it is elementary to check that the map x I+ a, defines a smooth section of T*(M). This identifies a as an element of A 1 ( ~ ) completing , the proof of the following.


PROPOSITION 6.1.16. There is a canonical isomorphism,

of C" ( M )-modules.

REMARK.Similarly, X(M) = Horncm (A1(M), Cm(M)). Let cp : M -t N be smooth. If w E A1(N), define cp*(w) : M --+ T * ( M )by

If f E C m ( N ) , define cp*(f) = f lemmas will be Exercise (3).

o

cp E C W ( M ) . The proof of the following two

LEMMA6.1.17. If cp : M -+ N is a smooth map of manifolds and w E A' (N), then cp4(w) E A'(M) and, this defines a linear map

of vector spaces over R. Furthermore, i f f E Coo(M),

are smooth maps of manifolds, then ($ o cp)* = cp* A' (P).

o

$* on both C"(P)

and

REMARK.In the language of category theory, A' is a contravariant functor from the category of differentiable manifolds and smooth maps to the category of real vector spaces and linear maps. We explain this in a little more detail. A category is a class C of objects and of maps between these objects, called rnorphisms, with the property that compositions of morphisms, whenever defined, yields morphisms and, for each object 0 E C, ido E C. Thus, the class V of real vector spaces and linear maps between them is a category, as is the class M of differentiable manifolds and smooth maps. A functor is either a "homomorphism" or an "antihomomorphism" between two categories. In the first case, the functor is said t o be covariant and, in the second case, it is contravariant. More precisely F : C1 -+ C2 is a covariant functor if, whenever cp : 0 -+ 0' is a morphism in C1, then F(cp) : F(0) + F(0') is a morphism in C2 and (1) F(cp 0 $) = F(cp)0 F($), v morphisms cp, Q E C1, V object 0 E C1. (2) F(ido) = The functor F is contravariant if F(cp) : F ( 0 ' ) + F ( 0 ) and the first of the above properties is replaced with

In the case of a covariant functor, it is customary to write F(cp) = cp, and, in the contravariant case, to write F(cp)= cp*. Thus, Lemmas 6.1.17, 6.1.18 and the trivial fact that i d L = i d A ~ ( Mimply ) that A' : M V is a contravariant functor. By contrast, X(M) does not define -+

166


a functor since, generally, cp, : X ( M ) -,X ( N ) is not defined. For many purposes, this makes 1-forms significantly more useful than vector fields. Observe also that Coo : M + V is a contravariant functor.

LEMMA6.1.19. I' f E C m ( M ) , then df : M df,, Vx E M , is a 1-form on M .

-+

T * ( M ) , defined by df (x) =

Indeed, the fact that df is a smooth section of T * ( M ) is easily checked via local coordinates, where the formula is

Alternatively, it is an immediate consequence of Lemma 6.1.2 that

DEFINITION 6.1.20. If f E C o o ( M ) ,then df is called the exterior derivative of f . The R-linear map d : C m ( M ) + A 1 ( M )is called exterior differentiation. L E M M A6.1.21. If f , g E C m ( M ) , then d ( f g ) = f d g + g d f .

PROOF.Indeed, if X E X ( M ) , then

Since X E X ( M ) is arbitrary, the assertion follows.

[7

This lemma is a Leibnitz rule for exterior differentiation.

LEMMA6.1.22. If cp : M

N is smooth, then the diagram

is commutative. That is, d(cp*( f ) ) = cp*(df),V f E C m ( N ) . The property of d in this lemma is called the naturality of the exterior derivative. The proof, as the one for the preceding lemma, works itself and is left to the reader.

6.2. LINE INTEGRALS

EXERCISES (1) Exhibit a bundle isomorphism between T(M) and T*(M). In the case that M is an open subset of Euclidean space, show that there is a canonical choice of isomorphism. What is the difficulty about defining a canonical isomorphism in general? (In classical physics and advanced calculus courses, it is quite common not to distinguish tangent vectors and cotangent vectors. It is precisely because these treatments are carried out in Euclidean domains that this identification is legitimate.) (2) Prove that the dual bundle E*,as described in Exercise (2) on page 156, is canonically isomorphic to the dual bundle E* as constructed in this section. (3) Prove Lemmas 6.1.17 and 6.1.18. (4) Let X E X(M) and let @ denote the local flow generated by X. One defines the Lie derivative

@t*(w)- W , vw E A ~ ( M ) , t+O t taken pointwise on M. Recall from Section 2.8 the analogous definitions of the Lie derivatives L x ( f ) and L x ( Y ) for f E Coo(M) and Y E X(M). Prove the following identities for arbitrary f E C OQ ( M ), w E A (M), and Y E X(M). (a) t x ( d f ) = dLx(f)(b) Lx(fw) = L x ( f )w + f t x ( w ) . (c) L x (w(Y)) = Lx (w)(Y) + (Y)). Lx(w) = lim

6.2. Line Integrals

If w E A1(M) and s : [a,b] We can write s*(w) = f dt.

-+

M is a smooth curve, then s * ( w ) E A1([a, 61).

DEFINITION6.2.1. The line integral of w E A 1 ( ~ along ) a smooth curve s : [a,b]-+M is

f (t) dt. Line integrals are insensitive to orientation preserving changes of parameter and only experience a sign change under an orientation reversing repararnetrization. It is not even necessary to require that the change of parameter be nonsingular or monotonic. LEMMA6.2.2. Let s : [a, b] o = S O u. Then,

4

M and u : [c,4

-+

[a,b] be smooth. Set

(1) 2f u ( c ) = a and u(d) = b, J,w = JUw, Vw E A1(M); (2) if u(c) = b and u ( d ) = a, - Js w = Jm w, V w E A'(M).

6.

168

COVECTORS AND 1-FORMS

4.

PROOF.Let t denote the coordinate of [a,b] and r the coordinate of [c, Then

In case (1))the rule for change of variable in integrals gives

In case (2), the same rule gives

M and s2 : [c,4 -+ M be smooth paths with the same initial point and the same terminal point. That is, s l ( a ) = sz(c) = x and sl (b) = s2(d) = y . If f E CM( M ) , then

LEMMA 6.2.3. Let sl

: [a,b] -+

PROOF.By Lemma 6.2.2, we lose no generality in assuming that [a,b] = [c,dl. Then,

6.2. LINE INTEGRALS

169

This lemma is a 14imensional version of Stokes' Theorem. As the proof makes clear, it is just the fundamental theorem of calculus. DEFINITION 6.2.4. A form w E A 1 ( M )is said to be exact if w = df, for some f E Coo(M). Lemma 6.2.3 says that the line integral S,w of an exact l-form w depends only on the endpoints of the path s, not otherwise on s. In physics, the law of conservation of energy is a special case of this result. Lemma 6.2.3 is a part of Theorem 6.2.9, which will be stated and proven shortly. The notion of a line integral can be extended to allow integration of l-forms along paths s : [a, b] -+ M that are only piecewise smooth. That is, s is continuous and there exists a partition a = t o < t l < - . - < t , = b such that si = sl [ti-l, ti]is smooth, 1 5 i 5 q. We write s = sl sz . sq and define

+ + +

Since it is not assumed that the partition contains only points at which s is not smooth, it is necessary to observe that this definition is independent of the choice of allowable partition. This is elementary and is left to the reader. The proof of the following consequence of Lemma 6.2.3 is also left to the reader. COROLLARY 6.2.5. Let sl : [a,b] -+ M and sz : [c,d] -+ M be piecewise smooth paths with the same initial point x and the same terminal point y. Then,

LEMMA6.2.6. If w E A 1 ( M ) and iJ for every piecewise smooth path s, the integral Js w = 0 , then w = 0. PROOF. Otherwise, there is a point z E M and a tangent vector v E T , ( M ) such that w,(v) > 0. Let s : [ - E , E ] -+ M be smooth such that s(0) = z and i ( 0 ) = v. Choosing E > 0 smaller, if necessary, we can assume that

An elementary computation shows that s * ( w ) = ~ ~ , ( t , ( s ( t )dt, ) so

contradicting the hypothesis. DEFINITION 6.2.7. We say that w E A 1 ( M ) has path independent line integrals if, for every piecewise smooth path s : [a,b] -+ M , Js w depends only on s (a) and s(b) and not otherwise on s. DEFINITION 6.2.8. A piecewise smooth path s : [a, b] -+ M is a loop if s(a) = s(b).

170

6 . COVECTORS A N D 1-FORMS

THEOREM 6.2.9. For w E A1(M), the following are equivalent. (1) w is an exact form. (2) Js w = 0, for all piecewise smooth loops s. (3) w has path independent line integrals. PROOF.We prove that (1) + (2). If w = df is exact and s : [a,b] -, M is a piecewise smooth loop, s(a) = q = s(b), then Corollary 6.2.5 implies that

where q denotes the constant path q(t) = q, a 5 t 5 b We prove that (2) + (3). Let sl and s2 be piecewise smooth curves starting at the same point x and ending at the same point y. Without loss of generality, assume that sl is parametrized on [- 1,0] and s2 on [0, 11. Let u : [0, I] [O,1) be defined by u(t) = 1 - t. Then s 2 o u starts at y and ends at x and -+

is a piecewise smooth loop. By our assumption,

where we have used part (2) of Lemma 6.2.2 to write

We prove that (3) + (1) by using (3) to construct f E Coo(M) such that w = df. Without loss of generality, we assume that M is connected (otherwise, carry out the construction off on each component individually). Fix a basepoint xo E M. Given any point x E M , use connectivity to find a piecewise smooth path s : [a, b] -+ M such that s(a) = xo and s(b) = x. (In fact, the homogeneity lemma, Theorem 3.8.7, implies the existence of a smooth path, but the present claim is more elementary and is left to the reader.) Set

By the assumption of path independence, this is independent of the choice of piecewise smooth path s from xo to x. Remark that f (xo)= 0. We prove first that f : M -+ R is smooth. Let q E M be arbitrary and choose a coordinate chart (U, x l , . . . ,xn) about q in which q is the origin and U = int Dn, where D n is the unit ball in Rn. In these coordinates, we can write

6.2. LINE INTEGRALS

For each x E U, let s, : [O,1] express f J U by the formula

-+

U be defined by s,(t) = tx, 0 5 t

171

5 1, and

Thus, on U,

This is clearly smooth. Since q E M is arbitrary, f E Cm(M). Next, we prove that w = df. Let s : [a,b] -, M be an arbitrary piecewise smooth path. Let c < a and let s o : [c,a] + M be piecewise smooth such that s 0 ( c ) = xo and so(a) = s(a). Then

That is

It follows that the form 5 = w - df satisfies

for all piecewise smooth paths s. By Lemma 6.2.6, 5 = 0, so w = df.

DEFINITION 6.2.10. A 1-form

w E

A'(M) is locally exact if, for each x E M ,

there is an open neighborhood U of x such that wl U E A'(U) is exact.

EXAMPLE 6.2.11. On the manifold M = W2 \ ((0, 0)), define the 1-form

We claim that q is locally exact. Indeed, if q E M is not on the y-axis, a branch of 8 = arctan(y/x) is defined and smooth on a neighborhood of q. A direct computation gives d8 = q. Similarly, if q E M is not on the x-axis, select a branch of 8 = - arctan(x/y) and check that dB = q. Since no point of M is on both axes, this proves that q is locally exact. We claim, however, that q is not exact. Indeed, consider the smooth loop s : [O, 11 -+ M defined by s(t) = (cos 2nt, sin 27rt). Clearly, qs(t) = - sin 2rt

+ cos 27rt dys(t),

and s*(dx) = -2n sin 2 r t dt s*(dy) = 27r cos 2nt dt,

6. COVECTORS A N D 1-FORMS

172 SO

s*(7) = 2n(sin2 27rt

+ cos22xt) dt = 2n dt.

Thus

Remark that the form 7 in the above example cannot be extended to a 1-form on IR2. We are going to see shortly (Corollary 6.2.14) that, on R 2 , every locally exact 1-form is, in fact, exact. The above example reflects a topological feature of IR2 \ { ( O , O)}, the missing point, that distinguishes that space from IR2. DEFINITION 6.2.12. Let so,sl : Ia,b] -+ M be piecewise smooth loops. We say that so is homotopic to sl, and write s o $1, if there is a continuous map N

H : [a, b] x [ O , l ] and apartition a = to < tl

0, the vector spaces Z (M) and B1(M) are infinite dimensional, it frequently happens that H1(M) is finite dimensional. We will see, for instance, that this is the case whenever M is compact. Cohomology is a contravariant functor from the category of differentiable manifolds (smooth maps are the morphisms) to the category of real vector spaces (and linear maps). Indeed, by Lemma 6.1.22 a smooth map cp : M -+ N induces a linear map cp* : Z1(N) -+ Z1(M) and cp*(B1(N)) C B1(M), so cp* passes to a well defined linear map (of the same name)

It is trivial to check that (cp 0 $)* = $* o cp* and (idM)*= idH1(M). PROPOSITION 6.3.3. Let w, G E z l ( M ) . Then [w] = [G] E H1(M) zf and only if w = SsS as s varies over all piecewise smooth loops in M . These numbers are called the periods ofw and of the cohomology class [w].

Ss

174


PROOF.The locally exact forms w, S E z'(M) have the same periods Jsw = Js3, for every piecewise smooth loop s, if and only if Js(w - 5) = 0 for all such loops. By Theorem 6.2.9, this holds if and only if w - L? E B1(M). Equivalently, [w]=[W].

PROPOSITION 66.4. If fo, f l : M -+ N are smooth and homotopic (fo .- fl), then f; = ff : H'(N) -+ H ~ ( M ) . PROOF.Let [w] E H 1 ( N ) . If s : [a,b] -+ M is a piecewise smooth loop, then si = f, o s : [a, b] -+ N is &o a piecewise smooth loop, i = 0 , l . Let H : M x R + M be a homotopy of f o to fi. Then, the composition [a, b] x [O,1]

sxid

M xR

H 4

N

is a homotopy of so to sl, so

=

d.

w

(Proposition 6.2.13)

Since s is an arbitrary piecewise smooth loop, Proposition 6.3.3 implies that

Since [w] E H' (M) is arbitrary, f f = f; at the cohomology level, as desired. DEFINITION 6.3.5. A smooth map f : M -+ N is a homotopy equivalence if there exists asmooth map g : N + M such that f o g .- idN and g o f --idM. COROLLARY 6.3.6. A hornotopy equivalence f : M isomorphism f * : H'(N) -+ H1(M).

-+

N induces a linear

PROOF.Since f o g idN, it follows by functoriality and Proposition 6.3.4 that g * ~ f * = ( f ~ g ) * = i d & =Hi1(N) d N

Similarly, f * o g* = idH' ( M ) , SO f * and g* are mutually inverse isomorphisms on cohomology.

-

6.3. FIRST COHOMOLOGY

175

-

EXAMPLE 6.3.7. Let f : (0) Dn be the inclusion. Let g : Dn -+ {0} be the only map. These maps are smooth and g o f = id ~ 0 )(hence, g o f id{o)). Consider the map f o g : Dn + Dn having image (0). We claim that this is homotopic to idon. Indeed, let cp : R -+ [0, I] be smooth such that cp(0) = 0 and cp(1) = 1. Define H : Dn x R -,Dn by

Then H(x,O) = 0 - f(g(x)), V X E Dn, and

H(x,1) = x = idDn(x), V X E Dn. This establishes the desired homotopy and completes the proof that f is a hw mot opy equivalence, so

is an isomorphism. That is,

H'(D") = o

or, equivalently, every Iocally exact 1-form on D " is exact. A similar proof shows that Rn is homotopically equivalent to a point, and we recover Corollary 6.2.14.

EXAMPLE 6.3.8. Let

i : s1~f R2 \ {(O,O))

be the inclusion. Let g : R2 \

{(O,O)}

4

s1

be the map defined by

These maps are smooth, and goi=idsl. We claim that hence that i is a homotopy equivalence. Indeed, define

bv the formula

+

This is smooth since, llvll being strictly positive, so is t (1 - t) IIvII, 0 5 t 5 1. Then H ( v , 1) = v, V V E R2 \ {(O,O)}, and

v H(v,O) = - =i(g(v)), V V E EX2 llvll

\

{(0,0)}.

It follows that

H ~ ( R \ { ~( o , ~ ) }= ) ~l(sl).


176

PROPOSITION 6.3.9. There i s a canonical isomorphism H l ( S 1 )= IW. We will prove this via three lemmas. First define the map

by

p(t) = (cos 2rt, sin 2nt). This is the map that induces the standard diffeomorphism R/Z = S 1 . Define

a linear map.

LEMMA6.3.10. The linear m a p a passes t o a well defined linear map

PROOF.Indeed,

0

= p1 [O, 11 is a smooth loop and w E

B

'(s') implies that

by Theorem 6.2.9.

LEMMA6.3.11. T h e linear m a p a : H 1( S 1 ) R is injective. -+

PROOF.Let w E Z 1 ( S 1 )be such that a(w) = 0. We must prove that w E B' (5''). For n E Z, let T, : R -+R be the translation rn(t)= t n. Then, p o Tn = p,

+

SO T;

o p * = P* :

~ l ( s-+ l )A ~ ( R ) .

This and the change of variable formula for the integral gives

In particular, since a(w) = 0, we obtain

Define

fw E

C m ( S 1 )by

If this is well defined, it will be smooth. It will be well defined precisely if


If n = 0, this is obvious. If n > 0,

=

l+n 1 p* ('d)

t

=

P*(w).

A similar computation for the case n < 0 is left to the reader. If we define

f;= P*(fW), we get so the fundamental theorem of calculus and Lemma 6.1.22 give

But p : R 4 S1 is a local diffeomorphism, so w and df, are equal locally, hence globally. That is, w E B'(s') as desired. Recall the locally exact form

of Example 6.2.11 and let

where i is the inclusion map of S1 into B2 ((0,O)). For the loop a = p1[0,1], s = i o a is as in Example 6.2.11, and we showed that a*(ij)= o*(i*(7))= (i o a)*(7)= S* (7)= 27rdt.

LEMMA6.3.12. The linear map a : H1(S1)-+ B is surjective. PROOF. It is enough to show that a is nontrivial. But [ij] E H '(S1) and

The proof of Proposition 6.3.9 is complete.

EXAMPLE 6.3.14. Recall that the Brouwer fixed point theorem for smooth (in fact, continuous) maps f : D2 -+ D2 follows from the nonexistence of a smooth retraction p : D 2 + dD2 = s l . Recall that, for p to be a retraction, it is required that the diagram


commute, where L is the inclusion of the boundary circle. By the functoriality of cohomology, this produces a commutative diagram

That is,

commutes, which is absurd. This example illustrates the general philosophy of algebraic topology: use functors from a topological category to an algebraic category to "paint" an algebraic picture of a topological problem, then solve the algebraic problem and try to determine the implications of this solution on the original topological problem. This algebraic picture, to be useful, must leave out a lot of detail but not, of course, too much. For instance, H 1 cannot see the difference between D n , Rn, and a point, but if it could not distinguish these from S l , it would have been useless for proving the nonexistence of the retraction. Our computation of H1(S1) generalizes to the higher dimensional tori. You will be led through a proof of the following in Exercise (1).

PROPOSITION 6.3.15. There is a canonical isomorphism

EXERCISES (1) Let exp : Rn --, Tn be the homomorphism of abelian Lie groups defined by 27rixn exP(xl,.. . , xn ) = 27rix1 (cf. Exercise (7) on page 135). If a : [a,b] -+ Rn is piecewise smooth such that v', = a ( b ) - a(a) E Zn, then exp oa is a piecewise smooth loop on Tn. Prove Proposition 6.3.15, using this observation, as follows.


179

(a) Prove that every piecewise smooth loop on Tn is of the form exp o a as above and that the homot opy class [expoa] is completely determined by v', . (b) If w E Z1(Tn), define cp, : Zn R as follows. Given v' E Zn, choose a piecewise smooth path a : [a,b] -+ Rn such that v' = v', and set -+

vw(q=/

w-

exp o o

Show that this is well defined and that cp, : Zn uniquely to a linear functional of the same name

-+

R extends

(c) Prove that the assignment w cpw passes to a well defined linear injection cp : H ~ ( T " )-+(R")* = R". (d) Show that there are forms e l , . . . ,On E Z1(Tn) such that exp*(8') = dx',

15 i

< n.

Use this to show that

is also surjective, hence is the canonical isomorphism we seek. (2) If n 2 2, show that T n and Sn are not homotopically equivalent. (3) If the vector space H1(M) has finite dimension k, prove that there are piecewise smooth loops 01, . . . ,a k on M such that the map

defined by

is an isomorphism of vector spaces. (4) We will say that a locally exact form w E Z1(M) is integral if all of its periods are integers. For example, jj/27r E Z1(S1) is integral. By Proposition 6.3.3, w is integral if and only if every w' E [w] is integral, in which case we say that [w] is an integral cohomology class. We denote by H1(M; Z) c H1(M) the subset of integral cohomology classes. (a) Prove that H1(M;Z) is a subgroup of the additive group of the vector space H1(M). We call H1(M; Z) the integral cohomology of M . (b) If f : M N is smooth and w E Z1(N) is integral, prove that f*(w) E Z1(M) is also integral. Using this, show that integral cohomology is a contravariant functor from the category M of smooth manifolds and smooth maps to the category G of abelian groups and group homomorphisms. -+

6. COVECTORS AND

1-FORMS

(c) Referring to Exercise ( I ) , prove that H' (Tn;Z) is canonically the integer lattice cI W= ~ HI (T~).

zn

(d) To each smooth map f : Tn -t Tn, show how to assign canonically an n x n matrix Af of integers, depending only on the homotopy class of f , such that Af,, = ABAf(matrix multiplication). If f is a diffeomorphism of T n onto itself, prove that A is unimodular (i.e., has determinant f1). (e) Prove that every n x n unimodular matrix of integers occurs as the matrix Af assigned to some diffeomorphism f : Tn -+Tn.

6.4. Some Topological Applications To begin with, we will classify the smooth maps

up to homotopy. The first remark is that, since H l(S1) = R,f induces a linear map

f*:R-+R depending only on the homotopy class of f . Thus, f * is just multiplication by a certain constant af E R and af depends only on the homotopy class of f . By part (d) of Exercise (4) on page 180, a f is an integer, being the sole entry in the 1 x 1 integer matrix Af. We are going to give alternative ways to see that af E Z, proving in particular that this integer determines f up to homotopy. This will give a one to one correspondence between Z and the set of homotopy classes of smooth maps of S' to itself. LEMMA6.4.1 (LIFTINGLEMMA).If f : S1

smooth map

j :R

-+

-+

S1 is smooth, there exists a

R so that the dzagram

commutes. F u r t h e n o r e , some integer k .

f

is another such lip o f f if and only if j

=

f + k , for

PROOF.We first prove that, if g : R 4 S1 is a smooth map, then there exists a continuous map i j : R -+ R such that p o i j = g. The map i j will actually be smooth because of the smoothness of g and the fact that p is locally a diffeomorphism. Applying this to g = f o p, we see that j = lj is the required lift. First we define such a lift on an arbitrary compact interval [a,b] c R. Let J+ and J- be the complements in S1 c C of the points +1 and -1, respectively. Then p - ' ( J + ) is the disjoint union of all intervals of the form (k, k I ) , k t Z, while p-l(J-) is the union of all (k - 1/2, k + 1/2), k E Z. These intervals are

+

6.4.

TOPOLOGICAL APPLICATIONS

181

carried diffeomorphically by p onto J+ or J-, respectively. By the continuity of g and the compactness of [a, b], find a partition a = t o < tl < . . . < t, = b such that g[ti-l,ti] lies either in J+ or J-, 1 I i 5 q. Then gl = gl[to,tl] has a continuous lift ijl (depending only on the choice of (to) E p-l (g(to))),after which g;! = [ [t t2] can be lifted uniquely so that iz (tl) = gl (tl). Continuing in this way, we produce the lift of gl [a, b] as desired, depending only on the choice of the value of the lift at to = a. Remark that we could just as well have produced a lift by choosing its value at t, = b. Expressing R as the union of intervals [k,k 11, k E Z, we first choose a lift of g on [0, 11, then on [I,21 so that they agree at 1, then on [-I, 01 so that they agree at 0, etc. If f and j are two lifts of f : S1-9 S1,the fact that p is a homomorphism of the Lie group R onto the Lie group S implies that

+

That is f(t) - f(t) E Z, Vt E W. The continuity of this expression in t then implies that this integer value is a constant k, hence that f = f k.

+

PROPOSITION 6.4.2. Iff : S1 -+ S1 is smooth, then a j = j(1) - f(0) E Z, where j is any iifi off as in Lemma 6.4.1. PROOF.Let

f be a lift of f

as in Lemma 6.4.1. Then

SO

!(I) - f(0) = m E Z. Thus, let i j E zl(S1) be as on page 177 and compute

That is, af = m E Z. DEFINITION 6.4.3. If f : S1 -+ S1 is smooth, the integer af is called the degree of f and denoted deg(f ). In Section 3.9, we defined deg,(f) E Z2for smooth maps between manifolds (without boundary) of the same dimension. We will see that, for smooth maps of the circle to itself, degz(f) is just the residue class modulo 2 of deg ( f ) (Corollary 6.4.7). COROLLARY 6.4.4. Iff : S1 -* s1is smooth, if j is a lift as in Lemma 6.4.1, and if t E IR is arbitrary, then deg(f) = j ( t 1) - f(t).

+


182

PROOF.View p : R

-+

S1 c C as a group homomorphism. Then

so f(t + 1) - j ( t ) E Z, Vt E P.This function of t, being continuous and integervalued, is constant on R, hence equal to f(1) - j(0) = deg(f). 17

DEFINITION 6.4.5. The set of homotopy classes of smooth maps f : M will be denoted n[M, N].

-+

N

By the homotopy invariance of cohomology, we have defined deg : n [ s l , s']+ Z. We describe deg(f) in terms of regular d u e s . Let zo E S' be a regular value of f : S1 + S1. Then f-'(zo) = {zl, . . . ,z,). Here, if f -'(to) = 0, we take r = 0. Recall that deg2(f) = r (mod 2). Choose iiE B such that p(&) = t i , 1 4 i j r . The smooth map f : S' -+ S1 preserves orientation s t zi if ?(Hi) > 0 and reverses orientation at Z i if f'(4) < 0. Let ri = j'(ii)/l f (2,) 1 E {-1,l) and remark that this depends only on f and Z* .

PROPOSITION 6.4.6. With the above conventions, deg(f ) =

xi=,

€i.

PROOF.Choose a E R \ {p-'{zl, . . . ,r,)). Then p - l ( ~ i )n (a, a + 1) is a singleton and we choose this point as our Hi, 1 5 i 5 r. We will also use the fact that f(a 1) = f(a) deg(f ). Let p-'(ro) = {b + kIrez and consider the graph of s = f(t), a < t < a + 1, together with the horizontal lines s = b + k, k E Z. Each time the graph crosses a line s = b + k, the parameter t is equal to one of the Zi and e, records whether the graph crosses this lime while increasing (ci = 1) or decreasing (r; = -1). Figure 6.1 illustrates a case in which r = 7, €1 = €2 = €3 = €4 = €7 = 1, and €5 = r s = -1. The sum of the ti's pertaining to a single line s = b + k is 1, - 1, or 0, the net number of directed crossings. Ciearly, the sum of all these net numbers is

+

+

(In Figure 6.1, the degree is 3.)

COROLLARY 6.4.7. deg,(f)

= deg(f)

(mod 2).

6.4. TOPOLOGICAL APPLICATIONS

FIGURE6.1. Graph of

j

EXAMPLE 6.4.8. For each n E Z, let f, : S1 -+ s1be defined by fn(z) = zn. Here, of course, we view S1 C @. We can choose the lift fn : R -+ R to be fn(t) = nt, so deg(fn) = j n ( 1 ) - fn(0) = n. If z t. S1is a regular value of f n , then

where pl, . . . , ars the distinct nth roots of t . Of course, if n = 0, then f o is constant and f&'(z) = 0. If n > 0, all € i = +1 and, if n < 0, all s i = -1. Thus,

in all cases.

THEOREM 6.4.9. The function deg : r[S1,S1]-+ Z is bijective.

-

PROOF.For each integer n, deg(fn) = n, so deg is surjective. We must show that, if deg(f ) = deg(g) = n, then f g. Define

by the formula I;T(t, T) = T j ( t )

+ (1

-

~)j(t).

184


Since f7(t

+ 1, r ) - H(t, r ) = r n + (1 - r ) n = n, the formula

well defines a smooth map

Evidently, H(z,O) = g ( z ) and H ( z , l ) = f ( z ) , V z E S1,s o f - g . THEOREM 6.4.10. A smooth map f : S1 F : D2 S1 if and only if deg(f) = 0.

s1extends

to a smooth map

-+

PROOF.First suppose that the smooth extension F exists. That is, f = F o i where i : S1L) R2 is the inclusion. Then f * = i* o F* and

implying that f * = 0. Therefore, deg(f) = 0. For the converse, suppose that deg(f) = 0. By Theorem 6.4.9, it follows that f -- fo E 1. By the C" Urysohn trick, choose the homotopy

so that

Then H induces a smooth map of the disk to the circle as follows. Define a smooth surjection cp : S' x [0, 11 4 D~ C C by p(z, t ) = tz. Then cp carries S1 x (O,1] diffeomorphically onto D 2\ (0). Define

d}

This is well defined. It is smooth on D 2\ {O) and, on {w E R 2 ( (w(5 it is constant, so F is smooth. Evidently, F(z) = H(z, 1) = f (z), V z E S ', so F extends f . Recall that the mod 2 degree allowed us to prove the fundamental theorem of algebra for polynomials of odd degree (Theorem 3.9.14). The integer degree makes it possible to carry out essentially the same argument for all positive degrees. OF ALGEBRA). Let f : C -+ @. THEOREM 6.4.11 (FUNDAMENTAL THEOREM be a polynomial of degree n 2 1. Then there is zo E @. such that f ( z o ) = 0.


PROOF.We can assume that the leading coefficient is 1 and write

f (z) = zn

+ alzn-' + . - .+ a,-lr + a,.

Suppose this has no root. For each positive real number r, define a smooth function Fr : l I 2 -4 s1 by the formula

This is where we use the hypothesis that f has no roots. Let g, = F,.Isl. Then set Gr(z, t) = (rz)" t(al(rz)"-l . an)

+ +

+

and note that, if r is large enough, this vanishes nowhere on S1 x [0,11. Indeed,

approaches 1 as r of r and define

4

oo, uniformly on

s1x [0,11. Thus, fix a large enough value

H : s1x [ O , l ]

4

S'

by the formula

-

Then H ( z , l ) = g,(z) and H(z,O) = zn, V Z E S1. Thus, g, fn and deg(g,) = n > 0. But g, extends smoothly to F, : D 2-+ S1,a contradiction to Theorem 6.4.10.

LEMMA6.4.12. If f , g : s1 S1 are smooth, thendeg(f og) = deg(f)deg(g). Indeed, functoriality of cohomology implies that a f o g = &fas, SO the lemma is immediate. -+

COROLLARY 6.4.13. If f , g : S1 4 S1 are smooth, then f o g and g o f are homotopic. If f , g : S1 -+ S1 are smooth, define the pointwise product f g : S1 + S1 by the Lie group structure of S'. That is,

-

-

It is elementary that, whenever fo f l and go gl, then fogo fig1 (the homotopy is the pointwise product of the given homotopies). This defines a commutative multiplication on the set n[S1, S1].The constant map 1 determines a class [I]E n[S1, S1]which is an identity for this multiplication. If L : S1-+ S1 is the group inversion map, then each [f ] E n[S l , s']has an inverse [ L O f ] relative to this multiplication, so n [ s l , sl]is canonically an abelian group. N

PROPOSITION 6.4.14. The bijection deg : n[S l , S1]+ Z is an isomorphism of groups.

186

6.

-

COVECTORS AND 1-FORMS

+

-

-

PROOF.It is only necessary to prove that deg(fg) = deg(f) deg(g). But f fn and g f m , where n = deg(f) and m = deg(g). Then f g fnfm = fn+mMore generally, the Lie group structure on S1 makes n[M,S1]into an abelian group and one obtains the following. THEOREM 6.4.15. The map

defined by

x[fl = f *[7?/2nl7 is an isomorphism of groups. Here, the integral cohomology H'(M; Z) is defined as in Exercise (4) on page 180. The fact that [ij/2n] is an integral class implies that f *[ij/2n] is also integral by that same exercise. You will be led through a proof of this theorem in Exercise (2).

EXERCISES (1) Use degree theory to show that the group Diff (S1)has exactly two is* topy classes. (2) Prove Theorem 6.4.15. Proceed as follows. (a) Show that x is a group homomorphism. (b) Let w E Z1(M) be an integral form. You are going to define a S1 such that f: (7?/27r) = w. For this, no smooth map f, : M generality will be lost in assuming that M is connected (why?), so make that assumption and fix a basepoint xo E M . For each x E M , choose any piecewise smooth path s : [a, b] -+ M such that s(a) = xo and s(b) = x, and show that -+

depends only on x (and xo), not on the choice of path s. (c) Prove that f, : M -+ S' is smooth and that its homotopy class is independent of the choice of basepoint xo. (d) Prove that f;(ij/2n) = w . In particular, conclude that x is surjective. (e) Let f : M S1be such that w = f *(ij/2n) is an exact form. You are to prove that f 1, SO note that, again, no generality is lost in assuming that M is connected. In this case, show that f,, as defined in step (b), is actually well defined as a map f, : M -+ R and that there is a constant c such that f = p o (f, c ) . Conclude 1, hence that x is one to one. that f (3) Show that the correspondence f H A f of Exercise (4) on page 180 induces a one to one correspondence between the set of isotopy classes in Diff(Tn) and the set of n x n unimodular integer matrices. -+

-

-

+


187

(4) Let p = (a, b) E R2 and, in analogy with Example 6.2.11, define a locally exact 1-form qp on R2 \ {p) having periods exactly the set Z of all integers. If s = sl . . . s, is a piecewise smooth loop on R2 \ {p), one defines the winding number of s about p to be the integer

+ +

If s is smooth, show that the mod 2 residue class of w(s,p) agrees with the notion of winding number in Definition 3.9.18.

CHAPTER 7 Multilinear Algebra and Tensors

Smooth functions, vector fields and I-forms are tensors of fairly simple types. In order to handle higher order tensors, we will need some rather sophisticated multilinear algebra. The reader who is well grounded in the multilinear algebra of R-modules can skip ahead to Section 7.4, referring to the first three sections only as needed.

7.1. Tensor Algebra We will be working in the category M ( R ) of R-modules and R-linear maps, where R is a fixed commutative ring with unity 1. In order to study R-multilinear maps, we build a universal model of multilinear objects called the tensor algebra over R. In the typical applications in this book, R will be either the real field R or the ring Cm(M).

DEF~NITION 7.1.1. An R-module V is free if there is a subset B c V such that every nonzero element v E V can be written uniquely as a finite R-linear combination of eIements of B (terms with coefficient 0 being suppressed). The set B will be called a (free) basis of V.

If R is a field, every R module is free. Another example is the integer lattice 2Zk , a free Z-module. At the other extreme, the abelian group Z 2 , when viewed as a Z-module, is not free. A basis would have to contain 1 E Z2, but 0 E Z2 would then have infinitely many representations a . 1, a E 22. The following example will be very important. E -, M be a trivial n-plane bundle. Then r ( E ) is a free Cm(M)-module on a basis of n elements. Indeed, fix a trivialization E 2 M x Rn, let {el,. . . ,en) be the standard basis of R n , and define si E r ( E ) by the formula si(x) = (x,ei), 1 5 i 5 n. An arbitrary section s(x) = (x, f (x), . . . ,fn (x)) has the unique expression

EXAMPLE 7.1.2. Let

7r :

7. MULTILINEAR ALGEBRA

190

REMARKS.There are strong but limited analogies between vector spaces over a field and free R-modules. Here are some of the facts. (1) If V is free on the basis B , then R-linear maps cp : V W into arbitrary R-modules W correspond one to one to set maps p : B + W, the correspondence being p = cpl B. (2) If V is a free R-module, it can be shown that any two bases of V have the same cardinality, called dimR V. For example, dimz Zk = k. (3) On the other hand, there are important dissimilarities. A submodule W c V of a free R-module can fail to be free. Even when the submodule W is free, it may have no basis that extends to a basis of V. For example, the even integers 22 form a free submodule of the free Z-module Z,but neither of its two bases, (2) or (-21, extends t o a basis of Z. --+

Modules will not be assumed free unless that is explicitly stated. DEFINITION7.1.3. If Vl, V2,V3 are objects in M ( R ) , a map cp : Vl x V2 is R-bilinear if

--+

V3

are R-linear, Vvi E Vi, i = 1,2. REMARK.For fixed choices of Vl ,V2, V3 E M (R), the set of R-bilinear maps cp : Vl x V2 --+ V3 is itself an R-module under the pointwise operations. DEFINITION 7.1.4. A tensor product of R-modules Vl, V2 is an R-module Vl 8 V2, together with an R-bilinear map

with the following "universal property": given any R-module V3 and any Rbilinear map cp : Vl x V2 + V3, there is a unique R-linear map : Vl @ V2 -r V3 such that the diagram Vl

'v2

\ 1" @

v 2

v 3

commutes. Write @(v,w) = v 8 w. Thus, the R-module of R-bilinear maps Vl x V2 + V3 is canonically isomorphic to the R-module HomR(Vl @ V2, h)of R-linear maps Vl @ V2 --+ V3. THEOREM 7.1.5. Given R-modules Vl, V2 u s above, a tensor product Vl @ V2 exists and is unique up to a unique isomorphism. That is, if

7.1. TENSOR ALGEBRA

191

are two such t e n ~ o products, r there is a unique zsomorphism 0 : Vl@V2 -+ Vl6fi of R-modules such that the diagram 43

K

x v2

commutes.

PROOF.First we prove uniqueness. If 43 and are two tensor products, the universal property gives unique R-linear maps O1 and O2 making the following diagrams commute:

Then the diagram

also commutes, as does

By the universal property, we conclude that O2 o O1 = id. Similarly, O1 o O2 = id, so O1 and O2 are mutually inverse R-linear isomorphisms. Since 191 is unique, we are done.

192


The existence proof, though elementary, is a bit more long winded. Let W be the free R-module spanned by the set Vl x V2. The module W is just the set of all formal linear combinations

where a* E R and (vi, wi) E Vl x V2. This is an R-module under the obvious operations and each element 0 # w E W is uniquely expressed as an R-linear combination of finitely many members of the basis Vl x V2. Any linear combination with all coefficients 0 is equal to the 0 E W. Let R C W be the submodule spanned by all elements of the form (av

+ h,W) - a(v, w) - b(u, w)

where a, b E R and u, v, w are in Vl or V2 appropriately. We think of R as the submodule of bilinear relations and set

The cosets of the elements of the basis VL x V2 will be denoted by

and we define @:

v, x v2-+ v18 v2

Bilinearity follows immediately from the definition of R. For example,

Note that, as a special case of bilinearity, (av) @ w = a(v @ w) = v @ (aw) and, in particular, v @ 0 = 0 = 0 8 v. We establish the universal property. Let cp : Vl x V2 -+ V3 be an R-bilinear map. Since Vl x V2 is a free basis of W, there is a unique R-linear map

-

such that ~ ( vw) , = ~ ( uw), , V(v, w) E Vl x V2. Since rp is bilinear, it follows that cp vanishes on the generators of R, hence that plR 0. Consequently, cp passes to a well defined R-linear map

such that the diagram

7.1. TENSOR ALGEBRA

-.

commutes. Since V1@V2is spanned by elements of the form v@w, cp is unique. In a completely parallel way, one can consider R-trilinear maps and prove the existence and uniqueness of a universal R-trilinear map

sending

( v 1 , ~ 2 , ~-211 3)

@v2 @v3

It is a trivial exercise to check that the composition

(Vl x v2) x V3

Bxidv,

(Vl @VZ)x

v32 (&

@VZ)@ &

also has the universal property, as does

COROLLARY 7.1.6. If Vi is an R-module, i = 1,2,3, there are unique Rlinear isomorphisms Vl 8 (V2 @ V3) = (Vl @ V2) @ V3 = Vl @ V2 @ & identihing €3212 €9213, Vvi E V , , i = 1,2,3. vl @ ( ~ €9213) 2 = (vl @ v2) € 3 ~ = 3 More generally, for each integer k 2 2, there is a unique universal, k-linear map (over R)

vl x

X

"'Vk % & @ V 2 @ " ' @ V k

and canonical identifications

An obvious induction shows that all groupings by parentheses are equivalent, so parentheses can be dropped or used selectively as desired.

DEFINITION 7.1.7. An element v E Vl 8 . . @ Vk is decomposable if it can be written as a monomial v = v l @ . . - @ vk, for suitable elements vi E V,, 1 5 i 5 k. Otherwise, v is indecomposable. By the construction of the tensor product in the proof of Theorem 7.1.5, the decomposable elements span.

LEMMA 7.1.8. If V and W are free R-modules with respective bases A and B, then V 8 W is free with basis C = { a @ b 1 a E A, b E B).

194


An arbitrary element v E A @ B can be written as a linear combination of decomposables. A decomposable element v @ w can be expanded, via the multilinearity of tensor product, to a linear combination of elements of C, proving that C spans V @ W. It remains for us to show that, if PROOF.

where ai E A and bj E B, 1 5 i 5 p, 1 5 j 5 q then all c i j = d i l j . Subtracting one expression from the other, we only need to prove that

implies that all cij = 0. The bilinear functionals cp : V x W R correspond one to one to arbitrary functions f : A x B + R. The correspondence is cp * pl(A x B). Thus, the linear functionals $ : V @ W -+R also correspond one to one to these functions f : A X B + R. If ( a ,b) E A X B, let fa,b : A x B -+ R be the function taking the value 1 on (a, b) and the value 0 on every other element of A x B. to The corresponding linear functional will be denoted by $a,b. Applying equation (*), we see that all cij = 0 as desired. -+

By an obvious induction on the number of factors, this lemma generalizes to the following.

COROLLARY 7.1.9. If Vl, . . . , Vk are free R-modules on bases B1, . . . ,Bk, respectively, then Vl @ - - - @ Vk is a free R-module with basis

PROPOSITION 7.1.10. If Ai : V, a unique R-linear map

-,

W iis an R-linear map, 1 I i I k , there is

which, on decomposable elements, has the formula

Since the decomposables span, uniqueness is immediate. For existence, define the multilinear map PROOF.

by

X(v1,. . . ,uk) = Xl(vl) 8 .- - 8Xk(vk).

Then X1 @ . . . @ Xk is defined to be the unique associated linear map.

DEFINITION 7.1.11. The dual V * of an R-module V is Hom (V, R) , the module of R-linear functionals.

7.1. TENSOR ALGEBRA

195

LEMMA7.1.12. If V has a finite free basis (vl,... ,v,), then V * has a finite free basis {v;, . . . ,WE}, called the dual basis and defined by

PROPOSITION 7.1.13. There is a unique R-linear map

which, on decomposable elements, has the formula

If the R-modules V , are all free on finite bases, then L is a canonical isomorphism.

PROOF.Uniqueness is immediate by the fact that decomposables span. For existence, define the multilinear functional

by ~ ( v I , - -,-~ k , v l , .,vk) -= ~i(vi)qz(va). . .~k(vk).

By the universal property, this gives the associated linear functional

and we define L :

:v @"'@v~

(vl@"'@vk)*

If {viTl,... ,vi,mi} is a free basis of ir,, 1 5 i 5 k, let {vll,. . . ,},,:v be the ; @ - - - @ Vi dual basis. Let B and B be the respective bases of Vl@ - . - @ Vk and V given by Corollary 7.1.9. The formula

shows that L carries the basis B* one one onto the basis dual to B, so isomorphism.

L

is an

Let V be an R-module and view R as a module over itself.

LEMMA7.1.14. Scalar multiplication

induces canonical isomorphisms R 63 V = V @ R = V relative to which 1 @ v = v@l=v.


196

Indeed, scalar multiplication is R-bilinear, so there are canonical R-linear maps

These are inverted by the R-linear maps

respectively.

DEFINITION 7.1.15. Let V be an R-module. For each integer n 2 0, the n t h tensor power of V is

REMARK.By Lemma 7.1.14, TO(V)@ Tn(V) = Tn(V) = 7"(V) 8 To(V). When n and m are both positive, the identity I n ( V ) @ Tm(V) = 7 n + m ( V ) is given by the associativity of the tensor product. Set 7 ( V ) = {7n(V))r=o and note that 8 defines an R-bilinear map 7"(V) x 7 m ( V )

s Tn(V)

@ Tm(V) = Tn+m(v).

This makes 7 ( V ) into a graded algebra over R in the following sense.

DEFINITION 7.1.16. A graded (associative) algebra A over R is a sequence n of R-modules, toget her with R-bilinear maps (multiplication) {A 00

(written (a, b) H a b or, sometimes, (a, b) +t that the compositions

An x (Am x AT) are equal, V n, m, r 2 0.

ab) which is associative in the sense

id x .

An x Am+T

DEFINITION 7.1.17. The graded algebra A is connected if A0 = R and A0 x Am 3 Am

c A"

x A'

are equal to scalar multiplication, V m 2 0. Remark that a connected graded algebra has unity 1 E R = A'.

DEFINITION 7.1.18. If V is an R-module, then 7 ( V ) , with multiplication 8, is called the tensor algebra of V.

7.1. TENSOR ALGEBRA

197

It is clear that the tensor algebra T ( V ) is connected. DEFINITION 7.1.19. A homomorphism cp : A -+ B of graded R-algebras is a collection of R-linear maps cpn : An B n , V n 2 0, such that the diagrams -+

commute, V n, m 2 0. The homomorphism cp is an isomorphism if cpn is bijective, VnlO. THEOREM 7.1.20. If A : V --, W is an R-linear map, then there is a unique induced homomorphism T(A) : T(V) -+I ( W ) of graded R-algebras such that T0(A) = idR and I1(A) = A. This homomorphism satisfies

V n 2 2, V v i E V, 1 2 i 5 n. Finally, this induced homomorphism makes I a covariant functor from the category of R-modules and R-linear maps to the categov of graded algebras over R and graded algebra homomorphisms.

PROOF.The formula on decomposable tensors is imposed by the requirement that T(A) be a homomorphism of graded algebras, together with the stipulation that T1(A) = A. Existence and uniqueness of the linear maps Tn(A) are given by A) preserves 63 multiplication is immediate. Proposition 7.1.10. The fact that I( The final assertion amounts to the obvious identities

The following is an elementary consequence of Corollary 7.1.9 and Proposition 7.1.13. THEOREM 7.1.21. If V is a free R-module with basis { e 1, . . . ,e m ) then

is a free basis of T k ( V ) and Tk(V*) = T k ( v ) *. REMARK.In particular, if V is a free R-module with dim R V = m, then

TERMINOLOGY. The established terminology "covariant tensor" and "contravariant tensor" in geometry is inconsistent with "covariant" and "contravariant" as used in category theory. For later reference, here are the geometer's definitions. Let V be a finite dimensional vector space over the field IF.


198

DEFINITION 7.1.22. For each integer r 2 0, Tr(V*), viewed as the space of r-linear maps VT IF, is called the space of covariant tensors on V of degree r and is denoted T,'(V). -+

DEFINITION 7.1.23. For each integer s 2 0, T s ( V ) , viewed as the space of s-linear maps (V*)' -+ IF, is called the space of contravariant tensors on V of degree s and is denoted by 'T;(V). DEFINITION7.1.24. The space of tensors on V of type (r,s) is the tensor product T(V) =G(V)0C(V).

A tensor a

E c ( V ) is said to have covariant degree

r and contravariant degree

s.

Obviously, c ( V ) is the space of (r f s)-linear maps

EXERCISES (1) Let V be an R-module, V* its dual. (a) Exhibit a canonical R-linear map a : V * 8 V + R. (b) If V is free, prove that a is a surjection. If, in addition, V has a basis with one element, prove that a is a bijection. (c) If V and W are R-modules (not necessarily free), exhibit a canonical R-linear map P : V * @ W -,Hom (V, W ). (d) If R is a field, prove that P is injective. Do not assume that V and W are finite dimensional. is surjective if and only if either (e) If R is a field, prove that dimR V < oo or dimR W < oo. (2) Let A be a connected, graded R-algebra. (a) Show that there is a unique homomorphism

= idR and = idA1. of graded algebras such that (b) Define a suitable notion of 2sided ideal I 5 A so that A / I = ( A n / I n ) ~ = ois again a graded R-algebra. (c) If A is generated, as a graded algebra, by A', show that there is a canonical ideal I c 7(A1), with I' = (0) = 11,and a canonical isomorphism

such that yo = idR and y' = idA1.

7.2. EXTERIOR ALGEBRA

199

7.2. Exterior Algebra

Let R be any commutative ring with unity 1 such that f E R. That is, if 2 = 1 + 1 E R, then E R has the property that f . 2 = 1. In the case that R = IF is a field, this means that the characteristic of F is not 2. LEMMA7.2.1. Let V be an R-module, v E V. Then v = -v H v = 0. PROOF. Evidently, v = 0 =+v = -v. For the converse,

Let C be the group of permutations of {1,2, . . . ,k) , a group of order k! . DEFINITION 7.2.2. The sign of a E C k is (-1)O =

1, -1,

a an even permutation, a an odd permutation.

D E F I N ~ T7.2.3. ~ ~ N Let V and W be R-modules. An antisymmetric k-linear map cp : Vk + W is a k-linear map such that

Remark that this definition will be useful only because 1 # -1 in R. As in the definition of tensor product, for each k 2 2, define a universal antisymmetric k-linear map

written A(vl, ... ,Vk) = 211 A " ' A Vk. Here, the subspace 72 of relations is generated by the k-linear relations and all elements (211,

.. 7 ~ k -) ( - l ) u ( ~ o ( l ) ,. - , ~ u ( k ) ) 0, E Ck.

Existence and uniqueness, up to unique isomorphism, are established exactly as for tensor product. One sets AO(V)= R and A1(V) = V. DEFINIT~ON 7.2.4. The R-module Ak(V) is called the kth exterior power of V. The connected graded R-algebra A(V) = {A with multiplication

(~)}r=~

is called the exterior algebra of V.

200


As before, R-linear maps X : V + W induce canonical homomorphisms of graded algebras A(A) : A(V) -+ A(W) such that AO(A)= id^ , A' (A) = A, and

This makes A a covariant functor from the category of R-modules and R-linear maps to the category of graded algebras over R and graded algebra homomorphisms.

DEFINITION 7.2.5. An element w E A k ( v ) that can be expressed in the form vl A 712 A . - * Auk, where vi E V, 1 5 i 5 k, is said to be decomposable. Otherwise, w is indecomposable. It is clear that A k ( v ) is spanned by decomposable elements, but generally there are plenty of indecomposable elements as well. It will be useful to relate A(V) more directly to 'T(V).

DEFINITION 7.2.6. The ideal M(V) c T ( V ) is the 2sided ideal generated by all elements in T ~ ( v )of the form vl 8 v2 vz @ vl, vl, v2 E V.

+

Thus,

ak(v)= span

U

TP(V) @ a 2 ( v ) @ 7 (V)

p+q=k-2

Let cp : Vk -+ W be an antisymmetric k-linear map. As a k-linear map, cp can be interpreted as a linear map which, for clarity, we denote by (p : 7 k(V) -+ W.

LEMMA7.2.7. If cp : Vk

-+

W is antisymmetn'c, then (p(Mk(V)) = {O}.

ak(v).

PROOF. It will be enough to show that @ vanishes on a set spanning Thus, if w E 7p(V), u E Tq(V), p q = k - 2, and v l , v2 E V , we will show that

+

But the antisymmetry of cp implies that

and the assertion follows from linearity.

7.2. EXTERIOR ALGEBRA

COROLLARY 7.2.8. There is a canonical isomorphism A(V> = 7(V)/'21(V) of graded algebras.

PROOF.For k = 0,1, it is clear that Ak(V) = T k ( v ) / 2 l k ( ~ )For . each k

> 2,

consider the k-linear map

where 7r is the quotient projection. Given an arbitrary antisymmetric k-linear map cp : Vk -+ W, we obtain the commutative triangle

(v) = 0. Thus, jinduces q : Tk(v)/21k(v)--+ W making the following and @lak diagram commutative:

That is, the triangle

commutes. Since Tk(v)/ak( V) is spanned by

and commutativity of the diagrams forces

we see that q is the only linear map making the triangle commute. That is, T o 8 : Vk -+Tk(V)/21k(V)has the universal property for antisymmetric, klinear maps, hence is uniquely identified with A : V k Ak(V). Finally, this identification gives --+

so it is an identification of graded algebras.

T]


202

DEFINITION 7.2.9. A graded algebra A is anticommutative if a

and

E

E

AT + crp = (-l)kr@a.

COROLLARY 7.2.10. The graded algebra A ( V ) is antico~mmutative. PROOF.It is enough verify Definition 7.2.9 for decomposable elements of A k ( v ) and A r ( V ) . But that case is an elementary consequence of the case k = r = 1, and this latter case is given by VAW=V@W+%~(V) = -w @ v - -W A v ,

+ g2(v)

VV,WEV.

COROLLARY 7.2.11. If w E A2"+'(v), then w A w = 0. PROOF.Indeed, w

w = (- 1 )(2'+1)(2r+1)

/\ w = -w

A W.

By Lemma 7.2.1, w A w = 0.

COROLLARY 7.2.12. If w

E

A k ( v ) is decomposable, then w A w = 0.

For the remainder of this section, we specialize to the case in which V is a free R-module on a basis ( e l , . . . ,em).

LEMMA7.2.13. If V is as above, then {eil A ei2 A

A eik)15il 0.

Rm are open subsets, and p : U

Y* = p* o d : AP(V)+ AP+'(U),

-+ V

8.1. EXTERIOR DERIVATIVE

223

PROOF. In the following computation, the third equality is by the above remark: d(cp*( f dyZ1A

- - A dyip)) = d(cp*( f )cp* ( d y i l )A . . A cp* (dy'p)) = d(cp*(f d(cp*(yil)) A . . A d(cp*( y a p ) ) ) = d(cp*( f )) A d(cp*( y " ) ) A - - - A d(cp*(y")) = cp* ( d f ) A cp* (dyi') A = cp* (df A dy" A

. . A cp* (dyip) *

- . - A dy")

= cp*(d(f dydl A

A dyip)).

Since every q E AP(V) is a sum of forms of the type used in the above computation, the claim follows. We want to extend the exterior derivative to an R-linear operator

on all manifolds M and for all nonnegative integers p. We take an axiomatic approach, requiring that this operator satisfy the following: ( 1 ) f E A O ( M )and x E X ( M ) + d f ( x ) = x(f). ( 2 ) w E AP(M) and 77 E A Q ( M )+ d(w A 77) = dw A 77 (-1)pw A dq. (3) d2 = 0. ( 4 ) U c M open and w E A p ( M ) + (dw)tU = d(w1U). (5) cp:M-+Nsmooth~cp*od=docp*.

+

REMARK.If one only considers manifolds that are open subsets of Euclidean spaces, then all of the axioms hold for the exterior derivative as already defined. For d : A O ( M )+ A 1 ( M ) ,which has been defined on all manifolds, we have seen the truth of the first axiom (Lemma 6.1.2). DEFINITION 8.1.5. If an W-linear operator d : A P ( M ) -+ Ap+l(M), defined for all smooth manifolds M and all integers p 0, satisfies the above axioms, it is called an exterior derivative.

>

THEOREM 8.1.6. There is a unique exterior derivative. PROOF. First we prove uniqueness. By Axiom (4), it will be enough to show that, whenever U M is a coordinate neighborhood and w E A p ( M ) , the form d(w/U)f A ~ + ' ( u )is uniquely determined. Let cp : U + Rn be a diffeomorphism onto an open subset V and set cpi = cp*(x') = x' o cp, 1 5 i < - n. Also, by functoriality, (cp-')* = (cp*)-.l, which is to say that cp* : A * ( V ) + A * ( U ) is an isomorphism. Thus, since dcp" cp*(dxi) (Axiom ( 5 ) ) , the set

(U)-module AP(U). Thus, is a free basis of the Coo

224

8. INTEGRATION AND COHOMOLOGY

and

where the second equality uses Axioms (2) and ( 3 ) . But dfi,...ipis uniquely specified by Axiom (1). Thus, d(wlU) is unique and, as remarked above, the uniqueness in general of the operator d follows. Remark that all five axioms have been used in this argument. We turn to the proof of existence. Let

be the maximal coordinate atlas for M . The diagrams

make sense and commute, V a, p E a, since everything is written for open subsets of Euclidean space. For w E AP(M), define

If U, n Up# 0, we check that the two definitions agree on U, n Up.Indeed, by the commutative diagram,

Thus, the local definitions of dw piece together to give dw E Ap+'(M). Since the axioms are true for open subsets of Euclidean space, they are true locally on M , hence globally.

>

DEFIN~TION 8.1.7. A form w E AP(M), p 0, such that dw = 0 is called a closed p-form on M . Closed p-forms are also called p-cocycles and the real vector space of all such forms is denoted by Z P ( M ) . We set Z* ( M ) = { Z P ( M ) } g o . DEFINITION 8.1.8. A form w E AP(M), p > 1, is said to be exact if there is a form Q E A p - l ( M ) such that dq = w. Exact p-forms are also called pcoboundaries and the real vector space of all such forms is denoted B p ( M ) . For p = 0, we define BO(M) = { 0 ) c A'(M). Finally, B * ( M )= { B P ( M ) } E o .


225

REMARK.A form w E AP(M) is locally exact if every point x E M has an open neighborhood U such that wlU E Bp(U). One version of the Poincark Lemma (Section 8.3) asserts that the set of locally exact p-forms on M is precisely Zp(M). In particular, our earlier definition of Z (M) (Definition 6.3.1) agrees with our present one. In anticipation of the Poincarb lemma, you will be invited to prove a special case of this last assertion in Exercise (1). The formula d2 = 0 is equivalent to the inclusion BP(M) C ZP(M), p 0.

>

>

DEFINITION 8.1.9. For each integer p 0, the pth (de Rham) cohomology space of M is the real vector space HP(M) = ZP(M)/BP(M).

If cp : M -+ N is smooth, the formula cp* o d

=d o

cp* implies that

so cp* induces an R-linear map cp* : HP(N) + HP(M).

As usual, we have functoriality. LEMMA8.1.10. The pth cohomology HP is a contravariant functor from the category of diflerentiable manifolds and smooth maps to the category of real vector spaces and linear maps. If w E ZP(M) and q E Zq(M), then

That is, the exterior product of a closed p-form with a closed q-form is a closed ( p + 9)-form. Thus, Z*(M) is a graded algebra over R under exterior multiplication. Both A* (M) and Z*(M) are anticommutative in the sense of Definition 7.2.9. LEMMA8.1.11. The graded R-algebra Z*(M) is connected i f and only i f M is a connected manifold. PROOF.The space ZO(M)consists of all f E C m ( M ) such that df = 0. That is, ZO(M) is the space of locally constant, real-valued functions on M. Identifying R with the space of const ant functions in C "(M), we have R & zO(M). The product in Z*(M) of a constant function and a form becomes naturally identified with scalar multiplication. But locally constant functions are all constant if and only if M is connected. COROLLARY 8.1.12. The space H O ( M ) is one-dimensional i f and only i f M is connected. I n this case, HO(M) = W canonically. Generally, HO(M)is a direct product of copies of R, one for each component of M. PROOF. Indeed, HO(M) = zO(M)/BO(M) = ZO(M),the space of locally constant functions, and all claims follow easily. LEMMA8.1.13. If dim M = n, then HP(M) = 0 , V p > n.

226

PROOF.


Indeed, Ap(M) = 0 for all integers p greater than the dimension of

M.

LEMMA8.1.14. The graded subspace B*(M) Z*(M) is a 2-sided ideal, hence H * ( M ) = Z*(M)/B*(M) is a graded, untiwmmutative algebra over the field R. PROOF.If w E ZP(M) and Ap-'(M), hence

Q

f BQ(M),q

> 1, then 77 = d a for some a E

Since A w = ( - l ) P q ~ ~ 7 ] ,it follows that B*(M) is a 2sided ideal in Z*(M).

[7

Since smooth maps cp : M + N preserve exterior multiplication and pass to well defined maps in cohomology, we see that

is a homomorphism of graded algebras. We have completed the proof of the following.

THEOREM 8.1.15. The graded cohomology construction defines a contruvariant functor H* from the category of diflerentiable manifolds and smooth maps to the category of unticommutative graded algebras over R and graded algebra homomorphisms. The graded algebra H*(M) is connected if and only i f M is connected. The graded algebra H*(M) is called the (de Rharn) cohomology algebra of M. Whether or not it is connected, H '(M) has a unity, namely the constant function 1 E ZO(M)= H'(M).

DEFINITION 8.1.16. The space of compactly supported p-forms on M is denoted by AE(M). It is clear that the exterior product of two compactly supported forms is compactly supported. Indeed, it is enough that one of them be compactly supported. Thus, each Af(M) is a module over Coo(M) and these assemble into a graded algebra AZ(M) over Cm(M). It is also a graded algebra over R, which is more to our purpose. Furthermore,

so one can define the space

and the vector subspace


227

If we use the common convention that A-'(M) = A;l(M) = 0, the above definition of BZ(M) includes the case p = 0. As before, Z,*(M) is a graded subalgebra of A;fi(M) and B,* (M) Z,*(M) is a 2sided ideal.

DEFINITION 8.1.17. The (de Rharn) cohomology algebra with compact s u p ports is H,*(M)= Z,*(M)/B,*(M). Remark that H*(M) = H,*(M) if and only if M is compact. At the other extreme, if M has no compact component, the space z ~ ( M ) of compactly s u p ported, locally constant functions on M is trivial, so Hz(M) = 0. In any event, each element of Z:(M) will vanish on all but finitely many components of M. These observations establish the following.

LEMMA8.1.18. The vector space H:(M)

is isomorphic to a direct sum of copies of R, one for each compact component of M . Note the different conclusions in Lemma 8.1.18 and Corollary 8.1.12. Each element of a direct sum has terms from only finitely many summands, while elements of a direct product are allowed to have terms from infinitely many of the factors.

REMARK.The graded algebra H,*(M) generally does not have a unity unless M is compact. DEFINITION 8.1.19. A smooth map cp : M set C N , the set cp-l(C) is also compact.

+N

is proper if, for each compact

For example, id : M + M is always proper. If M is compact, cp is always proper. In any event, the composition of proper maps is proper, so the class of differentiable manifolds and smooth, proper maps between them is a category. If cp : M + N is proper and if w E A!(N), then cp* (w) E Az(M). As usual, cp* o d = d o cp*, so we get an induced homomorphism of graded algebras

THEOREM 8.1.20. Cohomology with compact supports is a contmvariant functor H,* from the category of diflerentiable manifolds and smooth, proper maps to the category of anticommutative graded algebras over R and graded algebra homomorphisms.

EXERCISES (1) Define a manifold M to be contractible if it is homotopy equivalent to a point. Prove that the following three versions of the Poincar6 Lemma for 1-forms are equivalent, and verify (c) when n = 2. (Here, Z '(M) denotes the space of closed 1-forms, not the space of locally exact ones.) (a) The form w E A1(M) is locally exact if and only if it is closed. (b) If M is contractible, then z'(M) = B'(M). (c) Z I ( I W= ~B ) ~(R~)(2) Prove that HE (R) S R. This is the one-dimensional case of the Poincar6 Lemma for compactly supported cohomology.

228


8.2. Stokes' Theorem and Singular Homology In this section, we define integration of forms and give a detailed treatment of two versions of Stokes' theorem. As an application of the second (combinatorial) version, we define the singular homology of a manifold and relate it to de Rham cohomology, stating the celebrated de Rham theorem. A detailed sketch of the proof of this theorem will appear in Section 8.9. Throughout what follows, we assume that dim M = n and that M is oriented. We also allow d M # 8. THEOREM8.2.1. For each oriented n-manifold M , there is a unique R-linear functional

called the integral and having the following property: i f (U, cp) is an orientationrespecting coordinate chart, iif w E AF(M) has supp(w) c U, and if cp-'*(w) = dxl A - A dxn E A: (p(U)), then

g JM

(the Riemann integral).

= J,,u,

PROOF.First we prove uniqueness. Let {(U,, cp,)},€%

be a smooth Wn-atlas be a smooth partition of unity on M respecting the orientation. Let subordinate to the atlas. If w E A,"(M), then X,w E AF(M) and X,w # 0 for only a finite number of a E a. This is because supp(w) is compact and the partition of unity is locally finite. Thus,

and this sum is actually finite. Then, if

JM

exists, linearity gives

and supp(X,w) = supp(X,) n supp(w) is a compact subset of U, property of J M , each X,w is uniquely given as

IM

.

By the local

where g, dxl A A dxn = cp;'*(wlU,). We give one way to define establishing existence. This will depend on a choice of orientation-respecting coordinate atlas {U,, and of subordinate partition of unity {Xcr},Ea. (One could remove some of this arbitrariness by requiring the atlas to be the maximal one, but the choice of partition of unity still could not be made canonical.) We appeal to the uniqueness already proven

L,

8.2. STOKES' THEOREM

229

to show independence of the choices. If w E A,"(M), only finitely many X,w are not identically 0. Define

where g, dxl A

... A dxn = (P;l*(~[U,).

Then define

a finite sum. Defined in this way,

is an R-linear map. We must check that, if supp(w) c U, where (U, cp) is an arbitrary orientationrespecting coordinate chart (not necessarily in our atlas), and if

then

JM l(u) g*

=

First remark that

(A,

0

cp,l)ga

since supp(w) c U. By Exercise (2) on page 220,

for each a E %. The fact that the charts are compatibly oriented is essential. Thus,

230


REMARK.By Exercise (2) on page 220 and the above proof,

The orientation of M induces an orientation of d M in the following way. Let {U,, x,1 , . . . , x : ) , ~ ~ be an Wn-atlas on M respecting the orientation. By the definition of Wn, x: 5 0, wherever defined, and x i = 0 exactly on U, n dM, V a E !2l. Let !2l' = {a E !2l1 U, n d M # 0 ) and consider the Ren-'-atlas

gzp

of dM. Let gap and denote the respective changes of coordinates for these atlases. At x E U, T) Up n 8 M ,

Here, since x: decreases with xi, the upper left hand entry in this matrix is strictly positive. Since det(Jg,p). > 0, it follows that d e t ( ~ ~ z > ~ 0. ) , Thus, this Itn-l-atlas on d M defines an orientation of dM. DEFINITION 8.2.2. The orientation of dM, produced as above, is said to be induced by the given orientation of M. We always assume that, when M is oriented, d M has the induced orientation. The following fundamental result asserts that exterior differentiation is the "adjoint" of passing to the boundary. For this reason, d is often referred to as the "coboundary operator".

THEOREM 8.2.3 (STOKES'THEOREM), Let M be an oriented n-manifold and let i : a M --, M be the inclusion. Then, i f w E A;-l(M),

where, i f d M = 0 , the right hand side is interpreted as 0. PROOF.First we prove the local case. That is, M = W n and d M = iRn-'. Any w E A:-'(HIn) can be written as

-

where fj has compact support, 1 5 j 5 n, and dxj indicates that this term is omitted. Then

and


231

By the fundament a1 theorem of calculus and the compactness of supp( f ) , for j =2,

..., n,

Therefore,

Now we can prove the global case. Let

denote the respective boundary inclusions and let

be the orientation-respecting atlases on M and d M , respectively, a s chosen above. If {A,),,ca is a smooth partition of unity subordinate to the atlas on M, then {A, o ~ ~ is a smooth ) partition ~ ~ of unity ~ subordinate t to the atlas on d M . Note that pgl o zwn = iMo (&)-', v a E a'. For w E A:-'(M), the fact that w = C , X,w is a finite sum gives finite sums


232

Therefore,

The last equality is by the local version proven above. If dM = 0, then

and this integral vanishes. Otherwise, we get

igw. =L M

THEOREM 8.2.4. Let M be an oriented n-manifold with a M = 0. Then

is a well defined, R-linear sudection.

PROOF.Since A:+l(M) = 0, we have Z:(M) = A:(M). If w = dv E B:(M), then Stokes' theorem and the fact that dM = 0 imply that

Thus, the linear map

r


induces a well defined linear map

To prove surjectivity, we only need prove that this map is nontrivial.

Let

(U,xl,. . . ,xn) be a compatibly oriented chart and let X E Coo ( M ) have compact support contained in U ,with X 2 0 everywhere and X > 0 somewhere. Thus w = X dx' A

. A dxn can be interpreted as an elemcnt of Z,"(M)

and of AY(Rn),

SO

A deeper fact, to be proven later (Theorem 8.6.4), is that, if M is both oriented and connected, then JM is a bijection from H,"(M) to W. In order to integrate p-forms, where p < dim M , it is necessary to define suitable p-dimensional domains of integration. For the case p = 1, we have already studied line integrals, the domain of integration being a (piecewise) smooth curve in M. In general, it is convenient to use singular psimplices (defined below) as domains for integrating p-forms. A singular l-simplex is simply a smooth curve. Recall that a subset A c RP is convex if, for each pair of points v, w E A, the straight line segment joining v and w lies entirely in A. If C C_ R P is an arbitrary subset, the convex hull of C is defined to be the smallest convex set containing C. Since an arbitrary intersection of convex sets is convex, and RP is itself convex, is just the intersection of all convex sets containing C.

DEFINITION 8.2.5. The standard psimplex A p c R P is the convex hull of the set {eo,e l , . . . , e,}, where ei is the ith standard basis vector, 1 < i 5 p, and eo = 0.

Thus, A, = {O}, a single point, and A = [0,1]. The cases p = 2 and p = 3 are pictured in Figures 8.1 and 8.2, respectively.

FIGURE 8.1. The standard 2simplex

A more explicit definition of the standard psimplex is


el FIGURE 8.2. The standard 3-simplex It is sometimes convenient to set A -l = 0.

DEFINITION 8.2.6. A singular p-simplex in a manifold M is a smooth map

s:Ap+M. Thus, each point of M can be thought of as a singular Osimplex and smooth curves, up to parametrization, are singular lsirnplices. One could also define piecewise smooth singular psimplices, but we will not do so.

DEFINITION 8.2.7. For 0 5 i 5 p, the ith f x e of the standard psimplex Ap is the singular ( p - 1)simplex Fi: Ap-l + A, defined by

The oth face of A. is considered to be defined but empty. If s : A, + M is a singular psimplex, the ith face of s is the singular (p - l)-simplex ais = s o F,. It is clear that Fi: ApPl + A, is a topological imbedding and that the image of Fiis exactly the subset ordinarily thought of as the "face" of A, opposite the vertex ei. The ith face dis of a singular psirnplex s is essentially the restriction of s to the ith face of Ap, but parametrized on the standard Ap-1.

DEFINITION 8.2.8. If s : A, 4 M is a singular psimplex and w E AP(M), then s*(w) has the form g dxl A . A d x p and we set

where the right hand side is the Riernann integral. If s : (0) + M is a singular Osimplex and w = f E AO(M),the integral is interpreted to mean


235

There is a combinatorial version of Stokes' theorem, according to which the integral of an exact p-form dq over a singular psimplex s is equal to the integral of q over the "boundary" of s. THEOREM 8.2.9 (COMBINATORIAL STOKES'THEOREM).If s : Ap + M is a singular p-simplex and 77 E Ap-I (M), then

REMARKS.The signs in the combinatorial Stokes' theorem are dictated by comparing the standard orientation of A I with the induced orientation of as a part of the boundary of Ap. We write the formal expression

,-

and express Stokes' formula as

This highlights the analogy with Theorem 8.2.3 and agrees with established usage in algebraic topology. For f E AO(M)and a smooth curve s : [0, I] + M , Theorem 8.2.9 asserts that

which is just Lemma 6.2.3. That lemma was a thinly disguised version of the fundamental theorem of calculus and Theorem 8.2.9 is a somewhat less thinly disguised version of the same fundamental theorem. PROOFOF THEOREM 8.2.9. It is clearly sufficient to prove that

where q is a (p - 1)-form defined on an open neighborhood of A, in RP. We can write

and, by the linearity of the integral, prove the formula for each term of the sum. That is, without loss of generality, we assume that 1 5 j 5 p and

By the local formula for exterior differentiation,


236

and we are reduced to proving the formula

The right hand side of equation (8.1) can be simplified. For this, it will be helpful to let xi denote the coordinates in Rp and zi the coordinates in RP-'. Remark that

and, if i

> 0,

One obtains the formula

and, if i

> 0,

-

F ; ( f d x 1 ~ . . . ~ dAx. 3. .

A ~ X P= )

i f i # j, (f o F j ) d ~ ' A . . - A d ~ P - 'i f i = j .

Substituting these terms in the equation (8.1) and multiplying both sides by (- l)j-I reduces us to proving

which, rewritten in terms of the Riemann integral, becomes

The linear change of coordinates in IRp-', defined by


237

preserves volume (i.e., the Jacobian determinant is f1) and carries morphically onto itself, as the reader will easily check. Also,

diffee

While this coordinate change reverses orientation when j is odd, the Riemann integral is insensitive to orientation, so (8.2) becomes

To avoid notational confusion, replace the dummy variables w with zi:

Let

A; = Fj(AP-1) = {(XI,.. . , b )E Ap the face of Ap opposite the vertex e j , 1 5 j 5 p. Since j the linear diffeomorphism

=

01,

> 0, it

is evident that

Fj : Ap-1 + A;

preserves ( p - l)-dimensional volume, so the three integrals in (8.3) can be computed, respectively, by the multiple integrals

With these substitutions, (8.3) is checked by standard manipulation of iterated integrals and an application of the fundamental theorem of calculus. The proof of Theorem 8.2.9 is complete.

COROLLARY 8.2.10. A f o n w E AP(M) is closed i f and only if every singular ( p + 1)-simplex s in M .

Sgsw = 0, for

238


PROOF. If w is closed, then

For the converse, suppose that dw = rj- # 0. Choose a point x E M such that A rj-, # 0. Choose vectors v l , . . . ,vp+l E T,(M) such that rj-,(vl > 0. These vectors must be linearly independent, so we can find a local coordinate chart (U, x l , . . . ,xn) about x in which vi is the value of the ith coordinate field ti = d/dxi at x, 1 _< i 5 p 1. By making this chart sufficiently small, we can guarantee that rj-(Jl A ... A Jp+l) > 0 on all of U. Let s : Ap + U be any orientation-preserving, smooth imbedding into the coordinate ( p + 1)-plane {(xl, , . . ,x n ) E U I X P + ~= . . . = xn = 0). It follows that

+

Singular simplices are used to detect topological features of a manifold. Recall, for instance, how a piecewise smooth, closed curve s = sl s, in the punctured plane It2 \ {x) can detect the missing point, provided that s has nonzero winding number w(s, x) about the point. The closed curve s is assembled from the singular l-simplices s l , , . . ,sqwhich join together, end to end, to form a "l-cycle". The winding number itself was found by integrating a certain locally exact (hence, closed) 1-form 77, over s (Exercise (4) on page 187). Piecewise smooth closed curves s in IR3 \ {x) do not detect the missing point. Indeed, it can be shown (Exercise (2)) that any such loop s is homotopic in IR3 \ {x) to a constant loop. However, a map s : S2 4 IR3 \ {x) can snag the missing point. One effective way to use this observation is to triangulate S2 (Section 1.3) and form singular 2simplices s l , . . . , s, by restricting s to these triangles. It is only necessary to assume that each si is smooth, so we get a piecewise smooth map s : S2 + It3 \ {x) and write s = s l + + s, by analogy with the case of loops. We call this a "singular 2-cycle". One should test whether or not the singular 2-cycle has snagged the missing point x by integrating a suitable closed 2-form over this cycle:

+

+

The possibilities for singular 2-cycles are richer than for I-cycles. For instance, triangulations of T2 and corresponding piecewise smooth maps s of T~ into M define '%oral" singular 2-cycles s = s l + . . s, in the manifold M and such a cycle might well detect a topological feature that would be missed by a "spherical" cycle. Again, a test of what this cycle detects is made by integrating closed 2-forms over the cycle. These remarks are extended and made precise by defining the singular homology of a manifold, a covariant functor H , from the category of smooth manifolds to the category of graded vector spaces over R. The celebrated de Rham theorem asserts that this functor is dual to de Rham cohomology. We sketch the

+


239

main facts, illustrating the importance of the combinatorial Stokes' theorem for algebraic topology.

DEFINITION 8.2.11. The set of all singular pimplices in M , p 2 0, is denoted by Ap(M). The space Cp(M) of singular pchains on M is the free Rmodule (real vector space) generated by the set Ap(M). By convention, if p Ap(M) = 0 and Cp(M) = 0.

< 0,

Each p-form w E AP(M) can be viewed as a linear functional

as follows. An arbitrary p c h a i n c E Cp(M) can be written uniquely (up to terms with coefficient 0) as a linear combination

where

sj

E Ap(M), 1 5 j 5 m. One then defines the value of w on c to be

The face operators 8,s = s 0 Fi have already been defined, V s E Ap(M), and can be viewed as set maps

ai : Ap(M) -+ Cp-l (M). Since Ap(M) is a basis of Cp(M), these set maps extend uniquely to linear maps

DEFINITION 8.2.12. The boundary operator d : Cp(M) -4Cp-l(M), p 2 0, is the linear map P

For w E AP(M) and c E Cp(M), the combinatorial Stokes' theorem asserts that

That is, the operators d and d are adjoint to one another. The boundary operator is an algebraic analogue of the geometric notion of a boundary. The following crucial property can be viewed as the algebraic analogue of the fact that the boundary of the boundary of a manifold is empty. The proof is a nice exercise in combinatorics, left to the reader as Exercise (1).

LEMMA8.2.13. The composition

is trivial (a2= 0).


240

DEFINIT~ON 8.2.14. The space Z p ( M ) C_ C p ( M )of all p-cycles is the kernel of the boundary operator d : C p ( M ) + Cp-l ( M ) . The space B p ( M ) C p ( M )of all p-boundaries is the image of the boundary operator a : C p + 1 ( M )+ C p ( M ) . An immediate corollary of Lemma 8.2.13 is that B p ( M )

Zp(M).

DEFINITION 8.2.15. The pth singular homology of M is the vector space

If z f Z p ( M ) ,the homology class of z is the coset [ z ] E H p ( M ) represented by the cycle z.

EXAMPLE 8.2.16. Since C - l ( M ) = 0, the boundary operator vanishes identically on C o ( M ) . That is, Z o ( M ) = C o ( M ) is the real vector space with basis the set of points of M. Define c : Z o ( M ) 3 R by

where all E R and all xi f M . If s E A 1 ( M ) , € ( a s ) = ~ ( s ( 1-) s ( 0 ) ) = 0 , so B o ( M ) C_ k e r ( ~ )and c passes t o a well defined linear map Z : H o ( M ) 4 R. We assume that M # 8, so there is a point x E M and Z([x])= 1, proving that 2 is a surjection and [x]# 0, Vx E M. If M is connected, then every two . . + s , and points x, y E M can be joined by a piecewise smooth path s = sl a s = y - x. This implies that [x] = [y], hence that H o ( M ) has basis consisting of a single element [a]. We have proven that the oth singular homology of a nonempty, connected manifold is canonically isomorphic to R.

+

cpt

EXAMPLE 8.2.17. Let M be contractible (cf. Exercise ( 1 ) on page 227). Let : M 4 M be the contraction, a homotopy of cpo = idM to a constant map

91.

Using this contraction, we are going to define linear maps

>

-

-

V p 0, with a remarkable property. Rp+l restricts to an inclusion A p AP+l The standard inclusion RP there is a unique which is just the face map Fp+l. For each point v E point v' E A p and a unique number t E [ O , 1 ] such that v = teP+l ( 1 - t ) v f ,and every point of WP+' of such a form is a point in If s : A p + M is smooth, define a smooth map L p ( s ) : Ap+l 4 M by the formula

+

The fact that this is well defined when t We view s I+ L p ( s ) as a set map

==

1 is due to

cpl

being a constant map.

and take the linear map L p to be the unique linear extension of this set map to all of C p ( M ) .In Exercise (6), you are invited t o check that


24 1

>

provided that p 1. This is the remarkable property promised above. If z E Zp(M) and p 2 1, it follows that

hence that Zp(M) Bp(M). The reverse inclusion also holds, so we have the result that the singular p-cycles and the singular p-boundaries in a contractible space are exactly the same, V p 2 1. That is, Hp(M) = 0 in all degrees p > 0. The above two examples give THEOREM 8.2.18. If M is a contractible n-manzfold, then

In particular, this is true for M = Rn. PROPOSITION 8.2.19. If w E ZP(M) and z E Zp(M), then the real number Jz w depends only on the cohomology class [w]E HP(M) and the homology class [ ~ El HP(M). PROOF.Indeed, [w] is the set of all closed pforrns w + d ~ where , q E Ap-' (M). We have by Stokes' theorem and the fact that z is a cycle, so

Similarly, [z]is the set of all pcycles of the form z Since

+ ac, where c E Cp+l (M).

we obtain

Thus, we can define an R-linear map

8.2.20 (THEDE RHAMTHEOREM).The linear map DR is a canTHEOREM onical isomorphism of vector spaces. This is a deep result. For the case in which M is compact, the proof will be discussed in some detail in Section 8.9. In that case, the vector spaces are finite dimensional (Theorem 8.5.8), so we also get Hp(M) = HP(M)*. The following corollary generalizes Proposition 6.3.3.

242


COROLLARY 8.2.21. Let w,3E ZP(M). Then [w] = [GIzf and only zf Jz w = Jz 3 as z ranges over all singular p-cycles in M . These numbers are culled the periods of w and of the cohomology class [w]. In particular, w is an exact form if and only if all of its periods are 0, which generalizes the equivalence of properties (1) and (2) in Theorem 6.2.9.

EXERCISES (1) Prove Lemma 8.2.13. (2) Let s be a piecewise smooth loop in Rn \ {x), where n 2 3. Show that s is homotopic to a constant loop. Without loss of generality, assume that x = 0 and proceed as follows. (a) Show that s is homotopic to a piecewise smooth loop on S c Wn \ (0). (b) Show that every piecewise smooth loop on Sn-' misses a point y E Sn-l, provided that n 3. (Hint: Sard's theorem.) (c) Show that Sn-' \ {y) is contractible and draw the desired conclusion. Note that, by the de Rham theorem, it follows that H1(Wn \ (0)) = 0 = H1(Rn \ {0)), n 3. (3) Let M be an n-manifold and let z E Zn+l(M). Assuming the de Rharn theorem, prove that there is a chain c E Cn+2(M)such that z = dc. (4) Show that singular homology is a covariant functor, proceeding as follows. (a) If f : M + N is a smooth map between manifolds, exhibit a canonical way to induce a linear map f # : Cp(M) + C,(N), v p 2 0. (b) Prove that the diagram

"-'

>

>

commutes, V p 2 0. Conclude that the linear map f # passes to a linear map f, : H, ( M ) + H, (M) of graded vector spaces. (c) Verify the properties (f o g), = f, o g, and id, = id. (d) Under the de Rham isomorphism of H k ( ~with ) Hk(M)*,show that

are adjoint to each other. (5) Without appealing to the de Rharn theorem, extend the argument in Example 8.2.16 to show that Ho(M) is a direct sum of copies of R, one for each connected component of M. (6) Prove the formula for d o L, asserted in Example 8.2.17.

8.3. POINCARE LEMMA

243

8.3. The Poincard Lemma In the following discussion, R will stand for any nondegenerate, compact interval [a,b] or for R. We consider an arbitrary n-manifold M , not necessarily orientable. If M has nonempty boundary, M x R will always denote M x 113, thereby avoiding manifolds with corners. Homotopies, therefore, will be understood in the sense of Definition 3.8.9 whenever convenient. We will agree to denote the standard projections by

and p: M x R 3 R .

The coordinate of R will be denoted by t and p*(dt) A1 ( M x R) will be denoted by dt (an abuse). An atlas {(W,, x , ) ) , ~ ~on M determines an atlas {(W, x R, (x,, t ) ) ) a E ~ on M x R. Here, if d M = 0 and R = [a,b], we model (n + 1)-manifolds with boundary on W n x [a, b] instead of on HIn+'. subordinate to { Wcr)aE%, determines A smooth partition of unity a smooth partition of unity

on M x R subordinate to {W, x R)aE'ZL.Here, -

L ( x , t) = Xa:(x). Ifw E A&(Mx R), let

u, = ul(W, x R) and write

where we use the conventions

and

Since

we write

u= For each cr E a,choose

0, : M

+

[O,l]


244

with supp(8,) and

c W,

and 8,) supp(A,)

1. Let

8, = n*(8,).

--

-

Then, A d , = A,

In each open set n-I (Wo) = Wp x R, only finitely many indices a E 21 correspond to nonzero terms in the expression for w (n-' (Wp). Also, for q = k or k - 1, 8, dx2 A . - ~ d x = z x* (v), for some 7 E AQ(M), so we have proven the following.

LEMMA8.3.1. Each form w E A k ( M x R) can be expressed as a locally finite sum of k-forms, each being one of the following two types:

We construct an important operator which "integrates out" the dt component of forms on M x R.

LEMMA8.3.2. For each r E R and each integer k 2 0, there is a unique R -linear map S, : A*(M x R) A ~ - ~ (xMR) which is additive over locally finite sums and satisfies

(Here we understand that A-'(M x R) = 0, so ST : A'(M x R) -, 0 is trivial, consistent with the fact that all forms in A O ( Mx R) are of type ( i i ) . )

PROOF. By the existence of a decomposition of w

E

A (M x R) into a locally

finite sum of forms of the types ( i ) and (ii),the stipulated properties of ST force that operator to be unique, provided that it exists. But, if we fix the choice of atlas {W,, x , ) , ~ ~ of , subordinate partition of unity {A,),cat, and of the functions (6a}aE%,we then have an algorithm for producing a locally finite decomposition of w f A k ( x~R) into the desired types of summands. We use (a) and ( b ) to define ST on each of these summands and remark that the result is a locally finite system of ( k - 1)-forms on M x R, hence that their sum is a well defined element ST(w) E A*-'(M x R). It is clear that S T , defined in this way, is R-linear. The crucial fact that it is also additive on Iocally finite sums is left as Exercise (1). Uniqueness shows that the definition of S , is really independent of the choices. For each T E R, let iT : M -t M x R be given by i,(x) = ( x , ~ )One . version of the Poincarb Lemma is that, at the cohomology level, n * and i: are mutually inverse isomorphisms. Indeed, it is clear that i: o n* = (ro iT)* is the identity at the level of forms, so it remains to show that .rr * 02: is the identity on cohomology. The main step is the following.

LEMMA8.3.3. O n A k ( x~R), the operator S, sat$es the identity

The reader is invited to check this formula in Exercise (2). THEOREM 8.3.4 (POINCARE LEMMA,VERSIONI). The map n* : H*(M) -t H*(M x R)

is an isomorphism and its inverse is i:, level, i: is independent of r.

QT

E R. In particular, at the cohomology

PROOF.As remarked above, we only need to prove that, at the cohomology level, n* 0 i: = id. If w E ZP(M), we apply Lemma 8.3.3 to obtain w - r *(2: (w)) = d(S, (w))

+ ST(d(w))

= d(S7(~)).

That is, w and n*(i:(w)) differ by a coboundary, and we are done.

THEOREM 8.3.5 PO IN CAR^ LEMMA,VERSION11). If f o , f l : M homotopic, then f ; = f ; : H * ( N ) + H *(M).

+

N are

PROOF.Let F : M x 113 + N be the homotopy. Then f o = F o io and fi = F 0 il . By functoriality,

fi = 28 o F* f ; = i; o F*.

But 2; = ii by Theorem 8.3.4, SO f i = fi. Here are four more versions of the Poincark Lemma. The first of these is immediate by Theorem 8.3.5, and each implies the next. All of the implications are rat her obvious. THEOREM 8.3.6 PO IN CAR^ LEMMA,VERSION111). I f f : M -t N is a homotopy equivalence, then f * : H *(N) 4 H*(M) is an isomorphism of graded algebras. THEOREM 8.3.7

PO IN CAR^

LEMMA,VERSIONIV). For a contractible man-

I n particular, this holds for M = Rn.

THEOREM 8.3.8 PO IN CAR^ LEMMA,VERSIONV). For k > 0, every closed k-form on a contractible manifold is exact. Since manifolds are locally contractible (each point has a neighborhood diffeomorphic to Rn), the next version follows. THEOREM 8.3.9 PO IN CAR^ LEMMA,VERSIONVI). If k > 0, a k-form on a manifold M is closed i f and only if it is locally exact. Thus, the definition of H1(M) given in Chapter 6 agrees with our current definition. It can be shown that Version VI implies Version I, so all versions are mutually equivalent. We will not prove this.

246


DEFINITION 8.3.10. If d M = 8 and fo, fl : M

N are proper smooth maps, they are said to be properly homotopic if there is a proper smooth map -t

such that F(x,O) = fo(x) and F ( x , 1) = fl(x), Vx E M. The map F is called a proper homotopy between f and f 1. For compactly supported cohomology on manifolds without boundary, define the term "homotopy" using R = [O, 11 and remark that both n * and i: are proper maps. Theorem 8.3.4 continues to hold in this situation and a suitably reworded version of Theorem 8.3.5 also holds (Exercise (3)). One could call this the Poincarh Lemma, but what usually goes by that name for compact cohomology takes a rather different form. This is our next topic. First note that w E A: ( M x R) is a finite linear combination of forms of the types

(i) f (x, t) d t ~ n * ( q )where , f (x, t) is compactly supported, but 7 E Ak-'(M) may not be compactly supported. (ii) f (x, t)rr*(7), where f (x, t ) is compactly supported, but q E A *( M ) may not be compactly supported. One then defines an R-linear map

called "integration along R" , by requiring that

As before, there is a unique R-linear operator n, with these properties. Remark that requiring the operator to be additive over locally finite sums is no longer necessary. It will also be convenient to define A;Q(M) = 0, V q > 0,to agree that d (= 0) is defined on this trivial module, hence to have H - Q ( M ) defined and trivial.

LEMMA8.3.11. With the above definitions, n, antiwmmutes with d. That is,

The proof is a straightforward computation on forms of types (i) and (ii). It is analogous to Exercise (Z), only easier.

COROLLARY 8.3.12. The linear map n, passes to a well defined linear map

8.3. POINCARE LEMMA

247

We want to prove that this map is an isomorphism, so we need a candidate for its inverse. Choose a compactly supported function b : IR -P W such that b(t)dt = 1. Let /iJ = b(t)dt E A:@). Finally, for each k E Z, define

It is practically immediate that

and we draw the following conclusion.

L E M M A8.3.13. The linear map P, passes to a well defined linear map

P* : H,L(M)

-+

H,'+'(M x W),

V k E Z. Clearly,

and we will show that, at the level of compact cohomology, P, identity. Once again, we construct an operator

o

n, is also the

such that doS+Sod=id-@,on,. We define S. Set

and define S on forms of type ( i i ) by

and, on those of type (i),by

Remark that this form is, indeed, compactly supported. This is obvious in the x variable and, for t 1 -00, it is also clear. But, as t f oo, the function in the parentheses ultimately becomes

so the support is bounded in all directions, hence compact.


248

LEMMA8.3.14. The formula

holds on A,k(M x R), V k E Z. The somewhat tedious computations that prove this lemma are relegated to Exercise (4). As in the proof of Theorem 8.3.4, this is all that is needed to establish the following.

THEOREM 8.3.15 (POINCARE LEMMAFOR

COMPACT SUPPORTS).

The map

is a canonical isomorphism with inverse P,, V k E Z. In particular, p, does not depend, at the level of compact cohomology, on the b(t)dt = 1. choice of the compactly supported function b(t) such that

Jrm_

REMARK. The operators S, and S, used to prove the Poincare lemmas for ordinary and compact de Rbam theory, are examples of cochain homotopies in algebraic topology. One says that i d A . ( ~ ~ is R )cochain homotopic to n* o i:, writing n* 0 i: idA*(MXR) . N

Similarly,

id^^

P* 0 n*

(MX

W)

.

Exactly as in the proof of Theorem 8.3.4, cochain homotopic maps induce the same map in cohomology. In the present situation, since one of the maps is the identity, they both induce the identity. In Example 8.2.17, we used a chain homotopy between the identity and a map that is 0 in positive degrees to show that the singular homology of a contractible manifold is trivial.

COROLLARY 8.3.16. For each integer n 2 0, H , ~ ( w "= )

R, k = n 0, otherwise.

PROOF. This is clearly true for n = 0. Inductively, suppose that it is true for a given value of n 0 and appeal to Theorem 8.3.15 to get

>

In Exercise (2) on page 227, you proved this for the case n = k = 1.

COROLLARY 8.3.17. The linear map

is an isomorphism.

8.4.

EXACT SEQUENCES

249

Indeed, since Rn is orientable, we have seen that this is a surjection (The* rem 8.2.4). Since the cohomology space is onedimensional, it is an isomorphism.

EXERCISES (1) Prove that the operator ST in Lemma 8.3.2 is, indeed, additive on locally finite sums. (2) Prove Lemma 8.3.3. (3) Show that Theorem 8.3.4 makes sense and holds for compactly s u p ported cohomology, provided that d M = 8 and R = [a,b]. Using proper maps and proper homotopies, formulate and prove the analogue of Theorem 8.3.5. (4) Prove Lemma 8.3.14. 8.4. Exact Sequences

A basic tool for computing cohomology will be the Mayer-Vietoris sequence (Section 8.5). In order to develop and apply this sequence, we will need some properties of exact sequences. This purely algebraic section may be a review for many readers. At any rate, the proofs are elementary and will be relegated to exercises. We fix a commutative ring R and consider modules A, B , C , etc., over R. All maps cp : A + B will be R-linear. We also consider graded R-modules A *, B*, etc., over R, in which case cp : A* --, B* will denote a homomorphism of graded R-modules. We will generally assume that the grading is indexed by Z rather than just Z+. No generality is lost since A* = { A * } E ~can be replaced by { A k } ~ - , by setting A-P = 0, V p > 0. DEFINITION 8.4.1. A sequence

of module homomorphisms is said to be exact at B if im(a) = ker(P). If a sequence of module homomorphisms is exact at each module (except the first and last), it is called an exact sequence. An exact sequence of the form

is called a short exact sequence. Similarly, the notions of exact sequence and of short exact sequence are defined for graded module homomorphisms. Remark that, in the short exact sequence, i is injective and j is surjective. LEMMA8.4.2 (THEFIVE a

LEMMA).

a

A - B - C - D - E

Let Y

5


250

be a commutative diagram in which the two rows are exact. I f / \ , p, p, and n are isomorphisms, then v is a n isomorphism. The proof is a mechanical diagram chase (Exercise (1)).

DEFINITION8.4.3. A cochain complex (A*,S) is a graded R-module, together with a sequence 6 ~ p ++ . . . -,6 Ap + l6 ~

p +! +

. . ~.

such that S2 = 0. Similarly, a chain complex (C,, a) is a graded R-module and a sequence

a cp a - . ---t

cp-la

-3

CP-2

a

+"'

such that d2 = 0.

As usual, one defines ZP = ker(S) n AP (respectively, Zp = ker(d) n Cp) and = im(S) n AP (respectively, Bp = im(d) n C,). In what follows, we explicitly consider cochain complexes, but everything goes through, with the obvious modifications, for chain complexes. In this book, we are mainly interested in the de Rham cochain complexes (A* (M),d) and (A: (M), d) and in the singular chain complex (C, (M) ,d) , although others will be mentioned on occasion. The condition that S2 = 0 implies that B* C Z* and the cohomology of the cochain complex is defined to be B p

a graded R-module. This can be viewed as a measure of the extent to which the sequence in Definition 8.4.3 fails to be exact. The corresponding construction for a chain complex (C,, d) is called the homology of the complex and denoted by H* (C*,a)-

DEFINITION 8.4.4. A homomorphism cp : (A*, S) 4 (C*,6 ) of cochain complexes is a homomorphism of the graded R-modules such that cp o S = 6 o cp.

-

Evidently, a homomorphism cp : (A*, S) (C*,S) of cochain complexes induces a homomorphism cp* : H*(A*,S) -,H*(C*,6) of graded R-modules.

DEFINITION 8.4.5. A homomorphism X p E Z is a sequence of R-linear maps

-00

:

H*(A*,S)

-+

H*(C*,6 ) of degree

< k < oo. This is sometimes written X : H*(A*,S) -+H*+p(C*,6).

For instance, in the Poincarb Lemma for compactly supported cohomology, we defined a homomorphism

of degree - 1.

8.4. EXACT SEQUENCES

L E M M A8.4.6. Let

be a short exact sequence of homomorphisms of cochain complexes. Then, there is canonically induced a homomorphism

S* : H* ( E * ,S )

4

H*+'

(c*, S)

of degree +1, called the connecting homomorphism. This homomorphism is "natural" in the following sense: i f

is a wmmutative diagram with both rows exact, then H* ( E * ,S )

6'

H*+'

(c*, 6)

also commutes. In the case of a short exact sequence of chain complexes, the connecting homomorphism has degree -1. We show how to find S* [el E H k + l ( C * ,S ) , where [el E H k ( E * ,6). Consider the commutative diagram

and choose e E E~ representing [el. In particular, S(e) = 0. Since j is surjective, choose et E Dk such that j ( e f ) = e. Then j ( 6 ( e f ) ) = S(j(e')) = 6(e) = 0 and exactness of the middle row implies that there is a unique c E C such that i(c) = S(el). Then i ( S ( c ) )= S(i(c)) = S(S(el))= 0. Since i is one to one, it follows that b(c) = 0, so we define S* [el = [c] E H ~ + ' ( C *6). , More diagram chasing proves that [c]is independent of the choices of e E [el and of e t E D k such that j ( e t ) = e. The "naturality" of S * , as defined in the statement of Lemma 8.4.6, is proven by yet another diagram chase (Exercise (2)).


252

-

PROPOSITION 8.4.7. A short exact sequence 0

( C * ,6 ) 5 ( D * ,6 ) -2. ( E * ,6 ) +0

of cochain complexes induces a long exact sequence

i n cohornology. The proof is a diagra,-n chase (Exercise (3)). One sometimes writes the long exact sequence more compactly as an exact triangle:

H* (C*,6 )

i*

A/

H * ( D * ,6 )

H* ( E * ,8 )

EXERCISES (I) Prove the Five Lemma. (2) Prove that the connecting homomorphism 6* of Lemma 8.4.6 is naturd. (3) Prove Lemma 8.4.7. (4) Let R be a field. If

A

~

B ~ C is an exact sequence of vector spaces over R, prove that the dual sequence

- i*

B* j C*, where i* and j* are the respective adjoints, is also exact. Find an example showing that this may fail for modules over a commutative ring.

A*

8.5. Mayer-Vietoris Sequences

Let U1 and Uz be open subsets of the n-manifold M and consider the inclusions

jl: Ul n U2 j2 : Ul n Uz

--

Ul U2

and

LEMMA8.5.1. The above inclusions give rise to a short exact sequence 0 -+ (A*(Ul U U 2 ) ,d ) 5 ( A * ( U l )d A*(Uz),d @ d ) f (A*(Ul ilU2),d )

-+

0

of cochain complexes, where i ( w ) = ( i ;( w ) ,ia ( w ) ) , V w E A* (U1 U U z ) , and j ( w 1 , w ~= ) jT(w1) - j;(w2), Vwt E A*(Ue), = 1,2.

8.5. MAYER-VIETORIS SEQUENCES

253

PROOF. Indeed, a nontrivial form on U1UU2 must be nontrivial on either U1 or U2, SO i is one to one. Since j! 0 2 ; = jzoi;, it is clear that im(i) C ker(j). For the reverse inclusion, let (wl, wz) E ker(j). Then wl J (Ul nU2) = w2I (Ul f7U2),SO these forms fit together smoothly to define a form w on U1 U U2 and (wl, up) = i(w). Finally we must prove that j is surjective. Let w be a form on U1 n U2. Let {XI, X2) be a partition of unity on U1 U U2 subordinate to {U1, Uz) and set wl = X2w, wp = X1w. (Note that, since X2 is supported in U2, X2w extends smoothly by 0 to all of U1. Similarly, Xlw is a form on U2.) Then, j ( w l , -w2) = W l + W 2 = w. 0

THEOREM 8.5.2. There is a long exact sequence

called the Mayer- Vietorzs sequence. Indeed, the cohomology of the cochain complex (A * (Ul) @ A*(Uz),d @ d) is clearly H *(Ul) @ H*(U2),so we apply Proposition 8.4.7. We turn to the Mayer-Vietoris sequence for compactly supported cohomology. Again, Ul and U2 are open subsets of some n-manifold M. Clearly, there are inclusions a e : A: (Ul n U2) --r A:(Ut), e = 1,2, and Pe : A: (Ut) A: (Ul U Uz), e = 1,2. It is evident that these inclusions commute with exterior differentiation, hence induce linear maps a;, fl; in compact cohomology, 4? = 1,2.

-

LEMMA8.5.3. The above inclusions induce a short exact sequence

of cochain complexes, where a ( w ) = (a1(w), - a 2 (w)), V w E A: (Ul n Uz), and /3(wl,wz) = PI(WI) PZ(WZ), vut E A;(u~), e = 1,2.

+

PROOF.Everything is clear except, perhaps, the fact that /3 is a surjection. If w is a compactly supported form on U1 U U2 and {Xi, X2) is a partition of unity on Ul U Uz subordinate to {U1, U2), then Alw has compact support in U1 and X2w has compact support in U2 (note the switch from the proof of Lemma 8.5.1). Then, /3(Xlw,X2w) = w and /3 is surjective. REMARK.We have chosen the signs differently than in Lemma 8.5.1. This is not necessary for our present needs, but will be useful in our treatment of Poincark duality. THEOREM 8.5.4. There is a long exact sequence

called the Mayer- Vietoris sequence for compactly supported cohomology.


254

DEFINITION 8.5.5. Let M be an n-manifold without boundary. An open cover {Ua)aEQ of M is said to be simple if it is locally finite and every nonempty, finite intersection U = U, n U,, n . . +n U, is contractible and has H,*(U) = Hz (Rn). By Theorem 8.2.18 and Theorem 8.3.7, simple covers have the following property. LEMMA8.5.6. If U is a simple cover of M and U is any nonempty, finite intersection of elements of 24, then H*(U) = H*(Rn) H, (U) = H , (Rn). THEOREM 8.5.7. If M is a manifold with d M = 8, then every open cover of M admits a simple refinement. We will postpone the proof of this theorem to Section 10.5, since it requires methods from Riemannian geometry. The idea is to produce a locally finite refinement by geodesically convex open sets and to prove that a geodesically convex open set U has the property in Definition 8.5.5. Since finite, nonempty intersections of geodesically convex sets are geodesically convex, we obtain a simple refinement. Using Theorem 8.5.7 and Mayer-Vietoris sequences, we obtain the following interesting result. THEOREM 8.5.8. If M admits a finite simple cover, then H *(M) and H,*(M) are finite dimensional. I n particular, if M is a compact manifold without boundary, then H*( M ) is finite dimensional. PROOF.Select a finite simple cover {Ui)T=l of M. We proceed by induction on r . If r = 1, then M = U1 has the ordinary and compact cohomology of R n. Thus, H*(Ul) = H*(singleton), hence is finite dimensional. For H,*(Ul), the assertion is given by Corollary 8.3.16. Suppose, then, that it has been shown that H,*(N) and H*(N) are finite dimensional whenever N has a simple cover by r - 1 elements, some r _> 2. Let M have the simple cover {Ui)F=l, let U = Ui and remark that {Ul n Ur, Uz n Ur, . . . , U,.-l n Ur) is a simple cover of U n U,.. We consider the compactly supported case. By the inductive hypothesis, H,*(U) and H,*(U n U,) are both finite dimensional as, of course, is H,*(U,).Since M = U UUr , the Mayer-Vietoris sequence gives an exact sequence

:u::

Hf (U) @ Hf (Ur) Z Hz (M)

d*

-+

H,'" (U n u,.).

By standard linear algebra,

Since p* has finite dimensional domain and d * has finite dimensional range, the assertion for H,*(M) follows. The proof for ordinary cohomology uses the appropriate Mayer-Vietoris sequence in the same way.

8.5. MAYER-VIETORIS SEQUENCES

255

REMARK. Even if the compact manifold M has boundary, it is true that H*(M) is finite dimensional. One way to prove this is to show that int(M) has a finite simple cover and that M and int(M) are homotopically equivalent. There is also a Mayer-Vietoris sequence for singular homology. The proof is similar to those for cohomology except for one technical point, the proof of which is very tedious and would take us too far afield. Since we will need this sequence for the proof of the de Rham theorem, we derive it here, referring the reader to standard references in algebraic topology for the bothersome technicality. The inclusions

and

induce an exact sequence

where j(c) = (jl#(c), -j 2 # (c)) and i(c1, cz) = il#(cl) +i2# (CZ)(and the induced homomorphisms il#, jl#, etc., are as in Exercise (4) on page 242). The exactness is immediate. If (*) were a short exact sequence, the Mayer-Vietoris sequence for singular homology would follow immediately, but it is generally false that i is a surjection. This brings us to the technical point.

DEFINITION 8.5.9.Let U = {Ua)aEQbe an open cover of the manifold M. A singular psimplex s : Ap -+ M is said to be Usmall if, for some a E I#, ( A ) U The set of U-small singular psimplices is denoted by A: (M). The vector subspace of Cp(M) spanned by A:(M) is denoted by C;(M). By the definition of the singular boundary operator, it is immediate that

DEFINITION 8.5.10. The chain complex ( C ~ ( M )a) , is called the complex of U-small chains. The homology of this complex is H ~ ( M ) the , Usmall homology of M. It is clear that the natural inclusion of the space of Usmall chains into the space of all chains is a homomorphism of chain complexes

so there is induced a canonical homomorphism in homology

We arrive at the technical result.


256

PROPOSITION 8.5.11. The homomorphism zy is a canonical isomorphism

Proofs of Proposition 8.5.11 will be found in the standard references in algebraic topology, such as f12, pp. 85-88] and [43, pp. 207-2081. The idea is to subdivide the singular simplices in each cycle z until all simplices in the subdivision are U-small. With appropriate choices of signs, there results a U-small cycle z' with [x'] = [ z ] . Thus, homology can be computed using Usmall chains, for any open cover U of M. In our situation, 24 = {U1, U2) is an open cover of the manifold U1U Uz and we replace the sequence (*) with the short exact sequence

THEOREM 8.5.12. There is a long exact sequence

called the Mayer-Vietoris homology sequence. Using the result of Exercise (4) on page 252, we obtain dual Mayer-Vietoris sequences. THEOREM 8.5.13. The Mayer- Vietoris sequences dualize to exact sequences

j'

- - .-, Hp(Ul)*@ Hp(U2)*

Hp(Ul U U2)*

where 'i is the adjoint of i* (respectively, of i,),

at

Hp-l(Ul n U2)*

...

etc.

EXERCISES (1) If the manifold M is connected, but not necessarily compact, prove that the real vector spaces H * (M) and H , (M) have dimension at most countably infinite. You may use the de Rbam theorem. (2) Prove that

H k ( s n )= for all n 2 1.

R, Ic = 0 , n 0, otherwise,

8.6. COMPUTATIONS OF COHOMOLOGY

257

8.6. C o m p u t a t i o n s of Cohomology

In this section, we compute the top dimensional cohomology of connected manifolds.

LEMMA8.6.1. If the open subsets U1, U2, and Ul n U2 of an n-manifold M all have the same compact cohomology as Rn and are coherently oriented, then

is an isomorphism.

PROOF.Consider the diagram

where A(t) = (t, -t) and S(s, t) = s+t, V s, t E R. Commutativity of the diagram is obvious (coherency of orientations is essential), exactness of the bottom row is obvious, and the top row is exact by Theorem 8.5.4. The map JuInu2 is an isomorphism by the hypothesis that H," (Ul n U2) = R (as in Corollary 8.3.17). Similarly, JU, @ Ju2 is an isomorphism. Since the diagram can be extended harmlessly by a commutative square of 0's on the right, it follows from the Five Lemma that Jul uu2 is an isomorphism.

LEMMA8.6.2. Let {Ua)aEQ be a simple open cover of a connected, oriented n-manifold M without boundary, each U, being oriented coherently with the orientation of M . Let w,, wp E A: ( M ) have respective supports in U, and Up, . Then [w,] = [wp] if and only if J, w, = JM wp. a,

PROOF.If [wo] = [wp], then we know that w, = JM wp. For the converse, assume equality of the integrals. Since M is connected, we can find a sequence of indices a = a o , a l , . . . , a , = ,f3 in 24 such that U,,-, n U,, # 8, 1 5 i 5 r. Choose wai E AZ(M) such that supp(w,,) c U,,, 1 5 i 5 r, and such that Wao = W, War = wp, and

This is clearly possible. By Lemma 8.6.1,

JUai-

UUa,

: HP(U,,-,

U U,,)

-+ R

is

an isomorphism, 1 5 i 5 r, so [ L J , ~ - ~ ] = [w,,] E H,"(Uai-, U U,,). That is, there U Uai) such that woi = wai-, dq. These forms all is a form q E A:-l(Uai-, live in A,*(M), so [w,,] = [ w , ~ - ~ ]E H,"(M), 1 5 i 5 r. In particular, [w,] = [wp] as desired.

+


258

LEMMA8.6.3. If (Ua)acO is a simple cover of the connected, oriented nmanifold M with a M = 0, then, for each a0 E 8, the natural inclusion e : A;(U,,)

-

A! ( M )

induces an isomorphism

Sum,

Indeed, since : H,"(Uao) + W is an isomorphism, [w] E H,"(Ua0) is nontrivial if and only if JM w = J w # 0. Since JM vanishes on B:(M), it uao follows that e*[w] # 0,so e* is injective. We prove surjectivity. Let w E A:(M) and use a partition of unity subordinate to the simple cover, to write PROOF.

where A,, w E A: (U,, ), 1 5 i

< r. Choose wi E A:(U,,)

so that

Let r

and remark that, by the above lemma, [w,] = [X,,w] E HF (M), 1 5 i 5 r. Thus, as classes in H,72(M) ,

=

[w].

That is, viewing [Z] E HP(Uao) and [w] E H,"(M), we have proven that e*[S]= [ w ] , SO e* is surjective.

THEOREM 8.6.4. If M is a connected, oriented n-manifold without boundary, then the linear map

is an isomorphism.

8.6. COMPUTATIONS O F COHOMOLOGY

259

PROOF. Fix a simple cover and let U be an element of that cover. Consider the commutative diagram

Since e* and Ju are isomorphisms, so is JM. COROLLARY 8.6.5. If M is a compact, connected, oriented n-man2fold without boundary, then JM : H n ( M ) + R

is an isomorphism. THEOREM 8.6.6. If M is a connected, nonorientable n-manifold with empty boundary, then H,"(M) = 0. I n particular, if M is also compact, H n ( M ) = 0.

A:-'

PROOF.Let w E AF(M). We must show that w = dB for suitable B E (M). Choose a simple cover {U,),Ea. By a partition of unity argument,

write w as a finite sum of forms, each compactly supported in one or another element of the cover. If each of these is the exterior derivative of a compactly supported form, we are done. Thus, without loss of generality, we assume that supp(w) c U,, . By nonorientability of M , there is a sequence U,, ,U,, , . . . ,U,, of elements of the simple cover and orientations p, of Uai, 0 5 i 5 r, with the following properties: (1) u*,-l nu,, # 0, 15 5 r; (2) pi-1 and pi restrict to the same orientation of Uai-, n Uai, 1 5 i 5 r; pr = -Po. (3) U,, = Ua0 Choose forms wi E A:(U,,), 0 5 i 5 r, such that w = wo and

15 i

5 r . Thus, wi = wi-1

+ d~i-1, -

That is, wr = wo

On the other hand,

implying that

1 €

+d

UU

~ ,q € A~-'(M).

) ,I 5 i 5 r.

260


Combining these equations, we conclude that w = wo = d8 for suitable 8 E A;-'(M). o If M is compact, any simple cover, being locally finite, is finite. If M is noncompact, it may or may not admit finite simple covers, but it always admits infinite ones. The proof of the following is carried out by the same geometric arguments that prove the existence of simple covers in general.

LEMMA8.6.7. If M is noncompact and connected, d M = 8, then there is a countably infinite simple cover {Ui)El such that Ui n Ui+' # 0, 1 5 i < 00. EXERCISES (1) Use Lemma 8.6.7 to prove that, if M is a noncompact, connected, boundaryless n-manifold (orientable or not), then H (M) = 0. (2) If M is an n-manifold with d M compact and nonempty, prove that M and int(M) are homotopically equivalent. If M is connected, conclude that H n ( M ) = 0. (Hint: There is a compactly supported vector field on M, pointing inward and nowhere 0 along d M . This generates a "half flow", parametrized on [0,GO) and stationary outside of a neighborhood of dM.) 8.7. Degree Theory

Let M and N be connected, oriented n-manifolds without boundary and f : M -+ N a proper map. Let y E N be a regular value of f . Then f -'(y) is compact and discrete, hence finite. Set f -l(y) = {yl, . . . ,y,) and let E.j

=

+1 if f,,i preserves orientation, - 1 if f,,, reverses orientation,

DEFINITION 8.7.1. With the above conventions, deg,( f) = called the local degree of f at the regular value y.

xP=,

ri. This is

The canonical isomorphisms JM and allow US to identify H,"(M) and H,"(N) with R unambiguously. This identification is understood in the following discussion.

PROPOSITION 8.7.2. If f : M 4 N is proper, where M and N are connected, oriented n-manifolds without boundary, and if y E N is a regular value, then f * : H," (N) + H," (M) is multiplication by degl (f). PROOF.By Theorem 3.9.1, there is an open, connected neighborhood U C N of y such that

f-'(u)

=

u, U - . . u u,,

a union of disjoint open sets such that y i E Ui and f carries Ui diffeomorphically onto U ,I 5 i 5 q. If [w] E HP(N), we can choose a representative n-form w so that supp(w) is a compact subset of U. Then, wi = f * (w)lUi is compactly

8.7. DEGREE THEORY

supported in Ui,1 5 i 5 q, and f : Ui-+ according as c i = 1 or - 1. Thus,

It follows that f * : H,"(N)

-+

26 1

U preserves or reverses orientation

H,"(M) is multiplication by deg,(f).

REMARK. This has several obvious consequences: (1) f * : H,"(N) -+ HF(M) is multiplication by an integer. (2) deg, ( f ) = deg(f ) (the degree of f ) is independent of y. (3) deg(f ) is a proper homotopy invariant of f . (4) If M is compact, deg(f ) is a homotopy invariant of f . (5) deg(f g ) = deg(f deg(g)THEOREM 8.7.3. Let W be an oriented (n + 1)-manifold with nonempty, connected boundary. Let N be a connected, oriented n-manzfold without boundary and let f : dW -+N be proper. Iff extends to a proper map F : W -+ N, then deg(f) = 0. PROOF.Suppose that f extends to a proper map F : W -+ N. If w E A:(N), then F*(w) E A:(W) and d(F*(w)) = F*(dw) = 0, since dw E A:+'(N) = 0. Thus, by Stokes theorem,

By Proposition 8.7.2, it follows that deg(f) = 0. This theorem partially generalizes Theorem 6.4.10. Let Sn C lRn+' be the unit sphere and let a, : Sn 4 Sn denote the antipodal interchange map

PROOF.Since an is a diffeomorphism, every x E Sn is a regular value and has pre-image a singleton. Thus, the question reduces to whether a n preserves or reverses orientation. The linear extension A : lRn+l -+ Rn+l of a, is represented by the matrix -In+1 with determinant (- 1)n+l. This transformation, therefore, is orientation-preserving if and only if n is odd. The restriction of the transformation to the unit ball Dn+l is orientation-preserving if and only if n is


262

odd. But Sn = L3Dn+l has orientation induced by the orientation of D n , hence AISn = an : Sn -, Sn is orientation-preserving if and only if n is odd. THEOREM 8.7.5. The sphere Sn has a nowhere zero tangent vector field if and only if n is odd. PROOF. If n is odd, you constructed such a vector field in Exercise (2) on page 110. Suppose, therefore, that s : Sn -,Rn+' \ (0) is smooth with s(u) 1.v, V v E Sn. Equivalently, s is a nowhere zero section of T(Sn). Note that v cos 8

+ s(v) sin 9 # 0,

V 8 E R, so we can define a smooth map

by F(v, t) =

v cos t.lr + s(u) sin t.lr Ilv cos t.lr + s(u) sint.lrl1'

Then,

so an

-

F(v, 0) = v F ( u , 1) = -v

} vu,

sn,

id and 1 = deg(crn) = (-l)n+l, implying that n is odd.

COROLLARY 8.7.6. Every smooth flow on S2nhas at least one stationary point. For the sphere S2,Theorem 8.7.5 and its corollary are sometimes stated facetiously as "you can't comb the hair on a billiard ball". (People whose sensibilities .) are offended by hairy billiard balls substitute LLcoconut" THEOREM8.7.7. If f : Sn -, Sn is smooth and deg(f) has a jixed point.

# (-I)"+',

then f

PROOF.In fact, we will prove that, if f has no fixed point, then f N a,, hence deg(f ) = (- 1)n+l. We claim that t(f (v) v) # u, Vv E Sn,0 5 t 1. Otherwise, t # 0 and

+

Given [w] E H1(M), select a representative w E [w]. By simplicity, H1(U,) = 0, so the restriction w, = wlU, of the closed 1-form w is exact, V a E %. Thus, we can choose fa E AO(U,) such that w, = df,, V a E a. On U,, n U,, # 0, d(fao - f a l ) = w - w 0, so fao - fal is locally constant on U,, n U,,. The cover being simple, this set is connected, so fa, - f,, = ,c E R is a constant. This defines a Cech lbcochain c E ~ ' ( 2 - 4 ) .But

on U,, fl U,, n U,, , so c E 2' (U). If [c] E we can set

g1(2.4) depends only on [w] E

H' (M),

cp([wl) = lcl. LEMMA8.9.9. The class [c] defined above is independent of the choice of representative w E [w] and of the choices of f, E Ao(U,) such that df, = w,. Consequently, cp is a well defined linear map. The proof of Lemma 8.9.9 will be Exercise (2). We define a linear map

For this, we will need to fix the choice of a smooth partition of unity subordinate to U. Given [c] E H'(u), choose a representative cocycle c E [el. For each a0 E a, define f,, E A0(u,,) by

Then, on U,, fl U,, # 0,

2 74


It follows that df,, = df,, on U,, n U,, , so these exact forms assemble to give a well defined locally exact 1-form w f Z1(M). If [w] E H' (M) depends only on [c] E H' (u),we can set $([el) = [w].

LEMMA8.9.10. The class [w] defined above is independent of the choice of representative c E [c]. Consequently, $ is a well defined linear map. The proof of Lemma 8.9.10 will be Exercise (3). LEMMA8.9.11. The homomorphisms cp and $ are mutually inverse.

PROOF.Given [c] E H'(u), the definition of $([c]) = [w] produces functions f, E A0(u,)such that wlU, = df,, V a E 2l. Using this choice in the definition of cp([w]) gives back the representative cocycle c. That is, cpo$ = id. For the reverse composition, the definition of cp([w]) = [c] selected the functions f , E A1(U,) such that all df, = wl U, and fa, - fa, = c,,,, . The definition of $ produces different functions

where

The closed form 3 obtained by piecing together the exact forms t o w by 3-w=dh,

dj,

is related

so [Z] = [w], proving that $ o cp = id.

In particular, although q!.J was defined relative to a choice of partition of unity, it inverts cp which did not depend on that choice, so q!.J is, in fact, independent of the choice also. We close this section with some remarks about triangulations and cohomology. Let S = {eo,el, . . . ,en) be the set of vertices of the standard n-simplex A,. The convex hull of any subset C C S of cardinality p + 1 is a p-simplex. It lies in the boundary of A, and will be called a p-face of A,. The natural ordering of the indices of the points ei, E C defines a canonical identification of this p-face with the standard p i m p l e x Ap. More precisely, this natural ordering defines a canonical linear imbedding A, r A, with image the given p-face. If s : A, + M is a singular nsimplex, its p-faces are the singular p-simplices obtained by restricting s to the p-faces of A,. Recall from Section 1.3 the fact that compact surfaces can be triangulated. A corresponding theorem for compact, differentiable n-manifolds also holds. That is, the manifold can be divided up into a union of smoothly imbedded nsimplices A:, . . . A:, any two of which either do not meet at all or meet along exactly one common lower dimensional face. This theorem is intuitively plausible, but rather difficult to prove. If A stands for a choice of triangulation of M , we obtain a chain subcomplex (C) (M), a) c (C, (M), a) by using only those singular simplices that are the

8.9. DE RHAM THEOREM

275

inclusion maps of simplices of the triangulation. (By the simplices of the trimgulation, we mean all of the p-faces of the nsimplices of A, 0 < p n.) This is called the simplicial chain complex associated to the triangulation A. Remark ) a finite dimensional vector subspace of Cp(M), 0 2 p 5 n, and that C ~ ( M is vanishes if p > n. Let

0. Let

10. RIEMANNIAN GEOMETRY

304

Then, [hij] = [ ? k i l l

[fkt]

[ytj] and it follows that

Also, det(y')ql A where y' = (y

T

)-I.

- A qn = w1 A

h wn,

Putting this information together, we obtain

Thus, the locally defined volume forms fit together coherently to define R as desired.

DEFINITION 10.2.7. Let (U, x l , . . . , xn) be a coordinate chart with coordinate fields ti= d/axi. Then the functions

are called the metric coefficients. Thus, in correctly oriented local coordinate charts (U, x element is given in terms of the metric coefficients by

',. . . ,xn), the volume

where g = det[gij]

DEFINITION 10.2.8. Let M be an oriented Riemannian n-manifold and let U M be a relatively compact, open subset. Then the volume of U is

REMARK. On the a-algebra of Borel sets of M , this Riemannian volume generates a measure, finite on the relatively compact Borel sets. Even if M is not orientable, R is defined locally up to sign and, for small, connected, open sets U,

This also leads to a Borel measure, finite on relatively compact Borel sets.

DEFINITION 10.2.9. A connection V on the Riemannian manifold M is a Riemannian connection if, for all X, Y,Z E X(M) ,

DEFINITION 10.2.10. A connection V on the Riemannian manifold M is a Levi-Civita connection if it is symmetric and Riemannian.

10+2. RIEMANNIAN MANIFOLDS

305

EXAMPLE10.2.11. Let M Rm be a smoothly imbedded n-manifold. The standard inner product (-, -) on R m , viewed as a Riemannian metric on the tangent bundle T(Rm), restricts to a Riemannian metric (-, -) on T ( M ) C_ T(Rm)IM. By Lemma 10.1.8, the connection V on M , constructed in the previous section and called there the Levi-Civita connection, is symmetric. The Euclidean connection D on R m clearly satisfies X (Y, Z) = (DxY Z)

+ (Y, D x Z) , V X , Y, Z E X(Rm).

If X , Y, Z E X(M), then DxY = VxY everywhere on M. Consequently,

+ W, where W E r(v(M)), hence W IZ

and, similarly, (Y,DxZ) = (Y,VxZ), . Thus, V is Riemannian, hence is a Levi-Civita connection in our new sense as well. In Exercise (I), you will be led through a proof of the following fundamental result. THEOREM10.2.12. A Riemannian manifold M has a unique Levi-Czvita con-

nection. In any local coordinate chart, the matrix [gij] of metric coefficients is nonsingular, so we can define

isk']= [gijl-l. The coefficients g k e are rational functions of the metric coefficients g y . By definition, they satisfy sirsk' = sf. DEFINITION10.2.13. A property of the Riemannian manifold M is intrinsic if it depends only on the metric. Otherwise, the property is extrinsic. It is geometric if it does not depend on choices of local coordinates. For example, the functions g g and sk' are intrinsic, but not geometric. By Exercise (2), the Christoffel symbols I?& for the Levi-Civita connection are also intrinsic, but not geometric. The Levi-Civita connection itself is both intrinsic and geometric. In particular, the Levi-Civita connection for M C R m is intrinsic, even though our initial definition of it used the normal bundle v(M), a structure that can be proven to be extrinsic. Our definition of "intrinsic" may seem a bit too informal by current standards. For a more formal definition, one needs the notion of an isometry. DEFINITION10.2.14. An isometry cp : M I + M2 between two Riemannian manifolds, with respective metrics (., .) i , i = 1,2, is a diffeomorphism such that

for arbitrary x E M I and v, w E Tx(Ml).

306


Now we see that a property of Riemannian manifolds should be called intrinsic if and only if it is preserved by all isometries.

EXERCISES (1) Given a Riemannian manifold M , proceed as follows to show that it has a unique Levi-Civita connection. ( a ) Uniqueness. Show that a Levi-Civita connection V must satisfy the identity 2 (VxY, 2) = X (Y, 2 )

+ Y (X, 2 ) - Z (X, Y)

+ I[X,YI, 2) + ([Z,XI, Y )+ ([Z,yl, X )

7

VX, Y, Z E X(M). (b) Ezistence. Use the identity in (a) to define

and prove that V is a Levi-Civita connection. (2) Let I'h denote the Christoffel symbols of the Levi-Civita connection and find a formula for r&that involves only the gk"s and first derivatives of the gij's. (3) If f : M + N is a diffeomorphism and V is a connection on N, show how to define the pull-back connection f *V on M . If f is an isometry between Riemannian manifolds and V is the Levi-Civita connection on N, prove that f * V is the Levi-Civita connection on M. This is the formal proof that the Levi-Civita connection is an intrinsic property of a Riemannian manifold. (4) Let v, w E X(s) . Show that the covariant derivative defined by the LeviCivita connection (indeed, by any Riemannian connection) satisfies

(It follows from this exercise that fields parallel along a curve s, relative to a Levi-Civita connection, make a constant angle with each other and have constant lengths along s.)

10.3. Gauss Curvature Throughout this section, unless otherwise stated, we assume that dim M = 2, that a M = 0, and that we are given a fixed imbedding M L, IR3 with normal bundle v(M). As usual, we use the metric induced on M by the Euclidean metric of IR3 and the associated Levi-Civita connection V. The Euclidean connection will be denoted by D. M of x and a smooth Given x E M, we find a connected neighborhood U section 6 E r ( v ( M ) IU)such that Il6ll E 1. Remark that 6 is determined up to sign. Remark also that one can interpret n' as a smooth map

10.3. GAUSS CURVATURE

DEFINITION 10.3.1. The map n' : U

+ S2is

called the Gauss map.

FIGURE 10.2. The flat plane

FIGURE 10.3. The flat cylinder Intuitively, the area of 6(U) & S2 seems to have something t o do with the curvature of M in U. Thus, if U is an open subset of a 2-plane in It3, Z(U) degenerates to a single point, hence has area 0 (Figure 10.2). We say that the plane has (Gauss) curvature 0. Similarly, if U lies in a right circular cylinder, n'(U) will lie in a great circle in S2 (Figure 10.3). Again the area of d(U) is 0 and we say that the cylinder has (Gauss) curvature 0. The point here is that the cylinder can be "unrolled" to a portion of a plane without any metric then n' = fid and the area of n'(U) is distortions. On the other hand, if U C s2, the same as the area of U, this being a positive number. We say that the sphere has positive (Gauss) curvature. It is not possible to flatten out any portion of S2 to be planar without distorting the metric properties. For similar reasons, every convex surface has nonnegative curvature everywhere (Figure 10.4). By introducing a notion of "signed area7', one obtains cases of negative curvature, a saddle shaped surface being the typical example (Figure 10.5 and Exercise (3)). In order to put these ideas into precise form, we introduce the Weingarten map.


t

FIGURE 10.4. Nonnegative curvature

FIGURE10.5. Negative curvature If v E Ty(M),some y E

U ,then Dun' E IR3 makes sense. The equation

0 = v (n',n') = (D,n',n')

+ (n', DUG)= 2 (Dun',5)

proves that DUG1n', hence Dun' E Ty(M). DEFINITION 10.3.2. The linear map L : T,(M)

-,Ty(M), defined

by

L ( v ) = Dun', is called the Weingarten map. This map is well defined up to sign. As soon as the sign of the Gauss map n' has been fixed, the sign of the Weingarten map is determined also. Remark that T, ( M ) = T6(,)(S2)in IR3, since these 2-planes have the common normal vector n'(y). Thus, we are allowed to view the Weingarten map as


LEMMA10.3.3. The Weingarten map L : T,(M) tial at y of the Gauss map.

309

+

T5(,)(s2) is the difleren-

PROOF.Represent an arbitrary vector in T , (M) as s(0) for a suitable arc s : ( - E , E ) 4 M. Then

LEMMA10.3.4. The Weingarten map is self adjoint. That is,

PROOF.Let v,w E T,(M) and, by making the open set U & M smaller, if necessary, extend these vectors to fields X, Y E E(U).Then (L(X)l,)'

= ( D ~ z l Y, = X (5,Y) -

(n', D x Y )

= - (n', DxY)

+

= - (5,[X7Y] D y X ) = - (G, DyX)

Thus, relative to any choice of orthonormal basis in Ty(M), the matrix of L is symmetric. By the diagonalization theorem for symmetric matrices, there is an orthonormal basis { e l ,e2) C T,(M) relative to which the matrix for L is

That is, ei is an eigenvector for L with eigenvalue number these so that 61 L ~ 2 .

~

i i ,=

1,2. We agree to

LEMMA10.3.5. As v ranges over the unit circle in T,(M), the quadratic form (L(v), v) takes minimum value ~1 and maximum value ~ 2 . Indeed, if

,

is the matrix of a quadratic form Q on EX2, it is standard that the extreme values of Q (v) on S1 are the eigenvalues of the matrix (the method of Lagrange multipliers). Remark that, for the eigenvectors e , , tci = (Vein',e i ) measures the rate at which the normal vector G is turning in the direction e i. This motivates the following definition.


310

DEFINITION 10.3.6. The numbers 61 and 1c2 are called the principal curvai tures of M at y. The product K I K ~= det L is called the Gauss curvature of M at y and is denoted by K (or ~ ( y ) ) . REMARK.There is another important kind of curvature, the m e a n curvature h of M, which we will not treat in any detail. It is defined by

Unlike the Gauss curvature, this quantity depends, up to sign, on the choice of the unit normal field v. It can be shown that surfaces of mean curvature h z 0 are exactly the ones that, in a certain precise sense, locally minimize surface area. Such surfaces are called minimal surfaces. They arise, for instance, when one considers the possible shapes of soap films spanning wire loops of various configurations. Such a soap film will be modeled by a 2-manifold S with boundary and M = S \ dS will be a minimal surface. Until recently, soap films were the main examples, but the work of Hoffman and Meeks ([17], [18], [19], et 61.)) inspired and illuminated by some spectacular computer graphics, has revealed an astounding array of complete, unbounded minimal surfaces. Let R' denote the Riemann volume form on S2 and let fl denote the volume form on M. Define the "signed area" of the Gauss map on U to be

and, a s usual, let the area of U be

As U shrinks down on { y ) , one can try to form the "Radon-Nikodym derivative" of the Gauss mar, Z:

This number, if it exists, is characterized as the unique number such that, for each E > 0, there is 6 > 0 for which

U

3 y and

IA(U)I < S +

< E,

The proof of the following will be an exercise. PROPOSITION 10.3.7. For each y E M , the above Radon-Nikodym derivative exists and In the above discussion, the normal field n' played a central role. The curvature of M was seen as a measure of how much this field "spreads" infinitesimally at a point. Thus, curvature appears to be an extrinsic property of the surface. But Gauss proved a remarkable theorem (he c d e d it his "Theorema Egregium") that showed the Gauss curvature to be intrinsic. A two-dimensional inhabitant


311

of the surface can take measurements leading to the computation of curvature. We turn to this theorem. - - Let X, Y, Z E X(U) and extend these to fields X , Y, Z E X(I?), where 6 & IR3 is an open set such that 6n M = U.

LEMMA10.3.8. For fields chosen as above, along U. PROOF.Along U, Lemma 10.3.4,

D,-Y

depends only on X and Y. As in the proof of

( D x K A) = - (L(X), Y) , so the component of Dx Y perpendicular to M is - (L (X), Y) A. By the definition of the Levi-Civita connection, VxY is the component of DxY tangent to M. The Euclidean connection D satisfies a simple commutator relation:

This is because D,- and DT operate on a vector field 2 by applying 2 and B respectively to the individual components of 2. It turns out that, on M , curvature is an obstruction to this commutator relation for V. DEFINITION 10.3.9. The curvature operator

is defined, for arbitrary X, Y E X(M), by

The fact that this operator is related to curvature is far from obvious. It is the content of the Theorema Egregium. By Lemma 10.3.8, at every point of U we have

Similarly, at every point of U, -

D ~ ( D ~=Z-vY(vXz) ) + (L(x),

Z) L(Y)

+ Y (L(X), Z) 6 + (L(Y),V x Z ) n'.

Findy, at every point of U,

- ~ ~= - V~[ X ,, Y +~~~(L([X, ~ Y]),z z)6.

312


The sum of left sides of these equations being 0, the tangential parts and normal parts of the right sides separately sum to 0. The first of these zero sums gives the Gauss equation:

Summing the coefficients of n' also gives 0. Applying the fact that V is a Riemannian connection and using the fact that Z varies freely over all tangent fields, the reader can obtain the Codazzi-Mainardi equation (10.4)

L([X,Y]) = VxL(Y) - VyL(X).

One immediate consequence of equation (10.3) is the following. LEMMA10.3.10. The expression R(X, Y)Z has value at y E U depending only on the vectors X,, Y,, 2,. Thus, this expression is a tensor, called the Riemann curvature tensor R. Indeed, the right hand side of equation (10.3) is clearly a tensor in all three vector fields. That is, R E T 3 ( M ) . Note also that, since V is an intrinsic and geometric property of the surface, so is the Riemann curvature tensor.

THEOREM 10.3.11 (THEOREMA EGREGIUM). Let y E U and let {e 1, e2) be an orthonormal basis of T,(M). Then

In particular, the Gauss curvature of a surface in IR3 is intrinsic and geometric. PROOF.By equation (10.3), @(el, e2I.2, el) = (L(ea),ea) (L(ei), el) - (L(el),ea) (L(ea),el) . Since the basis is orthonormal, it is true (and easily checked) that the corresponding matrix representation of L is the 2 x 2 matrix with (i,j) th entry (L(ej),et). Thus, (R(el, ez)ez, el) = det L = ~ ( y ) .

Since the Levi-Civita connection is preserved by isometries (Exercise (3) on page 306), we obtain the more formal version of the statement that n is an intrinsic property. COROLLARY 10.3.12. If f : M1 4 M2 is an isometry between two surfaces in IK3 and if I C ~is the Gauss curvature of M i , i = 1,2, then K.~(f (x)) = r;l (x), V X E M1. Theorem 10.3.11 suggests a definition of curvature for a general connection on an n-manifold M .


313

DEFINITION 10.3.13. Let M be an n-manifold and let V be a connection on M. The curvature tensor R of V is given, for each choice of X, Y, Z E X(M), by

If M is a Riemannian manifold and V is the Levi-Civita connection, then R is called the Riemann tensor R. The fact that this is a tensor in all three arguments will be Exercise (2). Consequently, for each x E M and u, v, w E Tz(M), R(u, v)w E Tz(M) is well defined. We will return to the study of this tensor later. It turns out that, in Riemannian geometry, the Riemann tensor is exactly the obstruction to the geometry being locally Euclidean. In the non-Ftiemannian geometry of spacet ime, there is an analogue of the Levi-Civita connection and Einstein represents gravity by the curvature tensor of this connection.

EXERCISES (1) Prove Proposition 10.3.7. (2) Verify that the curvature tensor R(X, Y)Z of a connection V is Cm(M)trilinear in the fields X , Y, Z. (3) Let M C R3 be the graph of the equation r = x 2 - y2. Prove that K(X,y, z ) < 0, V (5,y , r ) E M. Can you explain this negative sign intuitively? (4) Let M C R3 be a compact 2-manifold. You are to prove that it is not possible that K 5 0 on all of M. Proceed as follows. (a) By compactness, choose a point vo E M at which the function X : M + R, defined by X(v) = IJvl12,assumes its maximum. Prove that 0 # vo IT,, (M). (b) If s : (-E, E) + M is smooth with s(0) = vo and i(0) # 0, prove that (s(O),n'(vo)) is strictly negative. (c) Using the above, prove that the principal curvatures ~1 and ~2 are both nonzero and have the same sign at vo, hence ~ ( v >~0.) (5) One calls a point v E M at which K I = ~2 an umbilic point. Let U C M be the set of points that are not umbilic. (a) Prove that U is open in M and that and ~2 are smooth functions on U. (b) Prove that each v E U has a neighborhood V C_ U on which there is a smooth, orthonormal frame field (XI, X2) such that L(Xi) = KiXi, i = l,2. (c) For V C U and Xl , X2 E X(V) as in part (b), define f l , f 2 E CM(V)by


Prove that

and that [Xl,X2] = f l X l - f2X2. (d) Using the formulas in part (c), show that, if v E U is a critical point for both I C ~and 6 2 , then, at v,

(6) Let M C lR3 be a compact, connected 2-manifold with constant curvature IC = a. By Exercise (4), a > 0. You are to prove that M is a 2sphere, centered at some point wo E R3 and of radius l/&. Proceed as follows. (a) Prove that v E M is a point at which I C is ~ maximum if and only if it is a point at which I C ~is minimum. (b) Use part (d) of Exercise (5) to show that I C ~can be maximum only at an umbilic point. (c) Prove that every point of M is an umbilic and that IC 1 & = I C ~ . (Hint: This is the maximum value of K ~ . ) (d) Deduce the form of the Gauss map, drawing the desired conclusion. Be sure to make clear how you use the hypothesis that M is connected.

-

10.4. Complete Riemannian Manifolds

This section presents the Hopf-Know theorem and related matters. The author first learned this materid horn Milnor's beautiful exposition [29, pp. 55-64], and its influence will be evident in what follows. The goal here is to use the Riemannian metric to obtain a topologicd metric on M and to relate the topological notion of "completeness" to the problem of extending geodesics indefinitely. In the process, one also discovers, without the use of variational calculus, that geodesics locally minimize arc length. In Euclidean geometry, straight lines play a central role. They can be characterized as the unique smooth curves whose tangent fields are parallel. Here, parallelism under the Euclidean connection is clearly identical with the absolute parallelism in Euclidean space. Taking our cue from this observation, we define the notion of a geodesic for a general connection.

DEFINITION 10.4.1. Let M be an n-manifold with a connection V. A smooth curve s : [a, b] -+M is a geodesic for V if i(t) is parallel along s(t), a 5 t 5 b. We are interested in the case in which V is the Levi-Civita connection of a Riemannian manifold, so we make that assumption from here on. We emphasize that we are considering general Riemannian manifolds, not just surfaces in R3.

10.4. COMPLETE RIEMANNIAN MANIFOLDS

DEFINITION 10.4.2. A smooth curve o : [a,b] Ilb-(t)ll = C, a 5 t 5 b, for some constant c 2 0.

-, M

315

is evenly parametrized if

By Exercise (4) on page 306, the following is immediate.

LEMMA10.4.3. A geodesic s on a Riemannian manifold M is necessarily evenly parametrized. In local coordinates, the definition of a geodesic translates into a system of nonlinear, second order, ordinary differential equations. Indeed, write

and write down the parallelism condition for d ( t ) :

Equivalently, this is the second order system

By setting ui = xi, 1 5 i 5 n, we get an equivalent, nonlinear, first order system u. k

+

i

j

k rij = 0,

ie=ue,

l S k 5 n l 0 sufficiently small (as usual), let S6, be the spherical shell of radius 6' around ?(to). Let xb E S6t be a point with p(xb, y) minimum. Then,

10.4. COMPLETE RIEMANNIAN MANIFOLDS

as before,

SO

p(xb,y) = r - to - S' = r - (to +St). We will show that xb = y (to + 6'). Indeed,

But the path consisting of the segment of y from x to ?(to), followed by a minimal geodesic from ?(to) to xb, has length to S', hence is a piecewise geodesic, parametrized by arc length, joining x to xb and of minimal length. By Corollary 10.4.21, this path is a geodesic. It coincides with y on [0,to] and to > 0, hence it coincides with y on [0,to + 6'1. That is, xb = y (to 6') , as claimed. We have proven that

+

+

which is the assertion Ft,+61. Since 6' > 0 was arbitrarily small, there is a halfopen interval [to,q'), 77' < r , on which Ft holds. The union of all such intervals produces the maximal one [t0, 7). (c) Since F6 holds, let [ 6 , ~be ) the maximal half-open interval on which Ft holds. But the truth of Ft on ( 6 , ~ implies ) F,, by continuity. Thus, if 17 < r , we could apply (b) to obtain a contradiction to the maximality of [6,7). Consequently, v = r and F, holds.

PROOF O F THEOREM 10.4.13. We first assume that M is geodesically complete and we prove that M , as a metric space under p, is complete. For this, M is a pbounded subset, the it will be enough to prove that, whenever B closure B is compact. Choose any x E B and consider the continuous map

c

exp, : T,(M)

+ M,

defined because M is geodesically complete. Since B is bounded, there is a number r > s u p y c p(x, ~ 9). If D C T,(M) is the closed ball of radius r, then B exp, (D) and exp, (D) is compact. Thus, B exp, (D) is compact. Next, assuming that M is complete in the metric p, we prove that M is geodesically complete. Let x E M and let v E T,(M) have unit norm. Let (a, b) denote the maximal open interval about 0 in R such that exp,(tv) is defined, V t E (a, b). We must show that a = -w and b = w. If b < m, choose { t k ) g l c (a, b) such that tk b strictly. This is a Cauchy te - tk, whenever sequence. Set xk = exp,(tkv) and remark that p(xe7xk) I k < 4?. Indeed, the segment of exp, (t), tk 5 t 5 te, is a geodesic of length te - tk

c

c

326


joining these two points. Therefore, {x k)r=, is Cauchy in the metric p and, by the completeness of this metric, xk -+ y E M. Define y : [0,b] + M by

By the previous paragraph, y is continuous on [0, b]. It is smooth on [0,b). If y is also smooth at b, it is a geodesic and can be extended as a geodesic to [0,b 7)) some q > 0. This would contradict the maximality of (a, b), proving that b = w. In order to prove that y is smooth at b, choose a neighborhood V C M of y and a number E > 0 such that exp,(w) is defined, V z E V, Vw E T,(M) with 11 w 11 < 6. Fbr a Cauchy sequence {xk = exp, (tkv))p==l, chosen as above, xk E V and b - tk < c, for all sufficiently large values of k. Let k be large enough and set vk = j.(tk) E T,, (M). Since llvk11 = 1, exp (tvk) is defined for c,'. 0 5 t 5 b-tk < E. But, for 0 5 t < b-tk, this curve coincides with y ( t k + t ) . By continuity, y(b) = expXi( ( b - tk)vk), completing the proof that y is smooth at b. A completely parallel argument shows that a = -m.

+

EXERCISES (1) Let M C iR3 be a smoothly imbedded surface with the relativized metric and Levi-Civita connection. Let a : [a,b] -, M be evenly parametrized and suppose that im(a) C_ P n M , where P is a 2-plane in IW such that v o ( t ) ( M )c P, a t 5 b. Prove that a is a geodesic. (2) If M is a surface of revolution (as defined in freshman calculus), identify a natural, infinite family of geodesics. Discuss whether or when the "circles of latitude" are geodesics. (3) Show that every geodesic on the unit sphere s2 C IR3 must lie along a great circle. (4) Prove Lemma 10.4.17. (5) Let M be a complete Riemannian manifold, F a foliation of M, and let L be a leaf of F. The Riemannian metric (-, -) on M induces a Riemannian metric (-, -)L on L via the one to one immersion i : L M. Let p~ denote the corresponding topological metric on L. Generally, this is not the restriction of the metric p of M. Let { x k ) ~ c , L be pL-Cauchy. (a) Prove that { x k ) E 1 is p-Cauchy, hence that xk -+ x E M . yl, . . . ,yn) be a F'robenius neighborhood of x. Prove that (b) Let (U, all but finitely many of the points xk lie on the same F-plaque in U as x. (c) Using the above, conclude that, as a Riemannian manifold in the induced metric (-, - ) L , L is complete. In particular, a leaf L in a compact, foliated manifold (M, F) is complete in any metric (., -) that arises, as above, by restricting to L an arbitrary Riemannian metric (., -) on M. The leaf L need not, itself, be compact.

0 such that llvrll = 1 and x = expxo(txvx).Then, F : X x [O,1] --t X , defined by

is the desired contraction. It seems that open, star shaped sets U M are always diffeomorphic to Rn, but this is extremely difficult to prove. The problem is that the set theoretic boundary dU may be very badly behaved. For instance, the "radius function" r : s:,-I

+

[o, 001,

even if it only takes finite vdues, may not be continuous, let alone smooth. This function is defined on the sphere of unit vectors in Txo(M) and assigns to the supremum r (v) of the numbers T > 0 such that exp,, (tv) E U , v E S:,-l 0 < t 5 T . Keep the possible bad behavior of r in mind while formulating a proof of the the following (Exercise (1)).


328

LEMMA10.5.6. Let U M be open and star shaped with respect to xo E U and let C c U be compact. Then there is an open set V C U, also star shaped with respect to xo, such that is a compact subset of M and C c V c c U.

v

PROPOSITION 10.5.7. If U C M is an open set, star shaped with respect to xo, then Hz (U) = HE (Rn). PROOF.For a suitable value po > 0, the open ball Bxo(po)c T,,(M), centered at 0 with radius po, is carried by exp,, diffeomorphically onto an open set Up, c U with compact closure in U. Since Bxo(po)is diffeomorphic to Rn, so U induces is Up,. In particular, Hz (Up,) = H,*(Rn). The inclusion i : Upo homomorphisms

-

so it will be enough to prove that the second of these is an isomorphism. Let C c U be compact. Using Lemma 10.5.6, find open sets V c W c U, star shaped with respect t o xo and such that C W C W C U, and W are U C c V. Also, fix 0 < a < b < po and the corresponding compact, and open balls Ua c Ub c Up,. By the smooth Urysohn lemma, find f : U -r [O, 11 such that

v

upo

v

Let Z E X(U \ (xo)) be nowhere 0, tangent to the radial geodesics out of xo, and everywhere pointing toward xo. Let Ft : U --, U denote the flow, defined for all time t, generated by the compactly supported vector field f Z E X(U). Then, since C \ Upo is contained in the interior of the support of f 2, as is agpo, = F-,. Since F-, is there is a value T > 0 such that F,(C) C Up,. Set a compactly supported diffeomorphism of U onto itself, isotopic through such diffeomorphisms Ft to Fo = idv, it follows that q* : H,*(U) 4 H,*(U) is the identity. Let w E Z,P(U) and let C = supp(w). By the previous paragraph, we obtain : U --, U such that +* (w) E Z,P(Up, ) and +* [w] = [w] E H,P(U). It follows that [w] E im(i,), hence that i, carries Hz(Upo) onto Hz(U). Suppose that w E Z,P(Upo)and that i. [w] = 0. Choose a > 0 as above such that supp(w) C U,. Viewing w = i. (w) in Z,P(U), we find 7 E A!-' (u) such that w = dq. Let C = supp(q) and obtain : U -, U, as above, so that

+

+

+

but dqo = d$* (7) = +*(dq) = +*(w) = W. That is, [w] = 0 in H,'(Upo), proving that i, is one to one.

10.5. GEODESIC CONVEXITY

329

COROLLARY 10.5.8. Every open cover of M admits a simple refinement. PROOF. By Theorem 10.5.4, each open cover admits a refinement by open, geodesically convex sets. With a little care, one chooses this refinement to be locally finite (Exercise (2)). By Lemma 10.5.3, any finite intersection of elements of this refinement is also an open, geodesically convex set, hence star shaped. By Lemma 10.5.5 and Proposition 10.5.7, this refinement is simple. We turn to the proof of Theorem 10.5.4. Let x E W, as in the statement of the theorem. As in Section 10.4, choose a neighborhood V of x in W and a number e > 0 such that any two points y, z E V can be joined by a unique geodesic a , in M of length < E . As usual, a,,, is parametrized on [ O , l ] and depends smoothly on (y, z) E V x V. Choose 6 > 0 such that the open ball B,(6) c T,(M) of radius 6 is carried diffeomorphically by exp, onto a neighborhood U, C V of x. Let (vl, . . . , u,) be an orthonormal frame for T,(M) and coordinatize this vector space by (xl, . . . ,xn) * xiui. Under the diffeomorphism exp;l : U, -+ B,(6), these become coordinates on U, (called a normal coordinate system on U,). The corresponding coordinate fields are ti E X(U,), 1 5 i 5 n. If y E U, has coordinates (bl,. . . ,b,), then

If 0 < 6, < 6, if Ss, C U, is the spherical shell of radius 6,, centered at x, if Y = (bl,. . . , bn) E S6,, and if v = E Ty(Ss,), then

The key lemma for the proof of Theorem 10.5.4 is the following. LEMMA10.5.9. If 6,

E

(0,6) is small enough, then evey geodesic

such that y(0) = y E S g , and T(0) E Ty(S6,), has the property that p(x, y(t)) 6, for all suficiently small values of It1 # 0.

>

PROOF.As above, denote the normal coordinates of y by (b 1, . . . ,b,). Let 6, be so small that, for C:=, b: 5 6,, the symmetric matrix

is so close to [26ke]as to be positive definite. Let be a geodesic in M, tangent to S6,at y(0) = y = (bl,. . . ,b,), and let T(0) = a'&. Define n

F(t) = p(x, y(t))2 - 61 =

~ ' ( t -) ~ 61.

10. FUEMANNIAN GEOMETRY

330

For small values of It 1, this is a smooth function and

Since ~ ( t is) a geodesic, it satisfies

giving = 2((aa)2- aka'(birh(bl,

. . . , b,)))

the value of a positive definite quadratic form on the vector +(O) # 0. That is, Ft'(0) > 0. Plugging this data into the 2nd order Taylor expansion of F(t) about t = 0, we see that t2 F(t) = Ftf(0) O(t3) > 0,

+

for small enough values of It 1 # 0.

PROOFOF THEOREM 10.5.4. Choose N, = exp,(B,(G,)), where 6, is chosen by the above lemma. Let R N , x N, be the subset of all (y, E) such that ay,, lies entirely in N,. By the smooth dependence of this geodesic on its endpoints, R is an open subset. It is also clear that R # 8. If we prove that R is also a closed subset, then, by the connectivity of Nz x N,, R = Nz x N, and Nz is geodesically convex. Let {(gk,z k ) ) E l c R with limk+oc(yk,zk) = (yo, zO) in N, x N,. If (yo,zO)&I R, then ayo ,, meets dNz = Ss,. If a,,,, is tangent to the spherical shell at some point of intersection, an application of the lemma shows that a,,,, contains points in U,\ Nx. But smooth dependence on (yo, ro) implies that this remains true for all values of (y, r) sufficiently near (yo, .to), hence for (yk,rk), k sufficiently large. This contradicts the fact that (yk,zk) E R. But if an intersection point of a,, ,, with the shell is not a point of tangency, it is clear that ox,, contains points in 11, \ leading to the same contradiction. Thus, (xo, yo) E R, proving that R is closed in N , .

,

x,

EXERCISES (1) Prove Lemma 10.5.6. (2) Check the assertion in the proof of Corollary 10.5.8 that the refinement by open, geodesically convex sets can be chosen to be locally finite.

10.6. CARTAN STRUCTURE EQUATIONS

331

(3) Let x E U M where U is open in M and star shaped with respect to x. Let r : S,"-l -, [O,m] be the radius function for U. That is, 27,"-l c Tz(M) is the unit sphere and

U = {exp,(tv) I v E s:-' and 0 < t < r(v)}. If r is finite-valued of class Cm, prove that U is diffeomorphic to Rn. (4) Let x E U C_ M and r : 9,"-' + [0, m ] be as in the preceding exercise, but do not assume that r is smooth or even continuous. Prove that r is lower semicontinuous. That is, r ( a ,m ] is open in SF', V a E W. M and r : S,"-' -+ [0,m ] be as in the preceding exer(5) Let x E U cises. Construct an example in which r is finite-valued everywhere and discontinuous on a dense subset of S,"-'.

-'

10.6. The Cartan Structure Equations

We return to the torsion and curvature tensors that were introduced earlier for a connection V. The key to understanding the geometric significance of these tensors is a pair of equations, written in terms of differential forms, called the equations of structure. One application will be a proof that the Riemann tensor is exactly the obstruction to the integrability of the infinitesimal O(n)structure defined by the Riemannian metric. More, precisely, we will prove

THEOREM 10.6.1. Let M be a Riemannian manifold, V the Levi-Ciwita connection, and R the Riemann tensor. The following are equivalent. (1) R = 0. (2) There is a smooth atlas i n which the metric coeficients are everywhere gij &j. (3) There is a smooth atlas in which the Christoflel symbols are everywhere k G 0. r.. 13 (4) Each x E M has a neighborhood U, such that the holonomy of V around each loop a E S2(Uz,x) is the identity transformation. ( 5 ) The Riemannian metric, as an infinitesimal O(n)-structure, is integrable. A Riemannian manifold in which one, hence all, of these holds is said to be flat. In what follows, V is a general connection on the n-manifold M, n 2 2. To begin with, we will work in an open, trivializing neighborhood U C M for T(M) and fm the trivialization by a choice of a smooth frame (XI, . . . , X,) on U. Define Li' E A1(U) by Zi(Xj) = S;, 1 5 i,j 5 n. Then, each X E X(U) can be written x =Zi(x)x,. Define forms

6; E A1(U), 1 5 i,j 5 n, by

The fact that these are forms, i.e., that 6j(f X) = f r ) j ( ~ ) V, f E Cm(U), follows from the fact that VxXj is a tensor in X .


332

In order to express the torsion and curvature tensors of V in terms of this locd trividization of T ( M ) ,we introduce forms Ti, fij E A2(u) by the formulas

The fact that these are antisymmetric tensors (i.e., 2-forms) follows from the same properties of T and R. THEOREM 10.6.2 (CARTANSTRUCTURE EQUATIONS). The above f o m s satisfy the identities

PROOF. The proof is a computation. We carry this out for equation (10.7). The computation of equation (10.8) is more of the same and will be left as an exercise. For arbitrary X, Y E X(U),

Fi(x, Y)Xi = VXY - vyx - [X, Y] = vX(Z'(Y)X~) -

vY(Z'(X)X~) - ~ ' ( [ x Y])Xj ,

= (x(S~(Y))- Y(S'(X))

- ijj([x, Y ] ) ) x ~

+ s ( y ) v X x j - iZj(x)vYXj &j(x,y ) x j + s ~ ( Y ) B ~ ( x ) x ,- G ~ ( X ) ~ ; ( Y ) X , = (&;(x, Y) + ~ ( Y ) % ( x ) - ij-'(x)6;(y))xi. =

But we claim that

Indeed, the standard inclusion A2(u)

-

G2(U) takes

(Lemma 7.2.17). Thus, the coefficients of Xion each side of the above being equal, 1 5 i 5 n, we obtain

Since X and Y are arbitrary, equation (10.7) follows. 0


333

Using matrix notation, we can write the equations of structure more compactly. Set

The n-tuples 5 and ? can be thought of as Wn-valued forms. The matrices 6 and fi can be thought of as L(Gl(n))-valued forms. The tildes on these symbols are reminders that these forms are dependent on the choice of local trivialization. We will remove the tildes by lifting these local forms to the frame bundle of T(M), where they fit together coherently to define global forms.

DEFINITION 10.6.3. The Wn-valued forms i Z and 7 are called the trivializing coframe field and the torsion form, respectively. The L(Gl(n))-valued forms 8 and

6 are called the connection form and the curvature form, respectively.

The structure equations are

where we multiply matrices of forms by the usual rules of matrix multiplication, but use exterior multiplication for products of entries.

EXAMPLE 10.6.4. When the connection is associated to an infinitesimal Gstructure on M, its connection and curvature forms will be L(G)-valued. The Levi-Civita connections of Riemannian and "pseuddtiemannian" manifolds are important examples. Recall that the group O(k, n - k) c GI(n) consists of the matrices that leave invariant the quadratic form

+ + ( x ~ -) (~x ~ + ' ) ~. .- ( x " ) ~ .

Q ~ ( X '.,. . ,xn) = ( x ~ ) - ~- -

,

Recall from Example 3.4.15, that an infinitesimal O(k, n - k)structure on M is a smooth, nondegenerate, symmetric tensor (., .) E S2(M) of constant index n - k ( 2 . e., n - k is the maximal dimension of a subspace of Tz(M) on which this tensor is negative definite, Vx E M). Such a tensor is called an indefinite metric on M and we say that M, with this metric, is a pseudeRiemannian manifold. A particular example is a Lorentz manifold, a four-dimensional manifold with an indefinite metric of index 1. These are models of space-time in general relativity. Of course, Riemannian metrics are also a special case with k = n. For an indefinite metric, there is a unique associated Levi-Civita connection V (i.e.,

334


the torsion is 0 and X (Y, 2 ) = (VxY, 2 ) + (Y, VxZ)). The proof of this is by exactly the same argument as you used in Exercise (1) on page 306. Given such a metric, choose the locally trivializing frame (XI, . . . ,X,) to be "orthonormal" in the sense that 0, i#j 1, i=j k. This can be done by the Gram-Schmidt process. In Exercise (3), you are asked to prove that the forms and fi, computed relative to this frame, take values in the Lie algebra L(O(k, n - k)). In particular, for the Levi-Civita connection of a Riemannian metric, 6 and fi are L(O(n))-valued. The statement of Threm 10.6.1 will be true, with the obvious modification of parts (2) and (5), for Levi-Civita connections of indefinite metrics (see Exercise (5)), giving several characterizations of flat, pseuddtiemannian manifolds. The Lorentz manifold for special relativity is flat. The equations of structure depend on the choice of local trivializations or "framings" of T(M). In order to remove this dependence on local choices and thereby globalize the structure equations, we will lift them to the principal frame bundle. For the case in which V is a general connection, let p : P + M denote the frame bundle p : F(M) 4 M. If V is the Levi-Civita connection of a Riemannian manifold, we let P = O(M), the bundle of orthonormal frames. In the first case, P is a principal Gl(n)-bundle and, in the second, it is a principal O(n)bundle, namely, the O(n)-reduction of F ( M ) determined by the metric. Indeed, with little extra effort, we can include the case in which V is the Levi-Civita connection of an indefinite metric of index n - k. In this case, the associated reduction P = Ok(M) c F ( M ) is a principal O(k, n - k)-bundle. In order to discuss these cases simultaneously, we w ill denote the respective structure groups, GI(n), O(n), and O(k,n - k), all by G. We will lift equations (10.9) and (10.10), as promised, by finding R "-valued forms w and T on P and L(G)-valued forms 8 and fl on P satisfying

Furthermore, the choice of frame ( X I , . . . ,Xn) on U is a choice of smooth section a :U -,PIU and we will prove that

Thus, the local equations of structure are the a pull-backs of equations (10.11) and (10.12).


Let

C E P, x = p(C), and set P,

= p-'(x).

335

The vertical space at

C E P will

be = Tc(Pz) c Tc(P).

DEFINITION 10.6.5. The vertical subbundle V C T ( P ) is V = elements E E I'(V) are called the vertical fields on P.

UcEPVC. The

Each E E L(G) can be viewed as a vertical field on P. Indeed, E is an n x n matrix and etE is the one parameter subgroup of G generated by E. Using the right action P x G -+ P , we obtain, for each C E P, a curve C - e t E which, when t = 0, is at C. This curve lies in the fiber of P through C, so the corresponding infinitesimal curve is C - E = (C - etE),=, E Q. As varies over P, this defines a vertical field on P. The mapping E H F is a canonical linear injection L(G) -+ I'(V). From now on, we denote F by E, identifying L(G) as a vector subspace of I'(V). M is an open, triviaiizing neighborhood for P, the trivializations If U P(Ur U x G are in one to one correspondence with the sections a E I'(PJU). Indeed, a(x) - B e, (x,B). Fix the choice of a. Since B E G C Gl(n) is a nonsingular n x n matrix and, as remarked above, each E E L(G) is an n x n matrix, the value of the vertical field E at ( = (x, B) E P(Uis C . E = (x, B E ) , where B E is the matrix product. This is because E, as a left invariant vector field on G, has value at B E G given by BE. Thus, this way of viewing a matrix E E L(G) as a vertical field on P is quite analogous to the way that E is viewed as a left invariant field on G. The right action P x G -+ P induces a linear action

via the differential. If X E X(P) and B E G, it will be natural to denote this action by X H X . B. Consider also the automorphism Ad(B) : G -, G, called the adjoint act ion and defined by Ad (B)(A) = B - AB ( cf. Exercise (5) on page 144). The differential Ad(B) : X(G) -,X(G) restricts to an automorphism of L(G) where it will also be called Ad(B) and written Ad(B)(E) = B-'EB. The following is practically immediate.

'

.

-

LEMMA10.6.6. Under the inclusion L(G) I'(V), Ad(B)(E) = E B , VB E G, V E E L(G). I n particular, L(G) c r ( V ) is invariant under the right action of G.

REMARK.Any basis of L(G) c I'(V) gives a trivialization of V. If desired, it is possible to specify a canonical choice of this basis. In the case that G = Gl(n), this will be {Ej}15*,j 0 sufficiently small, Since @ ( x )= x - Q ( x ) ,it follows that

COROLLARY B.4. F o r each y E BVl2,there is a unique x E B,, such that @ ( x )= y.

+

PROOF.Define Ty on B, by T, ( 2 ) = y Q ( z ) . By the first inequality in Lemma B.3, it is clear that T, (B,) E B,. By this same inequality, so Ty is a contraction mapping on the complete metric space B,. Let x E B,, be the unique fixed point and remark that

is satisfied if and only if @ ( x )= y.

B. INVERSE FUNCTION THEOREM

Let Z = int(BVl2) and W = 8-'(2). V and U ,respectively.

355

These are open neighborhoods of 0 in

COROLLARY B.5. The mapping @ : W + Z as a homeomorphism. PROOF.We have shown that @ maps W oneone onto Z, so it remains to be proven that @-I is continuous. But the equations @(xl) = yl and @(xz)= y2 and the second inequality in Lemma B.3 imply that

proving the assertion. LEMMAB.6. The map @-'is differentiable at each point of Z and

PROOF.Let b = @(a)E Z, a E W. By differentiability at a, we can write

where lim Z(x, a) = 0.

2-+a

Since J@(a)is nonsingular, we can write x - a = J@(u)-'

- (@(x) - @(a))- Ilx - all J@(a)-' . Z(x, a).

Writing x = @-l(y), we obtain

where

But y + b if and only if x

+a

and we have the inequality

by Lemma B.3. That is, lim

67y, b) = O

y+b

and

is differentiable at b E Z with

Since b f Z is arbitrary, all assertions follow. COROLLARY B.7. The map Q-' is of class ckon Z.


356

PROOF.Since @ is of class c k , the entries in the matrix ( J @ ) - ~are functions of class Ck-I and Corollary B.5, together with Lemma B.6, implies that J ( @-') is continuous. That is, @-I is of class C1. If k = 1, we are done. Otherwise, feeding this new fact back into Lemma B.6 implies that is of class C2. Continuing in this way (forever, if k = oo), we complete the proof. We have proven the Ck version of Theorem 2.4.1, 1 5 k 5 oo. REMARK.It is not hard to adapt the above proof to work for mappings

where U C E is open and E, F are Banach spaces over R. One says that F is differentiable at p E U if there exists a bounded linear transformation

(the Jacobian of F at p) such that lim F(x) - F(P) - J F b ) . ( x - P) = 0. llx - PII

x-P

As usual, one shows that such a linear transformation is unique and that its existence implies the continuity of F at p. If this condition holds for all p E U, we obtain a map JF : U -,C(E, F), where C(E, F ) denotes the Banach space of bounded linear transformations from E to F. If the map JF is continuous, we say that F is of class C1 on U. As usual, one obtains the chain rule

for C1 functions, as well as the fact that a bounded linear transformation is its own Jacobian. Inductively, one defines F to be of class C k on U ,k 2 1, if JF is defined and of class Ck--' on U. If F is of class Ck on U, Vk 1, then F is of class CO" on U. If F is a Ck mapping of U, one-one onto an open subset V E F, k 2 1, and if F-I : V + U is also of class Ck,then F is said to be a c k diffeomorphism of U onto V. In our proof of the inverse function theorem, we chose 7 > 0 SO small that J@(x) is invertible, V x E B,. The usual determinant argument for this is unavailable in infinite dimensions, but it remains elementary that the subset of elements in L(E, F ) with bounded inverses is open (cf. 124, pp. 71-72]). Finally, the mean value theorem (equation (B.l)) is completely elementary for general Banach spaces (cf. [24, p. 107]), so the proof that we have given for the finite dimensional case of the inverse function theorem goes through unchanged.

>

THEOREM B.8. Let F and E be Banach spaces, U C E an open subset, and let F : U + F be of class Ck on U, 1 5 k 5 oo. If p E U and JF(p) is an isomorphism of Banach spaces, then there is an open neighborhood W of p in U that is carried ckdifleomorphically by F onto an open neighborhood F(W) of F(p) in F. The Jacobian of F-l at F(x) is the inverse of J F ( x ) , V x E W.


357

Here, of course, by an "isomorphism of Banach spaces" we mean a bounded linear transformation with bounded inverse. The final statement of Theorem B.8 is just the equation J(F-l) = (JF)-I o F-I (Lemma B.6). There is a corresponding version of the implicit function theorem (Corollary 2.4.8) for Banach spaces. This will give a remarkably elegant way of proving the smooth dependence on initial conditions in the fundamental theorem of O.D.E. In order to state this implicit function theorem, we will need some notation. Let E, F, and H be Banach spaces, U C E and V E F open subsets, and let F : U x V + H be of class Ck on U x V, I 5 k oo. Denoting the variables in U and V by x and y, respectively, let (p, q) E U x V and form the "partial Jacobian" J, F[p, q) E L(F, H) (respectively, JzF(p, q) E L(E, H)) by holding x (respectively, y) fixed and treating F as a function of the remaining variable.

0 smaller, if necessary,

LEMMAC.3.1. F o r each (x, s) f Rn x E(Rn), the bounded linear operator JsF ( x , s) has a bounded inverse. PROOF. Indeed, for arbitrary a E E ( R n ) ,

It follows that the operator L =

Jof

J X o s has norm 11 Lll 5 C M < 1, hence that

converges. That is, R E C(E(Rn),E(Rn)) and

as asserted. Thus, the hypothesis of Theorem B.9 is verified.

COROLLARY C.3.2. The map cp : R n + E(Rn), deJined by p(x) = s,, is smooth of class Ck. In fact, we only need to know that cp is of class C1.

COROLLARY (2.3.3. The global flow 9 o n Rn is smooth of class at least C1. PROOF.By Corollary C.3.2, cp is smooth of class at least

where lim 6(h) = 0 in E(Rn).

h-0

That is, lim 6(h)(t) = 0 uniformly on [-c, c].

h+O

Thus, from

c', so

C.3. SMOOTH DEPENDENCE

we deduce that the partial Jacobian

exists and, for each h E Rn, is continuous in (t, x). By successively substituting h = ei, 1 5 i 5 n, we conclude that all entries of the matrix J,@(t, x) are continuous functions of (t, x) E [-c, c] x Rn. Since the flow lines are integral to X, we also see that

at

x, = X(@(t,x))

is continuous in (t, x). Thus, all entries of J@(t,x) are continuous on [-c, c] x R n and O is of class C 1there. The parameter interval [-c, c] of this local C 1flow being uniform for all initial values x E R n , we obtain the unique global C 1flow

generated by X .

REMARK.In order to prove Lemma C.3.1, we had to allow c to be chosen small enough. Our final conclusion, however, was that the global flow 9 is C l. We are going to prove inductively that, for 1 5 q 5 k, @ is of class Cq (the case q = 1 is Corollary C.3.3). At each step of the induction, c may have to be chosen smaller. Nonetheless, at each step the conclusion is about the global flow, so the induction works even for the case k = oo. Let 2 5 j 5 k and assume that it has been shown that the flow @(t,x) is of class Cj-l.We will show that d@(t,x)/dt and Jz@(t,x) are both of class Cj-l in (t, x) , concluding that 9(t, x) is of class Cj.

LEMMAC.3.4. The expression d@(t,x)/dt is of class Cj-' i n (t, x).

PROOF.Indeed, a q t , x)/at = x p ( t , x)) and X is of class ck, @ of class Cj-l,and j 5 k.

LEMMA(3.3.5. The expression J x 9 ( t ,x) is of class Cj-l in (t,x). PROOF.Since we know that @ is of class at least C l , we can differentiate

under the integral sign with respect to the variables x, obtaining

Then,

a

, = JX(@(t,x))Jx (@(t,x)), at Jx ( @ ( t2)) and this can be interpreted as a time dependent system of O.D.E. with parameters x. The unknown functions are the entries of the matrix Jx(O(t,x)), so the system is linear. The coefficients are entries of the matrix JX(O(t,x)), hence are

366

C. ORDINARY DIFFERENTIAL EQUATIONS

of class Cj-' in (t, x). Thus, this system is of class Cj-'. As remarked at the beginning of this appendix, time dependent systems with parameters are equivalent to autonomous systems without parameters, so the inductive hypothesis guarantees that the entries of Jz (O(t, 2)) are smooth of class Cj- .

'

The proof of Theorem 2.8.4 is complete. REMARK.It would have been possible to formulate and prove the O.D.E. t h e orem entirely in the context of Banach spaces (see, e-g., Lang [24, pp. 132-1451, who credits the idea of using the implicit function theorem to Pugh and Robbin), but we have avoided significant technical details by resisting the temptation to do so. On the other hand, judicious use of the implicit function theorem in infinite dimensions does seem to simplify the proof of the finite dimensional theorem. C.4. The Linear Case

In the proof of Theorem 10.1.13, we appealed to the theorem that a linear system of 0.D.E. has solutions that are defined on the largest parameter interval (b, c) on which the system itself is defined. Here is the formal theorem. THEOREM C.4.1. Let A : (b, c) -t 9Jt(n) be smooth of class system s(t) = A(t) .s(t), b < t < c, with initial condition s ( t o ) = a has solution s(t) defined for b < t

c'.

Then the

< c.

PROOF.Let (b', c') E (b, c) be the maximal open interval about t o on which the solution s(t) is defined. If c' < c, we deduce a contradiction as follows. Fix a basis {wl, . . . ,w,) of Rn and, for 1 5 i 5 n, let ai(t) be the unique solution of

defined on some interval (c' - E,c' + E) C (b,c). Choose t, E ( d - E, c'), so close to c' that {al(t,), . . . ,an(t,)) is also a basis of Rn. This is possible by the continuity of the solutions aim Thus, there are constants cl, . . . ,cn E R such that

By the linearity of the system, it is clear that

is also a solution and that a ( t ,) = s(t, ). By uniqueness of solutions, a and s agree on (c' - E , c'), so a extends the solution s to the larger interval (b', c' + E). This contradicts the maximality of (b', c'), proving that c' = c. Similarly, b' = b.

APPENDIX D Sard's Theorem

In this appendix, we prove Theorem 2.9.3 (following Milnor [28]). Accordingly, let : U + V be a smooth map, where U C_ R n and V C_ R m are open. The critical set C E U consists of those points x for which rank Ja, < m and we must prove that @(C)5 V has Lebesgue measure zero. As usual, write a = (a1,. .. , a m ) .

DEFINITION D.1. For each integer hi 2 1, Ck C C is the set of points x E U such that all mixed partials of Qi of order 5 k vanish at z, I 5 i 5 m. It is clear that we obtain a nest

of closed subsets of U.The basic estimates behind Sard's theorem are contained in the following proof.

PROPOSITION D.2. For k 2 1 suficiently large, the set @(Ck)has Lebesgue measure zero. PROOF.Let Q C U be a compact cube. Since Ck n U is covered by countably many such cubes, it will be enough to find a value of k, depending only on m and n (not on Q) such that @ ( C k n Q) has Lebesgue measure zero. Let S denote the edgelength of Q. Let p range over Ckn Q. The kth order Taylor series, expanded about p, takes the form @(P + v) - Q(P) = Rb, v), where the remainder term satisfies a uniform estimate for all p E Ck n Q and all v E Rn such that p + v E Q. Subdivide Q into r n subcubes of edgelength S/r and let Q' be one of these subcubes containing a point p E Ck. Every point of Q' has the form p v, where

+

368

D. SARD'S

THEOREM

By the above estimate on the remainder term, we see that @ ( Q f lies ) in a cube, centered at @(p) and having edgelength e/rk+', with e = 2 ~ ( b f i ) ~ + lThus, . @(Ckn Q) is covered by a union of at most r n cubes with total measure

, For large enough k, the exponent of r is negative and lim,

V ( r ) = 0.

Sard's theorem is trivial when n = 0, so we make the inductive assumption that the theorem has been proven for the case U C Rn-l, some n 1, and deduce the case U C Rn.

>

LEMMAD.3. Let p E C \ C1. Then there are coordinate neighborhoods o f p in U and of @(p) in V relative to which the formula for @ becomes

PROOF.By the definition of C1, p $ Cl implies that some first order partial of some coordinate function of @ fails to vanish at y. Suitably permuting the coordinates in Rn and Rm, we lose no generality in assuming that

Thus, by an application of the inverse function theorem, the map

is a diffeomorphism of some open neighborhood W of p onto an open subset @(W) 5 Rn. This defines a coordinate system with the required property.

PROPOSITION D.4. The set @(C\ C1) has Lebesgue measure zero. PROOF.It will be enough to show that, for each p E C \ C1, there is a neighborhood Wp of p in U such that @(Cn Wp) has measure zero. We assume that m > 2 since, when m = 1, C = C1 and the assertion is vacuously true. By Lemma D.3, we can assume that there is a neighborhood Wp in which each point ( t ,x 2 , .. . ,xn) is carried by @ into the hyperplane { t ) x Rm-l. For fixed t , the restriction of @ to Wp n ( { t ) x Rn-l) can be viewed as a map Bt of that set into the hyperplane { t ) x Ktm-'. Since the coordinate t is preserved, the critical set of at is C n Wp n ( { t) x Rn-l ) and the inductive hypothesis implies that this is carried by at onto a set of (m - 1)4imensional measure zero. Integrate with respect to the t coordinate, concluding by F'ubini's theorem that @ ( Cn W,) has mdimensional measure zero.

PROPOSITION D.5.For each integer k 2 1, the set @(Ck\Ck+l)has Lebesgue measure zero.

D. SARD'S THEOREM

369

PROOF. Again, it will be enough to show that, for each p E Ck \ Ck+l,there is a neighborhood W pof p in U such that Q(Ck n W p )has measure zero. Let cp : U + R be a kth order partial of a coordinate function Q r such that some first order partial of cp fails to vanish at p. Without loss of generality, we assume that Of course, cp vanishes identically on Ck. The inverse function theorem again gives a change of coordinates defined on some neighborhood W p of p, thereby coordinatizing Ck n Wp as a subset of the hyperplane {0) x Rn-l. Every point in this set is a critical point of the restriction Qo of @ t o U n ((0) x Ikn-I), so the inductive hypothesis implies that Q(Ck n W p )= QO(Ckn W p )has Lebesgue measure zero. COROLLARY D.6. For each integer k

2 1, the set @(C\ Ck) has Lebesgue

measure zero. PROOF.Indeed, C \ Ck = (C \ Cl) u (Cl \ C2)u - - - u (Ck-l

\

Ck).

By this corollary and Proposition D.2, the proof of the inductive step is complete.

APPENDIX E The de

ham-cech Cohomology Theorem

In this appendix, our goal is to prove a version of the de Rham theorem that shows the de Rham cohomology algebra H * ( M ) to be a purely topological invariant of the manifold M (Theorem 8.9.6). For this, we must first define the ~ e c cohomology h algebra of an arbitrary topological space.

E. 1. Cech cohomology This section is largely definitions and statements of basic properties. Few proofs will be given because they are mostly routine (and tedious) computations. The first step is to define the cohomology of an open cover U = {Ua)aEll of the space X. DEFINITION E. 1.1. If p 2 0 is an integer, a psimplex of (p+ 1)-tuple (U,, ,U, , . . . , Uap) of elements of U such that U,,

U is an ordered n . - - n Uap # I.

DEFINITION E.1.2. If p >_ 0 is an integer, an R-valued p-cochain on U is a function cp which, to each psimplex (Uao, U,, , . . . ,Uap), assigns an element

The set of all R-valued p-cochains on U is denoted by @(U; R). Evidently, the operations of "simplexwise addition" of cochains and "simplexwise scalar multiplication" make &(u;R) into an R-module. More precisely,

and

-

(aip)ooa~..-a,- a ( ~ a o o l . . . a ~ ) r V a E R a n d V c p , $ ' ~@(u;R). DEFINITION E.1.3. The coboundary operator 6 : CP(U;R) the R-linear map defined by the formula

-+

CP+

(U; R) is

372

E. DE R H A M - ~ E C HTHEOREM

Thus, we obtain a sequence

The following is a routine computation.

LEMMAE.1.4. The sequence of coboundary operators satisfies b2 = 0 .

DEFINITION E. 1.5. The module of ~ e c pcocycles h is

and the module of p-coboundaries is

BP(U; R) = im(6) n Cp(Lf;R). " P (U; R) c - i p ( u ; R). As usual, B

DEFINITION E.1.6. The

pth

~ e c hcohomology of 2.4, with coefficients in the

ring R, is HP(U;R) = BP(U;R ) / z ~ ( u ;R). We have defined H*(u; R) as a graded R-module. It is made into a graded algebra by the "cup product".

DEFINITION E.1.7. If ip E @(u; R) and $ E @(u; R), the cup product CP+Q(U;R) of these cochains is defined by

ip$

E

This makes c*(u;R) into a graded algebra. Another straightforward computat ion proves the following.

LEMMAE.1.8. If p E @(u; R) and $ E CQ(U;R), then

This is formally the same as the formula for the exterior derivative of the wedge product of a p-form and a q-form. As in that case, we get the following consequence.

LEMMAE.1.9. The graded module z*(u;R) of Cech cocycles is a graded subalgebra of c*(u; R) and B*(u; R) c z*(u; R) is a 2-sided ideal. Consequently, cup product is well defined on H*(u; R), making that graded R-module into a graded algebra over R. Finally, we will define the ~ e c cohomology h algebra of the space X to be the "direct iimit" H * ( xR); = lim H*(u; R) +

over finer and finer open covers. We make this precise. Let {V,*),Ea be a family of graded R-algebras, indexed by a directed set U. That is, there is a partial ordering a 5 ,O on U such that, whenever a,,O E U, 3y E U with a 5 y and ,O 5 y. Assume also that, whenever a 5 ,O, there is given a homomorphism p! : V,' -+ V; of graded algebras and that, whenever

E.1. CECH COHOMOLOGY

a j p j 7, then 9; o 9f = 92. We say that {V,', pf},,pEa of graded R-algebras. On the disjoint union

define the equivalence relation

-

373

is a directed system

v* = LI v;, a€%

generated by

where v E V,* and a j p. Then V * / - has a natural graded R-algebra structure. Indeed, scalar multiplication a[v] = [av] is clearly well defined. As for addition, if [v], [w] E V * /- are represented by v E V," and w E Vp' , find y E U, cr 5 y, p 5 7,and set [vl + Iwl = [9',(~) + 9;cw)l. It is trivial to check that this is well defined and that these operations make V* /N into a graded R-module. Similarly, the algebra multiplication passes to a well defined multiplication making V * /- into a graded R-algebra.

DEFINITION E.l.lO. In the above situation, we set lim V; = V*/4

and call this graded R-algebra the direct limit of the directed system of graded R-algebras.

EXAMPLE E.l.ll. For a differentiable manifold M, let U, denote the set of open neighborhoods U of x E M. This is a directed set under the partial order U5 V# U V. The graded algebras {A*(U))uEu, form a directed system under the restriction homomorphisms

>

where w E A*(U) and U 2 V. Then

-

A: = lim A*(U) is just the graded algebra of germs at x E M of smooth forms. Let D(X) denote the set of open covers of X. This is partially ordered by: U 5 V a V is a refinement of U. Since any two open covers of X have a common refinement, this makes D(X) a directed set. If U = {Ua),Ea, V = {Vp)pEB,and U 5 V , then there is a choice function i : 93 + U such that Vp C Ui(p), Q p E 93. This induces a homomorphism ir : c*(u; R) 3 c*(v; R) of graded algebras, where -

id(9)pobl...pP- Vi(po)i(~I)-.i(P,) The following is trivial.

374

E. DE RHAM-CECH THEOREM

LEMMAE.1.12. The homomorphism i d : c*(u; R) 4 c*(v; R), as above, satisfies i d 06 = 6 o i n . We cannot use i d as a homomorphism P(: : t"(u; R) -4 C* (V; R) for a directed system of algebras. The problem is that ill depends on arbitrary choices, so we could never guarantee that oP (: =. P ;( But the above lemma implies that ill induces i* : H*(u; R) + H*(V; R) and it turns out that, at the level of cohomology, the arbitrariness disappears.

(PF-

DEFINITION E.1.13. If U 5 V in D(X), if a, j

:B+Q

are two choice func-

tions, as above, and if p f Z, define

by the formula

As usual, if p - 1 < 0,we understand that CP-'(2.4; R) = 0 and S = 0. The following is checked by a routine (if somewhat tedious) computation, left to the reader.

LEMMAE.1.14. If i,, j, and S are as above, then

Consequently, i* = j* : H*(u; R) 4 H*(v; R). By this lemma, whenever U 3 V in D(X), we define a homomorphism

of graded algebras that is independent of the (allowed) choice of i : B + Q.

LEMMAE.1.15. If U jV jW in D(X), then (PYo P(;

=

(PU.

PROOF.Indeed, write U = {Ua)aEa, V = (Vp)pEs, and W = {W,),Ee and let i : B 4 U, j : C 4 B be suitable choice functions. Then i o j : C + 2l is an allowed choice function relative to the refinement U 5 W. But

Thus, we get a directed system { H* (u; R), ( over R.

P ~ } ~ , of ~ graded ~ ~ ( algebras ~ )

DEFINITION E.1.16. The ~ e c hcohomology of the space X with coefficients in R is the direct limit

-

H*(x; R) = lim H*(u; R), taken over the directed set D(X).

E.2. DE RHAM-CECH COMPLEX

37s

Let f : X + Y be a continuous map between spaces. Given UID(Y), define f (U) E D(X) by the usual pull-back construction. If we are given a psimplex (f-'(U,,), . . . , fd1(u,,)) of f-'(U), then it is clear that (U,,, . . .,U,,) is a p simplex of U. Consequently, each ~ e c hcochain 0 E CP(U;R ) has a natural pull-back f d(0) E CP( f -'(u); R ) . This defines a homomorphism

of graded algebras. LEMMAE.1.17. Iff : X + Y , as above, then f f l o 6 = 6 o ffl and there is canonically defined an induced homomorphism

of gmded algebras over R. This makes &h cohomology into a contravariant functor on the category of topological spaces and continuous maps. DEFINITION E.1.18. If I# is a directed set, a cofinal subset C E ?2iis a directed subset with the property that, whenever a E I#, 3 y E C such that a 5 y. Finally, the proof of the following lemma is a straightforward application of definitions. LEMMAE.1.19. Let {V,', cpg},,pEabe a directed system of graded R-algebras. If C 2 I# is a cofinal subset, then there is a canonical isomorphism

-

lim V,' = lim V; 4

of graded R-algebras, where the first limit is taken over a11 a E I# and the second is taken over a11 y E C. By Corollary 10.5.8, the family of simple covers of a differentiable manifold is a cofinal subset of D (M). COROLLARY E.1.20. The cech cohomology H*(M; R) of a diflerentiable manifold can be computed by tahng the limit only over the directed set of simple covers.

E.2. The de

ham-tech complex

The proof we will give of the de Rham theorem is essentially that of Andre Weil [45].The main step is to prove the following. THEOREM E.2.1. If U is a simple cover of M , there is a canonical zsomorphism : H*(u;R)

+ H*(M)

of graded algebras and, if U 5 V, where both are simple covers, then the diagram


is commutative. The equivalence of de Rham theory and &ch theory follows easily. Indeed, the Cech cohomology can be computed by passing to the limit over the simple covers only (Corollary E.1.20), so the isomorphisms $u induce a well defined homomorphism $ : H*(M;R) 4 H*(M). The fact that each $u is an isomorphism implies the same for I). THEOREM E.2.2 (DE RHAM).There is a canonical isomorphism

of graded R-algebras. In order to prove Theorem E.2.1, we will build an enormous cochain complex of graded algebras that includes both (A*(M), d) and (c* (u;B),6) as subcomplexes. If U is simple, we will prove that the inclusions of these subcomplexes induce isomorphisms in cohomology. Fix an open cover U = {Ua)aEll of M. For the following definitions, it is not necessary that U be simple. DEFINITION E.2.3. A ~ e c p-cochain h on U with values in Aq is a function cp which, to each psimplex (U,, , U,, , . . . , Uap) of U assigns

The set of all such cochains will be denoted EP*q(U)= CP(U;Aq). Although the coefficient ring AQ(U,, n U,, n . . . n U,, ) changes with each simplex, one can still add cochains simplexwise and multiply them by real scalars. These operations make EPjq(U) = CP(U; Aq) into a real vector space. There is also a bigruded multiplication

In defining this and other operations, we make the notation less bewildering to the eye by abusing it (the notation, that is). Whenever respective forms have been defined on respective open sets with common, nonempty intersection, addition of such forms and exterior products of such forms are understood to be defined on their common domain. For instance, if (U,, , U,, , U,,) is a 2simplex of U and wffiaj E AQ(U,, n U,,), then

Similarly, if $'a0,,

E Aq(U,, f U, l , ) and ,$ ,,,

E As (U,,

n u,,), then

E.2. DE RHAM-CECH COMPLEX

377

With these conventions understood, we define the bigraded multiplication as follows. If cp E Ep*Q(U) and @ C, Erys(U),then p@ E Ep+r*Q+s(U) k defined on a (P+ r )-simplex (Uao, . . . ,Uap, . . . ,Uap+,) by ~

o

f

f

p

+

qr r (-1)

A ,@ ,

..,

+ ,,

E Aq+'(Uff,n

.. - n u,,,,).

Often we suppress explicit reference to the simplex on which this formula is being evaluated and write cp$ = ( - l ) " p A @ .

We say that E**( U ) = { E P I ~ ( U ) ) Eis~ a= bigraded ~ algebra under this multih plication. Note how this operation combines the cup product from ~ e c theory with the exterior multiplication from de Rham theory. The strange sign ( - l ) q r will be needed in the proof of Lemma E.2.7.

DEFINITION E.2.4. The de Rham operator d

:

Ep79 ( U ) + Ep*q+l( U ) is d,

fined by setting V cp E CP(U;Aq) and for every psimplex ( U f f ,U,, , , . . . , UaP)of U .

DEFINITION E.2.5. The ~ e c operator h 6 : EP3Q(U)-+ E P + ' * ~ ( Uis)defined by setting

Vcp E CP(U;Aq) and for every p-simplex (U,, , U,,

,. . . ,U,,)

of U .

The following are evident:

a2=o b2=0 aos=-boa Remark that the sign ( - l ) p in the definition of d is responsible for the anticommutativity of d and 6.

FIGURE E. 1. The de

ham-~ech complex

E. DE RHAM-CECH

378

THEOREM

DEFINITION E.2.6. For each integer m 2 0, Em(U) = @P+q=m EP,q(U) and the operator D : Em(U) 4 E ~ + ' ( U ) is D = d + 6. It is a good idea to picture E**(U) laid out as a first quadrant array in the (p, q)-plane, having Ep*Q(U)at the point (p, q) of the integer lattice as in Figure E.l, with the de Rharn operators d as vertical arrows and the ~ e c h operators 6 as horizontal arrows. This array is called the de Rharn- ~ e c complex. h The total degree of an element cp E EPyq(U) is p q and Em(U) is spanned by the elements of total degree m. One can view Em(U) in this diagram as lying along the diagonal p q = m. If cp E EplQ(U),where p q = rn,then

+

+

+

LEMMAE.2.7. The pair (E*(U),D) is a cochain complex in which E*(U) is a graded algebra over R and

where cp E Em(U).

PROOF.Indeed, E*(U)= {Em(U))z=ois a graded vector space and it is clear that the bigraded multiplication in E ** (U) induces a graded algebra structure on E*(U). Because of the anticommutativity of d and 6,

Finally, it is enough to verify the Leibnitz formula for cp E EP*Q(U)and $ E ErtS(U),p q = m. Suppress reference to the ( p r)simplex (U,, , . . . , U,,,,) and compute

+

+

That is, Similarly, suppressing reference to the ( p + r we compute

+ 1)simplex (U,,, . . . , Uap+r+l),

That is,

By adding equation (E.l) and equation (E.2), we obtain the desired Leibnitz rule for D.

E.2. DE R H A M - ~ E C H COMPLEX

There are canonical inclusions

(A*( M I ,d ) -i, ( E *( U ) ,D )

(c* (u;R ) ) L ( E *( U ) ,D ) of subcomplexes. Indeed, if w E A Q ( M ) ,i ( w ) E CO(U;A Q )= E'VQ(U)assigns to each O-simplex (U,,) the element i(w),, = w(U,,. Likewise, if cp E R), j ( 9 ) E CP(U;A') assigns to each p-simplex (U,, , . . . ,U,, ) the &form on the open set U,, n . . . n U,, which is the constant function cp,o...,p E R. It is clear from these definitions that

@(u;

whenever w E A Q ( M )and 7 E A S ( M ) ,and that

whenever cp E CP(U;R) and @ E

c'(u;R).

LEMMAE.2.8. There are canonical homomorphisms i* : H* ( M ) + H* ( E *( U ) ,D ) and

j* : H*(u; R ) 4 H*(E*(U)D , ) of graded algebras. We augment the rows of the diagram in Figure E.l by i. That is, the new rows are 6 E O ' ~ ( U ) f E'.Q(u)+ . - . A (M )

Similarly, we augment the columns by j:

LEMMAE.2.9. The augmented diagram has exact rows.

PROOF.If w

E

A Q ( M ) ,we evaluate 6(i(w)) on the lsimplex (U,, ,U,, ),

getting

(S(i(w)))aoal= wlUolo n Ual - wlUa0 n ua, = 0. I f ip E c'(u;Aq) and 6 ( 9 ) = 0, then cp,, E Aq(UaO)and cp,, E AQ(U,,) must agree on U,, n U,, , i f this intersection is nonempty. Hence, the forms c p , E AQ(U,) piece together smoothly to give a form w E AQ( M ) such that i ( w ) = cp. This proves exactness at E O ~ Q ( U ) . We prove exactness at EP?Q(U) = CP(U;AQ)when p _> 1. Let {A,),Ea be a + Ep-l>Q(U) as smooth partition of unity subordinate to U . Define A : EP*Q(U) follows. Given p E &(u; A Q )= Ep*q(U),define A ( 9 ) E CP-'(U; AQ)to be the element whose value on the (p - 1)simplex (U,, , . . . ,U,,-, ) is


380

where each term of this locally finite sum is interpreted, in the obvious way, as a q-form on U,, n - . . nU a p - l .If 6 ( p ) = 0, the reader can check that cp = 6 ( A ( p ) ) , proving exactness at EP?Q(U).

LEMMAE.2.10. If the cover U is simple, the augmented diagram has exact columns.

PROOF.If C E CP(U;B ) , then, on an arbitrary p-simplex (U,, j(C),, ...,, E Ao (Uao n . . - n U a p )is constant, so

, . . . ,U a p ) ,

If (o E @(u; A O ( M ) )and a ( p ) = 0, then d((o,o...ap)= 0, for each psimplex (UaO7 . . . ,UaP).The fact that U,, n. .n UaPis connected implies that E AO(Uaon . . . n Ua,) is constant. Thus, we can define C E ~ ~ ( 2 .W) 4 ; by +

Differentiable Manifolds: A First Course

Differentiable manifolds

Differentiable Manifolds