aubin - a course in differential geometry.pdf

A Course inDifferentialGeometry

Thierry Aubin

Graduate Studiesin MathematicsVolume 27

American Mathematical Society

Selected Titles in This Series27 Thierry Aubin, A course in differential geometry, 200126 Rolf Berndt, An introduction to symplectie geometry, 200125 Thomas } iedrich, Dirac operators in Riemannian geometry, 200024 Helmut Koch, Number theory: Algebraic numbers and functions, 200023 Alberta Candel and Lawrence Conlon, Foliation I. 200022 Gfinter R. Krouse and Thomas H. Lenagan, Growth of algebras and Gelfand-Klrillov

dimension, 200021 John B. Conway, A course In operator theory, 200020 Robert E. Gompf and Andrda I. Stlpsics, 4-manifolds and Kirby calculus, 199919 Lawrence C. Evans, Partial differential equations, 199818 Winfried Just and Martin Weese, Discovering modern set theory. IT: Set-theoretic

tools for every mathematician, 199717 Henryk Iwanlec, Tbpies in classical automorphic forms, 199716 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator

algebras. Volume 11: Advanced theory, 199715 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator

algebras. Volume I: I oentary theory, 199714 Elliott H. Lieb and Michael Loss, Analysis, 199713 Paul C. Shields, The ergodic theory of discrete sample paths, 199612 N. V. Krylov, Lectures on elliptic and parabolic equations in Holder spaces, 199611 Jacques Dhunier, Enveloping algebras, 1996 Printing10 Barry Simon, Representations of finite and compact groups, 19969 Dino Lorenalni, An invitation to arithmetic geometry, 19968 Winliried Just and Martin Weese, Discovering modern set theory. 1: The basics, 19967 Gerald J. Janusz, Algebraic number fields, second edition, 19966 Jens Carsten Jantzen, Lectures on quantum groups, 19965 Ride Miranda, Algebraic carves and Rlernann surfaces, 19954 Russell A. Gordon, The Integrals of Lebesgue, Denjoy, Perron, and Henstock, 19943 WIlliam W. Adams and Philippe Loustaunau, An introduction to Grdbner bases,

19942 Jack Graver, Brigitte Servatius, and Herman Servatius, Combinatorial rigidity,

19931 Ethan Akin, The general topology of dynamical systems, 1993

A Course inDifferentialGeometry

Thierry Aubin

Graduate Studiesin MathematicsVolume 27

Editorial BoardJames Humphreys (Chair)

David SaitmanDavid Sattinger

Ronald Stern

2000 Mathematics Subject Clamfiication. Primary 53B05, 53C05, 53C22, 53C40, 58A17,580.05, 58C25, 58C35, 58305.

Assmc-r. This book provides an introduction to differential geometry, with prinicpal emphasison Riemannian geometry . It covers the essentials, concluding with a chapter on the Yamahaproblem, which shows what research in the Said looks like. It is a textbook, at a level which isaccessible to graduate students. Its aim is to facilitate the study and the teaching of differentialgeometry. It is teachable on a chapter-by-chapter basis. Mary problems and a number of solutionsare included; most of them extend the course itself, which is confined to the main topics, suchas: differential manifolds, submanifolds, differential mappings, tangent vectors, differential forms,orientation, manifolds with boundary, Lie derivative, integration of p-direction field, connection,torsion, curvature, geodesics, covariant derivative, Riemannian manifolds, exponential mapping,and spectrum.

Library of Congress Cataloging-In-Publication DataAubin, Thierry.

A course in differential geometry / Thierry Aubin.p. cm. - (Graduate studies in mathematics, ISSN 1065-7339; v. 27)

Includes bibliographical references and index.ISBN 0-8218-2709-X (alk. paper)1. Geometry, Differential I. retie. II. Series.

QA641.A795 2000516.3'6--dc21 00.068275

Copying and reprinting. Individual readers of this publication, and nonprofit librariesact lag for them, are permitted to make fair use of the material, such as to copy a chapter for usein teaching or research. Permission is granted to quote brief passages from this publication inreviews, provided the customary acknowledgment of the source is given.

Republication, systematic copying, or multiple reproduction of any material in this publicationis permitted only under license from the American Mathematical Society. Requests for suchpermission should be addressed to the Assistant to the Publisher, American Mathematical Society,P.O. Box 6248, Providence, Rhode Island 02910.6248. Requests can also be made by e-mail toreprint-porsissionSaas.org.

2001 by the American Mathematical Society. AU rights reserved.The American Mathematical Society retains all rights

except those granted to the United States Government.Printed in the United States of America.

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Visit the AMS home page at URL: http;//vsv.ara.org/10987654321 060504030201

A mon ProfesseurAndre' Lichnerowicz

(in memoriam)

Contents

Preface ixChapter 0. Background Material 1

'Ibpology 1Tensors 3Differential CalculusExercises and Problems

Chapter 1. Differentiable Manifolds 19Basic Definitions 19Partition of Unity 25Differentiable Mappings 27Submanifnlds 29The Whitney Theorem 30The Sard Theorem

-34Exercises and Problems asSolutions to Exercises 40

Chapter 2. Tangent Space 43Tangent Vector 44Linear T agent Mapping 46Vector Bundles 48The Bracket [X, Y] 49Exterior Differential 52Orientable Manifolds 55Manifolds with Boundary 58Exercises and Problems fi4Solutions to Exercise and Problems fib

Chapter 3. Integration of Vector Fields and Differential Forms 77

vii

viii -1. Contents

Integration of Vector Fields 77

Lie Derivative 79The Frobenius Theorem 81Integrability Criteria. 85Exercises and Problems 87Solutions to Exercises and Problems 93

Chapter 4. Linear Connections 99First Definitions 99Christoffel Symbols 100Torsion and Curvature 101Parallel Transport. Gecxlesics 103Covariant Derivative 105Exercises and Problems 107Solutions to Exercises 108

Chapter 5. Riemannian Manifolds 111Some Definitions 11.1Riemanmian Connection 114Exponential Mapping 117Some Operators on Differential Fbrms 121Spectrum of a Manifold 125Fxereisp-q and Problems 129Solutions to Exercises and Problems 145

Chapter 6. The Yamabe Problem: An Introduction to Resear ch 169Bibliography 177Subject Index 179Notation 183

Preface

This book provides an introduction to differential geometry, with principalemphasis on Riemannian geometry. It can be used as a course for second-year graduate students. The main theorems are presented in complete detail,but the student is expected to provide the details of certain arguments. Weassume that the reader has a good working knowledge of multidimensionalcalculus and point-set topology

Many readers have been exposed to the elementary theory of curves andsurfaces in three-space, including tangent lines and tangent planes. Butthese techniques are not necessary prerequisites for this book.

In this book we work abstractly, so that the notion of tangent space doesnot necessarily have a concrete realization. Nevertheless we will eventuallyprove Whitney's theorem asserting that any abstract n-dimensional manifoldmay be imbedded in the Euclidean space RP if p is sufficiently large.

In order to develop the abstract theory, one must work hard at the be-ginning, to develop the notion of local charts, change of charts, and atlases.Once these notions are understood, the subsequent proofs are much easier,allowing one to obtain great generality with maximum efficiency. For exam-ple, the proof of Stokes' theorem-which is difficult in a concrete context-becomes transparent in the abstract context, reducing to the computationof the integral of a derivative of a function on a closed interval of the realline.

In Chapter I we find the first definitions and two important theorems,those of Whitney and Sard.

Chapter II deals with vector fields and differential forms.

ix

x Preface

Chapter III concerns integration of vector fields, then extends to p-planefields. We cite in particular the interesting proof of the Frobenius theorem,which proceeds by mathematical induction on the dimension.

Chapter IV deals with connections, the most difficult notion in differen-tial geometry. In Euclidean space the notion of parallel transport is intuitive,but on a manifold it needs to be developed, since tangent vectors at distinctpoints are not obviously related. Loosely speaking, a connection defines aninfinitesimal direction of motion in the tangent bundle, or, equivalently, aconnection defines a sort of directional derivative of a vector field with re-spect to another vector. This concrete notion of connection is rarely taughtin books on connections. In our work we devote ten pages to developing theseideas, together with the related notions of torsion, curvature and a workingknowledge of the covariant derivative. All of these notions are essential tothe study of real or complex manifolds.

In Chapter V we specialize to Riemannian manifolds. The viewpointhere is to deduce global properties of the manifold from local properties ofcurvature, the final goal being to determine the manifold completely.

In Chapter VI we explore some problems in partial differential equationswhich are suggested by the geometry of manifolds.

The last three chapters are devoted to global notation, specifically to us-ing the covariant derivative instead of computing in local coordinates withpartial derivatives. In some cases we are able to reduce a page of computa-tion in local coordinates to just a few lines of global computation. We hopeto further encourage the use of global notation among differential geometers.

The aim of this book is to facilitate the teaching of differential geometry.This material is useful in other fields of mathematics, such as partial differ-ential equations, to name one. We feel that workers in PDE would be morecomfortable with the covariant derivative if they had studied it in a coursesuch as the present one. Given that this material is rarely taught, one mayask why? We feel that it requires a substantial amount of effort, and thereis a shortage of good references. Of course there are reference books such asKobayashi and Nomizu (5J, which can be consulted for specific information,but that book is not written as a text for students of the subject.

The present book is made to be teachable on a chapter-by-chapter basis,including the solution of the exercises. The exercises are of varying difficulty,some being straightforward or solved in existing literature; others are morechallenging and more directly related to our approach.

This book is an outgrowth of a course which I presented at the UniversitkParis VI. I have included many problems and a number of solutions. Someof these originated from examinations in the course. I am very grateful tomy friend Mark Pinsky, who agreed to read the manuscript from beginning

to end. His comments allowed me to make many improvements, especially inthe English. I would like to thank also one of my students, Sophie Bismuth,who helped me to prepare the final draft of this book.

Chapter 0

Background Material

In this chapter we recall some fundamental knowledge which will be used inthe book: topology, algebra, integration, and differential calculus.

Zbpology0.1. Definition. A topology on a set E is defined by a family O of subsetsof E, called open sets, such that

a) The set E and the null set 0 are open sets.b) Any union of open sets is an open set.c) Any finite intersection of open sets is an open set.

(E, 0) is a topological space.0.2. Examples. If 0 = {E, 0}, the corresponding topology is called trivial.If 0 consists of all subsets of E, the topology is called the discrete topology.

On Rn the usual topology may be defined as follows: Let x be a pointof R" and p > 0 a real number. We consider the open ball of center x andradius p, BS(p) = {y E R" I Hix - yul < p}. An open set in R" will be aunion of open balls or the empty set 0.

0.3. Induced topology. Let F be a subset of E endowed with a topology0. The induced topology on F is defined by the following set 0 of subsetsofF: AEOifandonlyifA=AflFwithAEO.0.4. Example. Let F be a finite set of points in R. The topology on Finduced by the usual topology on R" is the discrete topology. We will findother examples in 1.16.

1

2 0. Background Material

0.5. Definitions. A neighbourhood of a point x in a topological space E isa subset of E containing an open set which contains the point x.

We can verify that a subset A C E is open if and only if it is a neigh-bourhood of each of its points.

B C E is closed if A = E\B is open.A topological space E is said to be connected if the only subsets which

are both open and closed are the empty set 0 and the space E itself.The closure of a subset B C E is the smallest closed set containing

B. The closure B always exists-indeed, the intersection of all closed setswhich contain B (E is one of them) is a closed set according to b) in 0.1.0.6. Proposition. Any neighbourhood A of X E B has a nonempty inter-section with B.

Proof. Let Q C A be an open neighbourhood of x. If o n B = 0, then E\12is a closed set containing B; hence B C E\SZ and x f B, a contradiction.

00.7. Definitions. The interior B of B C E is the largest open set containedin B (B is the union of all open sets included in B.)

A topological space is separable if it has a countable basis of open sets{Ai}iEN. That means any neighbourhood of x contains at least one Ai withx E Ai.

A topological space is Hausdorff if any two distinct points have disjointneighbourhoods.

A family {Sl;}ieI of subsets of E is a covering of B C E if B C (Rt.A subcovering of this covering is a subset of the family, {S?L}1ej (with J c I),which itself is a covering. If J is finite the subcovering is said to be finite.

0.8. Definition. A subset A C E is a compact set if it is Hausdorff and ifany covering of A by open sets has a finite subcovering.

This definition implies the following necessary and sufficient condition:A c E, a Hausdorff topological space, is compact if and only if any familyof closed sets whose intersection is empty has a finite subfamily of emptyintersection.

0.9. Theorem. Let E be a Hausdorf topological space. If K C E is acompact set, K is closed. This condition is sufficient when E is compact.

Proof. We argue by contradiction. If K is not closed (K 96 7), thereexists x E K such that x 0 K. Now in a Hausdorff topological space, theintersection of the closed neighbourhoods of a point x is just the subset {x},which is closed. Indeed, for any y E E there exist disjoint open sets 9 and

312, neighbourhoods respectively of x and y. E\i2 is a closed set, which is aneighbourhood of x since A C E\fl, and y E\i2. Thus the traces on K ofthe closed neighbourhoods { V; },E I of x would have an empty intersection.So {E\Vf}jeI would be a covering by open sets of K.

Since K is compact, there would exist a finite set J C I such that{E\ Vi };EJ is a covering of K. Thus n,E J Vi =V would be a neighbourhoodof x and V fl K 0. Since x E I?, Proposition 0.6 gives a contradiction.

Suppose E compact and A C E closed. Then the closed seta for A areclosed sets for E, and compactness for A follows from the necessary andsufficient coo&tion for A to be compact (see above).0.10. Definition. Let E and F be two topological spaces. A map f of Einto F is continuous if the preimage f -' (f) of any open set 1 C F is anopen subset of E.0.11. Theorem. he image by a continuous map of a compact set is com-pact.

Proof. Let K C E be a compact set. Consider any covering of f(K) byopen sets 5 (i E I). (f -1(S2i)}iE j is a covering of K by open sets; thusthe exists a finite set J C I such that K C UiSJ f -'(W- So {ftifiEJ is acovering of f (K).0.12. Definitions. A continuous map is said to be proper if the preimageof every compact set is a compact set.

If E and F are Hausdorff topological spaces, a continuous map f of Einto F is proper if E is compact. Indeed, let K C F be a compact set. SinceK is a closed set. f -1(K) is a closed set. So f -1(K) is compact, since aclosed set in a compact set is a compact set (Theorem 0.9).

Let E and F be two topological spaces. A map f of E onto F is ahomeomorphism if it is one to one and if f and f -1 are continuous maps.

Tensors0.13. Definition. Let E and F be two vector spaces of dimension respec-tively n and p. The tensor product of E and F is a vector apace of dimensionnp, and is denoted by E F. A vector of E OF is called a tensor. Tb X E Eand y E F we associate x y E E F. This product has the followingpropertim-

a) (xI+xa)y=x1y+x2yandx(yi+ya)=xyl+x@y2,where x, x1, x9 belong to E and y, yl, y2 belong to F.

b) If a E R (in this course the vector spaces are on R), then(ax)y=xs(ay)=a(x(9 y).


c) If { ei } 1

The contravariant indices are up, and the covariant indices are down.When we change the basis of E, { ei } -r {ea }, we express the components

of a (p, q)-tensor T (in the basis lei }) in terms of its new components (inthe basis {e}) by means of the matrix ((&)) for the covariant indices andthe matrix ((a )) for the contravariant indices.0.16. Examples. In this book we will consider a Riemannian metric g. Ata point, g is a (2,0)-tensor on the tangent space which is isomorphic to R'.gij are the components of g in the basis {ei}. If ga,q are the components ina new basis {ea}, we get g,1 = bbga,6, since g = 9ifel 8i = 9ayaea Bb

We will consider also the curvature tensor R, which is a (3,1)-tensor.We get

R ' = b a,3bkbiR in the first basis ands ra the components

in the new basis.

0.17. Theorem. A system of nP+9 real numbers attachedto a basis 8`1O'2 . 8'p eiy+1. are the components of a (p, q)-tensor on E if and only if in a new basis gal 9p eal the system of real numbers ta,a2...app}1"'Qp+ satisfies

iP+I...ip+9= ba1bb ... bp ro+1 a y+9 aP,...ap+9tili2...ip it i2 Lp aap+1 ... ap+9 OIOt2...ap

So we have

41 i76i 1 g`2 ... ga O"P eip+1 , ...

talae...0pP+l...ap+901 9012 ... gap eap.}1 ... eap+

Often it would require long computations to verify the above equalities.Fortunately there are tensoriality criteria.

0.18. 11msoriality Criteria Let {wi}1


1 v

ip+,...;p+aX+t X'2... X`pwl w44i i2...b 1 s P w ... p+4

= ta, l .. . Xp p -1 9ap+l .. wo +v.Then T is a (p, q) -tenor.Proof. For 1 < j < p we have XX = aQX7 , and for 1 < k < q we have

bwa. Putting these expressions in the condition of Theorem 0.19, weget

4Ii2.

ipy,l...aal... aa;b

v+ t2 ... btr+v -Thus T is a (p, q)-tensor.

There are others tensoriality criteria.

0.20. Example. Suppose that a system T of n3 real numbers tip in thebasis {ei}, 11.0 in the basis {ea} satisfies the following condition: for anyvectors X and Y, t kX'X' = Zk are the components of a vector Z; that is,we also have Z"r Then T is a (2,1)-tensor.

Proof. We have X' = a,X, Y' = a' ,Y'3 and Zk = t42-f. Thus tk?a,a', _aryP... Multiplying both sides by bk, we get, after summation, tija' a'b _

since bleary = 67. So T is a (2,1)-tensor.0.21. Definition. The exterior product of two vectors x and y of E is theskew-symmetric tensor

xAy=xy-yOx.If {x'} and {y'} are the components of x and y in the basis {e;}, then x'yf --xxy' = z'j are the components of x A y in the basis {ei A eJ }, 1:5 i < j < n,which has Cn elements.

In this book we will consider especially the skew-symmetric p-forms onR" (or on a vector space E of dimension n), that are the skew-symmetric(p, 0)-tensors. If {x'}1

here 1 < Al < Az < < Ap < n and 1 < pi < 2 <

0. Background Material

0.25. Let f be a differentiable map of 11 C R" into RP. f is defined by meansof a system f', f 2, JP P of p differentiable functions on Sl (real-valued):the coordinates of f (x) in RP are f' (x), f 2(x), , fP(x). The differentialf'(x) is represented by the Jacobian matrix ((8f/8x'))x with n columnsand p rows.

Here {x`}1

Differential Calculus 9

Thus

and

1 1f (b) - f (a) = t(1) - t(0) = fo 4'(t)dt = fo f'(a + ty) o y dt

1

11f (b)- f(a)ll


Proof. Let us consider the mapping 9(x) = f'-1(xj) o f (x). We haveg'(xo) = Identity. Set h(x) = g(x) - x. h is C1 on f2, and since h'(xo) = 0,there exists r > 0 such that llx

- xoll < r implies llh'(x) 11 < 1. We choose rsmall enough so that rank f = non the ball b C fl of radius r and centerxo. According to the mean value theorem, for x and x' in B

11h(x)- h(x')II < 'Ilx - x'II.

Let B be a ball of center yo = g(xe) and radius smaller than r/2. Considerthe equation x + h(x) = y for y E B given. We define by induction thesequence xk+l = y - h(xk) (k > 0). We havellxk+1-xkll = llh(xk)-h(xk-1)ll 0, there is q > 0 such that 11X - xol1 < r)implies

II g(x) - 9(xo) - x + xoll < Ellx - xoll,since h'(xo) = 0. Thus

lIf'-1(xo) o (y - yo) - $-1(y) + -1(yo) II < 2elly - yo Il.This inequality proves that -' is differentiable at xo. Its differential isfi-1(xo), and rankt-1 = n at yo. As we can give the same proof atany point zo E 0, it follows that $-1 is differentiable on f (0). Since(V1)'(y) = f'-' [V'(y)l, it follows that V1 is C' on f (0). 4-1 is Ckif f is Ck0.30. Remark. The inverse function theorem holds in Banach spaces. Theproof is the same.

And now a global result.

0.31. Theorem. Let f be a C' -mapping of ft C B into b, B and B beingtwo Banach spaces. Suppose that

a) f is injective, andb) f'(x) is an isomorphism from B onto b for every x E ci.

Then f is a difeomorphism from Il onto f (S2) C B.Since f is injective, 4 = f It, : 0 -i f (Cl) is invertible; i.e., 4-1 exists.

According to 0.29, $'1 is continuous and C1-differentiable (these are localproperties).0.32. Let B be a Banach space, and let (t, x) -- f (t, x) E B be a con-tinuous function defined in a neighbourhood of (to, xo) in R x B. Since fis continuous, there exists a neighbourhood V of (to, xo) where f (t, x) isbounded.

Ilf(t,z)II < M for all (t,x) E V.We consider a closed ball f Z of radius r and center xo in B, and I =

[to - a,to + a] C R (a > 0) such that I x f2 C V, a and r satisfyingMa < r. Recall that a map h of an open set 0 of a Banach space B1into a Banach space B2 is Lipschitz in 0 if there exists k E R such thatiIh(a) - h(b)l < klla - bit for (a, b) E 0 x 0.0.33. The Caudh Theorem. The differential equation(*) a = f (t, x), z(to) = zo,has a unique continuous solution x(t) if f (t, x) is Lipschitz in x on I x A.The solution is defined on a neighbourhood J C I of to, and its values arein n.

Proof. First of all, a continuous function x(t) satisfying (*) satisfies theintegral equation

x(t) =xo+J f(u,x(u))du,and conversely. Let C(I, B) be the Banach space of all continuous functionsg on I into B endowed with the norm sups 11g11. Consider the mapping 0of C(I, 0) into itself defined by

t : C(l, fl) i) x(t) y(t) = xo + J f (u, x(u)) du.Since f is Lipschitz in x, there exists k such that, for t E I and a, b in 11,

Ilf(t,a) - f(t,b)II 5 klla - b11-Thus 0 is locally a contracting mapping. Indeed,

II$(x)--O(y)o


0.34. Remark. If f (t, x) is not Lipschitz, we cannot say anything in gen-eral. But if the Banach space is R", then there is a solution of (*), butit may not be unique. In that case there exist a solution greater than theothers, and a solution smaller than the others.

0.35. Example. n = 1, x' = 2 jxJ, x(0) = 0.

0

General solution ........ Greatest solution - - - - Smallest solution

0.36. Dependence on Initial Conditions. The solution of (*) (Theorem0.33) depends on the initial conditions (to, xo). Thus we can write it in thefoam x(t, to, xo).

If we choose (i,i) in a neighbourhood 0 of (to,x0), we can choose r anda small enough so that the closed ball f2j of radius r and center i, andI = [t - a, t + o), satisfy I x fit C V for any (t, i) E 0.0.37. Theorem. Let f be a continuous function, Lipschitz in x, as in The-orem 0.33. Then there exists a neighbourhood 0 of (to, xo) in R x B such thatthe solution x(t, t, i) of the differential equation

x' = f (t, x), x(t) = z,

exists on [t' -,3, t + 01, 3 > 0 being independent of (t, i). Moreover, (t, t, s)- x(t, t, x) is continuous. If f is C-, this map is C.

Exercism and Problems 13

Exercises and Problems0.38. Problem. Let t x(t) E R be a function defined on an interval ofR. Consider the family of first order differential equations E,\ (A E R):

X, = x2(1 + t2A2) - 1, x(0) = 2.

a) Show that E,\ has a unique maximal CO solution xa defined on aninterval la = (aa, b.,).

b) Integrate the equation Eo explicitly.c) Let f (t, x) and g(t, x) be two continuous functions on R2 with value in

R, and suppose that f and g are uniformly Lipschitz in x. Let (to, xo)be initial conditions, y(t) the maximal solution of the equation

x' = f (t, x), x(to) = xo(it exists on I), and z(t) the maximal solution of the equation

x = g(t, x) , x(to) = xo(it exists on J). If f (t, x) < g(t, x) on R2, prove that z(t) > y(t) forto y(t) for to < t E I f1 Jand that z(t) < y(t) for to > t E I fl J. Hint. Consider the family offunctions 9,, (t, x) = g(t, x) + .1 with n E N.

e) Show that bA is finite. What is the upper bound of ba?f) Prow that xa > 0 on I. Hint. Argue by contradiction, considering

the equation x' = x2(1 + t2A2).g) Establish that xa(t) < 2 for t < 0. Deduce from the previous question

that as = -oo.0.39. Problem. Let f (t, x) be a continuous map of I x 1 C R x R" into R",where I=[to,to+a] and S1= B,,, (r)CR"(a>0,r>0,xoER",B,,,, (r)the ball with center xo and radius r). Set M = sup I If (t, x) I I for (t, x) E I x n,and choose a .

a) Consider for t E [to - a, to] the functionyo(t) = xo + f(to,xo)(t - to)

and the functions yk ( < k E N) defined by yk(t) = yo(t) for t E[to - a, to] and yk(t) = xo + f t f (s, yk(s - ))ds for t > to. Showthat yk(t) is defined and continuous on [to - a, to + k], and then on[to-a,to+a].

b) Prove that the family yk(t) is equicontinuous on I-that is to say, forany e > 0, there exists q > 0 such that It-$I < q Ilyk(f)-yk(s)II < Ef o r all k > t ands belonging to I. Then apply Ascoli's theorem:


there exists a subsequence {yki} C {yk} which converges uniformlyon I to a function z. Show that z satisfies on I the equation E:

z' = f (t, z), z(to) = xo.c) From now on n = 1 (y E R and f (t, y) E R). Consider the equation

Ep (p E N):

Y, = f (t, y) + p y(to) = xo-Prove that Ep has at least one Cl-solution defined on I. Let p < q betwo integers, zp a solution of Ep and zq a solution of Eq. Prove thatzp(t) > zq(t) for t E I. Deduce that z(t) = limp,, , zp(t) is a solutionof E larger than any other solution of E. z is called the maximalsolution of E.

0.40. Problem. Let t - x(t) E R be a function defined on an open set ofR. Consider the differential equation(E) X, = A - f (t)ex,where f is a nonvanishing continuous periodic function on R with period 1,and A is a real parameter.

a) For to E R, show that there exists, in a neighbourhood of to, a uniquedifferentiable solution of (E) such that x(to) = xp, xo being a givenreal number.

b) Verify that if x is a solution of (E), then y = e-z is a solution of= f(*) y' + AY

c) When A 0, prove that

YAM =ea

1 1 ( ea" f (t + u) du0

is the unique differentiable solution of (*) which is periodic of period1.

d) What can we say about the existence of a periodic solution of (*) inthe case A = 0?

e) When f > 0, deduce from c) that (E) has a periodic solution (ofperiod 1) if and only if A > 0.

f) Show that Aya (t) = f (t). Deduce that (E) has a periodicsolution (of period 1) for all A < 0 if and only if f (t) < 0 for all t.

g) If fo' f (t)dt < 0, establish the existence of e > 0 such that (E) has aperiodic solution (of period 1) for any A E (-e, 0).

0.41. Problem. LetE={f ECl([O,11)1 f(0)=0and f(1)=1}.

What is the greatest real number m such that

M < I f'(x) - f(x)I dxkw any f E E? Hint. Consider the function fe-=.0.42. Problem. Let t -' x(t) E R be a function defined on an open set ofR. Consider the differential equation

(Ea) x'=A+l+t2, x(0)=0, AER.a) Prove that there exists Ao such that (Ex) has a solution on [0, oo) if

and only if A < Ao. What is the value of AO?b) When A > Ao, the maximal solution exists on [0, aa). Show that

sinh-

< as < sinh arFA-I.

0.48. Exercise. Let f be a real valued differentiable function at each pointof [a, b] C R, and suppose that f(a) < f'(b). Let It E (f'(a), f'(b)). Provethat there exists zo E (a, b) such that f'(xo) = yo.0.44. Problem. Let A (n E N) be the positive solutions of tan x = x.

a) Show thatO 1 1?=10'n=1

b) Consider the equation(E) -y" + b2y = f(x), y(O) = 0, 01) = y(1),

with b > 0 and f a continuous function. Is there a solution? Is itunique?

c) Find a function G(x, t) on [0, 1] x [0, 1] such that the solution of (E)is

1) = G(x, t.) f (t)dt.fo

y(xd) Is G(x, t) continuous? Does it satisfy G(x, t) = G(t, x)?e) If the equation

-1!"+by=1iy, y(0)=0, y'(l) =y(1),has a non-trivial solution, what can we say about ?


0.45. Exercise. Consider the differential equationx' =X 2 + t, x(0) = 0,

where t -s x(t) E R is a function on a neighbourhood of 0 E ita) Show that a solution exists on (-b, a) with a < 3.b) What can we say about b?

0.46. Exercise. Consider the differential equation y' = x - Then x -'Y(x) E R is a function defined on an interval of R. Prove that it has a uniquesolution on (0, oo) which is positive and which tends to zero when x -+ oo.0.47. Problem. In this problem the given functions and the solutions aredefined on R, with values in it They are even and periodic of period 2ir. Eowill be the space of continuous functions on R which are even and periodicof p e r i o d 2sr. For k E N, we have Ek = { f E Eo I f E &I. Let C,06 bethe space of the bounded continuous functions on R endowed with the normIIf IIco = sup if I, and P,.(x) the set of the functions in El whose Fourier serieshave vanishing coefficients ak when k > n (PR(x) is a linear combination ofthe functions cos kx with 0 < k < n).

Part Ia) Let h E Eo. Show that for the equation y" + y' cotan x = h(x) to

have a solution in E2 on the open set Cl C R, where tanx 0, it isnecessary and sufficient that h(x) sin x dx. What can we say aboutuniqueness?

b) For p a real number, verify thatF:y -y"+i'cotanx-py

is a map inPP(x).Let f, E PR(x), and prove that if p > 0 (which is assumed hence-

forth), then the equation(1) y"+y'cotanx-py= fR(x)

has a unique solution in E2 (that means that the function in E2satisfies (1) on fl).

c) Let f E El, and denote by fa the partial sum up to order n of theFourier series of f. For each n E N, we consider the solution yn inE 2 of equation (1). Given k p o i n t s x1, .Z, . , xk of 10, ir], show thatthere exists a subsequence {yp} C {y.} which converges at these kpoints.

d) Prove that there exists a subsequence of {y,} which is a Cauchysequence in COD. Deduce that the equation

(2) y"+y'cotanx-py= f(x)

Exercises and Problems 17

has a unique solution in E2.e) Since f E El, what is the regularity of the solution?

Part HWe next study the equation

(3) z" + z' cotan x + h(x) = f (x)e"=,where h, f belong to Ep and v E R, f 0 0 and v 96 0.

a) Reduce the study of (3) to the study of the equation(4) y"+y'cotanx+a= f(x)ey,

where a is constant and f E Eu. When f has a constant sign, establishthat for (4) to have a solution in R2, it is necessary that a have thesign of f .

When a = 0, verify that for (4) to have a solution in E2, it isnecessary that f changes sign and that fo f (x) sin x dx > 0.

b) For the rest of Part II we suppose a>0and f(x)>0forall xER.Exhibit two real numbers m and M such that f (x)em < a < f (x)eMfor all x E R. Then consider the sequence of functions defined byinduction as follows: cpo = m and, for k > 0, pk is the solution in E2of the equation

(5) Wk + Vk cotan x - f (x)elk-' -- a - Wk-1,where p > 0 is a real number. Prove that Wr > 00.

c) If p is chosen large enough, establish that the sequence {Wpk} is in-creasing and bounded by M.

d) Prove that {cpk} is a Cauchy sequence in C.Deduce that equation (4) has a unique solution in E2. What is

its regularity?Part III

In this part we suppose a < 0.a) Verify that for (4) to have a solution in E2, it is necessary that f

be negative at least somewhere. When f (x) - -2 and a = -2, theequation

(6) y" + y' cotan x + 2ey = 2has an obvious solution yo. But in fact there exists a one-parameterfamily of solutions yt of (6) in E2, yt being Cr in a neighbourhood oft. Find the equation (7) satisfied by w = (dyt/dt)t--o.

b) Find the solutions of (7) in E2. Let tJi be one of them, i1' 0.


c) Find solutions of (6) in E2 of the formy = k log [p(1 + E V,)],

where k, la and a are real numbers to be chosen. Find all the solutionsof (6) in F-1-

d) If equation (4) has one or more solutions in E2, let y be one of them.Prove the following identity:

ffIsir2xevix=_(1+a/2)jfev sin 2x dx.When a = -2, show that equation (4) does not always have a solution,even if the necessary condition found in III a) is satisfied.

Specialists will recognize the Kazdan-Warner condition for theso-called Nirenberg problem (see Aubin [2]).

Chapter 1

DifferentiableManifolds

One begins a new field in mathematics with some definitions, and this courseis no exception. There are many definitions, especially at the beginning. Thesubject of our study is differentiable manifolds. It is necessary to understandwell what a differentiable manifold is.

We give the proof of the theorem on partition of unity, very useful indifferential geometry. This proof needs point-set topology. The reader isassumed to know the definition of a topology and that of a compact set (seeChapter 0). But what is useful throughout the book is differential calculus.One must know what a differentiable mapping is, and the Cauchy Theoremon ordinary differential equations.

This chapter continues with the definition of a submanifold. To provethat a subset of a manifold is a submanifold. using the definition, seems tobe difficult; fortunately we have at our disposal Theorem 1.19, which willbe very useful for applications.

We end the chapter with two basic theorems, Whitney's and Sard's.We give the difficult proof of Whitney's theorem, because it is a beautifulapplication of the knowledge already acquired. The reader may skip theproofs of Whitney's theorem and the theorem on partition of unity.

Basic Definitions1.1. Definition. A manifold Mn of dimension n is a Hausdorff topologicalspace such that each point P of Mn has a neighbourhood Cl homeomorphicto Rn (or equivalently to an open set of Rn).

19

20 1. Differentiable Manifolds

More generally, we define a Banach manifold: each point has a neigh-bourhood homeomorphic to an open set of a Banach space. Here we willstudy only manifolds of finite dimension.

The notion of dimension makes sense because there is no homeomor-phism of Rn into RP if n 96 p. We do not prove this main result, becausewe will study differentiable manifolds, for which the proof of the notionof dimension is obvious. We will only consider connected manifolds. If amanifold has more than one component, we study one component at a time.

1.2. Proposition. A manifold is locally compact and locally path connec-ted.

By definition, locally compact (resp. locally path connected) means thatevery point has a basis of compact (reap. path connected) neighbourhoods(a family of neighbourhoods is a basis of neighbourhoods at P, if any neigh-bourhood of P contains a neighbourhood of the family). A set E is pathconnected if, given any pair of points P, Q in E, there is an are in E from Pto Q.

An arc of M is the image, by a continuous map, of [0,1] C R into Mn.If I' is an arc in R" and cp the homeomorphism of Definition 1.1, W-1(I') isan arc in Mn. If there exist an are from P to Q and another from Q to T,their union is an are from P to T.

Let P E Mn and let it be a neighbourhood of P homeomorphic to anopen set of Rn, fl i- R". In this chapter, B,. will be the ball of R" withcenter 0 and radius r. As often, we suppose without loss of generalitythat cp(P) = 0. r = 1, 2, ... , p.... (p E N). form a basis of

Basic Definitions 21

neighbourhoods of P which are compact and path connected.

'p- I

1.3. Proposition. A connected manifold is path connected.

Proof. Let P E M,,, and let W be the set of points Q of Mn for whichthere is an are from P to Q. W is closed. Indeed, let T E W, and U aneighbourhood of T homeomorphic to R". We have U f1 W 0 0; thus thereexist Q E U fl W and an are from P to Q; there is also an arc from Q to T.Hence TEW.

W is open. Indeed, let Q E W; Q has a neighbourhood f2 homeomorphic toR" and 0 C W. Since M" is connected and W 0 0 (P E W), W = M".1.4. Definition. A local chart on M" is a pair (ft, cp), where fl is an openset of M" and 9 a homeomorphism of i2 onto an open set of R". A collection((T.j, Wi)ie1 of local charts such that UfE1 "4 = M" is called an atlas. Thecoordinates of P E fl related to the local chart (12, co) are the coordinates ofthe point rp(P) in R".1.5. Definition. An atlas of class Ck (respectively CO', C)? on M, is anatlas for which all changes of charts are Ck (respectively C, C). That isto say, if (fL,, spa), and (f2q, p,3) are two local charts with QC n Qq 0 0, then

22 I. Differentiable Manifolds

the map FAQ o cp,1, called change of charts, of VS(f Halo) onto cp,(i2Q nits)is a diffeomorphism of class CA (respectively C,Cw).

Rn D Vo(0Q n its) % 12 n fts f?. W.(n. n its) c Rn

We consider the following relation of equivalence between atlases of classCk on Mn: two atlases (U;, P+)iEI and (We, *Q)QEA of Class Ck are said to beequivalent if their union is an atlas of class Ck. That is to say that gyp, o * 1isCkon*,,(UtnWQ)when U,nWQ#0.

1.6. Definition. A differentiable manifold of class Ck (respectively. C orCI) is a manifold together with an equivalence class of Ck atlases (respec-tively, C or CO).

On a manifold there need not always exist a differentiable atlas (of classCk), but if there exists an atlas of class C', then time are atlases of class C(which are Cl-equivalent to it) if the manifold is paracompact. It is possiblenow to talk about differentiable functions Ck on a Cr-differentiable manifoldwhen k < p. A function f on Mn (unless we say otherwise, a function takesits values in R) is Ck-differentiable at P E M. if for a local chart (U, ip)with P E U the function f o V-1 is Ck-differentiable at W(P). We easilyverify that this definition makes sense--the notion of differentiability doesnot depend on the local chart. Indeed, let (i2, 0) be another local chart atP; then f otji-1 = f orp-1ocpoY,-1 is Ck-differentiable at 10(P) since Vo,1,-1

is Ck-differentiable because k < p.

1.7. Remark. We can define complex manifolds M. Consider an atlas oflocal charts (f)j, cpj)iE1, where Bpi is a homeomorphism of li onto an openset in C. If any change of charts Vj o cpi 1 is holomorphic on pi(S2j fl S2j),M is a complex manifold of complex dimension m (n = 2m).

1.8. Example. An open set Il of a differentiable manifold M,, is a differ-entiable manifold. It is endowed with the atlas (Ut, ' ){E, obtained fromthe atlas (UU, j)jE, of m. by setting U1 = U; fl 12 and letting cpj be therestriction of cp, to U1.

The sphere S, is a compact analytic manifold.

Proof. Let us consider the unit sphere S, C Rn+1 centered at 0 E R"+1,with P and T the north and south poles of coordinates zi+1 = 1,zj = 0 for 1 < i < n in Rn+1 We define the charts (it, c) and (8, vp) asfollows: n = S, \{P}, 8 = &\{T}; for Q E 0, cp(Q) = Q1, the intersectionof the straight line PQ with the hyperplane II of equation zn+1 = 0 ({p isthe stereographic projection of pole P); and for Q E 8, r(Q) = Q2, theintersection of the straight line TQ with H.


Obviously (0, So) and (8, t1') form an atlas A for S,,: 0 U 9 = S and IIis identified with R". What is the class of A?

Let (r, a) be polar coordinates for Ql and (p,w) polar coordinates forQ2. Thus r = OQI, P = OQ2i a = w and rp = OP.OT = 1. On S2 f16, thefinction apoljr-1, which is defined by a = w and p r = 1, is analytic sincep and r are not zero on SZ (10. We can see that, in Cartesian coordinates{x'} for Q1i {y'} for Q2 we have y'/p = x=/r; thus

x'and x' - y`y

1(x)2 'j 1(yj)21.9. The real projective space P,,(R) is a compact analytic manifold.

Proof. Recall that Pn(R) = (Rn+1 - {O})/R, where R. is the followingrelation of equivalence: x - 1 in Rn+1- {O} if there exists p E R such thatx=pi.

Let U ; (i = 1, 2, , n + 1) be the set of points of Rn+1 whose Ohcoordinates are not zero.

The open sets Sgt = U,/1 cover P,,(R): L)7+1 SZt = ,,(R). We considerthe atlas A = {f?,, Sot}1

Partition of Unity 25

To simplify the writing let us consider p o +t If a E thenx = {x'} with x" and, xx"+1 nonzero. For j < n we have yin+1 = z^I /x"+tand p = xi/xn. But yn+I = xn/xn+1 = 1/y;

+ 1ltn+1 = y /y for j < n.This change of coordinates is analytic since yn 36 0.

To prove that Pn(R) is compact, we consider in each Sti a compact Kisuet that P"(R) _ I 1"+'K- . For instance, lets E Pn(R), x = {x'} and x'Obe s u c h that Ix'I Ix - I f o r all 1 < i < n + 1. In fib, we consider Kc,, theset of s such that Ix'I < Ix'-) for all i. In R" the set K defined by Iy'I < 1for 1 < j < n is compact; thus K,. = W-1 (K) is compact.

Partition of Unity1.10. To study a manifold, we will have to glue together the charts of anatlas, and use partitions of unity. This is why we will suppose the manifolds

compact (the weakest topological hypothesis which implies the existenceof partitions of unity).

A topological space E is paracompact if, to any covering of E by openset f2; (i E I), we can associate a covering, locally finite and thinner, byopen sets 0j. Locally finite means that any point has a neighbourhood Wsuch that Z n W # 0 except for a finite set of indices i. Thinner meansthat 9 C fl; (with this notation, some 0, may be the empty set). Eis countable at infinity if there is a family of compact sets Kq (q E N)

0 0such that KI CK2C ... C Kq CKq+tC ..., the union of the K. being E(E==1Kq)1.11. Theorem. A paracompact manifold is the union of a family of con-nected manifolds which are countable at infinity.

Proof. The manifold is the union of its connected components; let Vn beone of them. Let { flp } pE V be a family of open sets houteomorphic to R"with lip compact. Since V" is paracompact, there is a locally finite cover{0{} thinner than the cover {flp}. Pick 91 96 0 and set K1 = BSI; thenby induction we define Kq = U3EJa e', J. being the set of indices j forwhich 9 f n Kq_I 0. The set Kq is compact, as a finite union of compactsets; indeed we verify below that the set of open sets 0i which have anonempty intersection with a given compact set K is finite. Let Up be anopen neighbourhood of P such that 9; n Up = 0 except for a finite set ofindices i. The family { Up } covers K; thus there exist Up, , Up2,... , Up,which cover K. Now if 9, n K # 0, then 9, n up O 0 for at. least one Up,,and the set of these 0, is finite.


Moreover K._1 C UiEJq 9j =Kq, so W = t 1 Kq is open. But W isclosed too. Indeed, for P E W, let 9j, be one of the open sets of the coveringwhich is a neighbourhood of P. We have Wfl9;" 36 0, thus 0j,, has nonemptyintersection with a compact set Kq, and P E 0; C K9+1 C W = V, sinceV" is connected and W * 0. Thus it is proved that V" is countable atinfinity.

1.12. Theorem (Partition of unity). On a paracompact differentiablemanifold of class Ck (respectively C0), there exists a Ck (respectively CO)partition of unity subordinated to a given covering.

From now on, manifolds are always paracompact; we will mention it nomore. Let {9;bE1 be a covering of a manifold by open sets. A partition ofunity subordinate to the covering {9f} is a family of functions {(x;}jE j withthe following properties:

i) suppa; C 9;.ii) Any point P has a neighbourhood U such that Unsuppoj = 0 except

for a finite set of ai.iii) 0

Differentiable Mappings 27

Differentiable Mappings1.18. Definition. A Ck mapping f of a differentiable manifold Wp intoanother Mn is called C'-differentiable (r < k) at P E 9 C Wp if 0 o f o v-1is Cr-differentiable at ip(P), and we define the rank of f at P to be the rankof 1# o f o


In example (iii) the equation of the mapping for t < 0 is x(t) = t,y(t) = 0 and for t > 1 it is x(t) = 0, y(t) = t. In example (iv) theequation of the mapping is, in polar coordinates, 8 = -t, r = i for t > 1.(i) is not an immersion, rank f = 0 at 0. (ii) is an immersion; we haverank f = I everywhere; but it is not an injective immersion. (iii) is aninjective immersion, but it is not an imbedding. Indeed, f ((-e, +e)) for anye > 0 is not an open set in the induced topology. In the induced topology, aneighbourhood of 0 contains open sets of the form f ((-e, +e)) U f ((E , oo)).(iv) is an imbedding. Here there is no difficulty; in the induced topology,a neighbourhood of 0 contains open sets of the form f ((1, oo) ). (v) is animbedding.

1.17. Definition. Let M, M be two Ck differentiable manifolds. M iscalled a covering manifold of M if there exists a differentiable mapping II(called a projection) of M onto M. such that for every P E M:

i) II-1(P) is a discrete space F, andii) there exists a neighbourhood 11 of P such that 11' (Q) is diffeomor-

phic to flxF.

Each point P E 11-1(P) has a neighbourhood f2' C M such that therestriction II' of II to SY is a diffeomorphism of IT onto Q.

The map II is a 2-sheeted covering if F consists of two points.

Af

!1

Submanifakls 29

Submanifolds1.18. Definition. A submanifold of dimension p of a differentiable mani-fold Mn is a subset W of Mn such that for any point of W there exists alocal chart (fl, gyp) of Mn, where o(1l) is an open set of the form 8 x V with9 c RP and V c R"-P, such that V(flnW) = 8 x {0}. W is endowed withthe topology induced from Mn.

Thus there exists a system of local coordinates (x 1, , xn) on Sl suchthat Wp is locally defined by the equations xP+1 = xp+2 = ... = xn = 0. WPis endowed with a structure of differentiable manifold induced from Mn. Weconsider for Mn an atlas (fl,, V,);E, of local charts as above. Then (fli, p,),E jis an atlas for W of the same class with ft; = fli n w and Bpi the componenton 8i of the restriction of gyp; to f2,.

Considering this definition, it seems difficult to prove that a subset Wof Mn is a submanifold. Fortunately there is the following theorem, whichis very convenient.

1.19. Theorem. A subset W of M. defined by a set of n - p equationsf, (P) = 0, ... , fn-_p(P) = 0, where fl, , fn-p are C1-functions on Mn, isa differentiable aubmanifold Wp of Mn if the map of Mn into R"-P definedby P (f1(P), ... , fn-p(P)) is of rank n - pat any point P E W.Proof. Let xo be a point of W C Mn, and (fl, gyp) a local chart of Mnwith xo E fl, (x1, x2, ..., x") the corresponding coordinates. One of thedeterminants of the (n - p) x (n - p) submatrices of the matrix ((8f/9z-'))is nonzero at xe. Without loss of generality, let us suppose that it is the onewhere j =p+ 1,p+2, n.

According to the inverse function theorem, there is a neighbourhood8 C fl of x0 on which as coordinates of P E ewe can take y1 = x1, ... , yP =xP, f+l = fl,... , y" = fn-p. Let aP be the homeomorphism defined on 0by P --- {y1} E R", 0(8 n W) CRP x {O}. Here W is a differentiablesubmanifold of Mn.

1.20. Example. An open set 9 of a manifold M. is a submanifold of di-mension n. Let us consider an atlas (Sli,,p,),EJ for Mn; then (fl,, p,),EJ withfli = fls n e and 0, _ w, /f4 is an atlas for e. gyp; (fl,) is an open set of Rn.

The set Sn(1) of the points x = {x'} E Rn+1 satisfying fl(x)=,'(x')' - 1 = 0 is a submanifold of Rn+l. The rank of the matrix((8f1/8x1)) is 1 on Indeed, the derivatives are 8fi/8xa = 20, andthe matrix is never 0 on Sn (1) since (xi )2 = 1.

The set of matrices T (n, p) with p rows and n columns is a normedvector space. If aj are the components of the matrix M E T(n, p), we set


JIM 11 = sup la-,')- So the bijection of T(n,p) onto ilt"P defined by ((a=)) -{xk} with xi+"(i-1) = a; is a homeomorphism. Here one chart covers themanifold T(n, p).1.21. The set T(n, p, k) C T(n, p) of matrices of rank k is a submanifold ofT(n, p)

Proof. In order for the rank of M E T(n,p) to be greater than or equal tok. we have to have that one of the determinants DQ of the submatrices k x kof M not zero. The set Ea of the points satisfying Da # 0 is an open setof T(n,p), and their union is an open set of T(n,p); it is a submanifold ofT(n, p) which is of dimension np (an open set of a manifold is a submanifold).

Thus if k = inf (n, p), we have proved that T(n, p, k) is a submanifold ofT(n,p) of dimension np.

If k < inf (n, p), then at a point M E T (n, p, k) at least one of thedeterminants D(, is not zero. Let us suppose Dl 96 0; we have to show(and it is a necessary and sufficient condition) that all the (k + 1) x (k + 1)determinants of the type

1 k k+aa1 ... a1 a1

Da3(M) = al ak ak;k kk+8

... ak+Q ak+9

vanish with 1 < a < n - k and 1 < 0 < p - k.The map r : T(n,p) - W"-k)(n-k) defined by

M -i {D11(.M), 1 ) 1 2 ( M ) , . . . , D"-k,v---k(M)}is of rank (p - k)(n - k) on El. Indeed, the partial derivatives with respecttoxx with A = k+v+n(k+,u -1), 1 : 5 1

The Whitney Theorem 31

We will prove a weaker theorem: every connected C2 -differentiable man-ifold has an immersion in R2' and an imbedding in R2"+1. The steps of theproof are Propositions 1.25 and 1.27.

In the first lemma we will prove that we can perturb the map f a littlebit, locally in 9, so that the new map is an immersion on 11. With thesecond lemma we prove that if f is already an immersion on a compact K,we can do this perturbation so that the new map is still an immersion onK.1.23. Lemma. Let f (x) be a C2-differentiable map of an open set Sl C R"into RP (with p > 2n). Then for any c > 0 there exists an n x p matrixA = ((a; )) with Iai I < e for all i and j so that x f (x) + A.x is animmersion.

Proof. If J(x) is the Jacobian matrix of f at x, we want J(x) + A to beof rank n for all x E S1. That is to say, A must be, for any x, different fromthe matrices of the form B - J(x) with B a matrix of rank < n.

The map F of 11 x T(n, p, k) into T(n, p) defined by (x, B) -p B - J(x)is C'. When k < n the dimension of fl x T(n, p, k) is n+k(n+p- k) < n+(n - 1) (p + 1) < np - 1 (by setting k = n - 1, then taking p > 2n).

Therefore when k < n the image of St x T (n, p, k) by F is of zero measurein T(n, p), identified with R"" according to a well known theorem of measuretheory:

The image by a C1-mapping of an open set of R" into R' (n < p) is ofzero measure in R".

Recall that a set A C RP is said to be of zero measure if for anyc > 0, there exists a sequence of balls A. such that A C U 1 Bi andi= , vol Bx < E. In particular, if the measure of A is zero, no open set,except the null set, is included in A. Thus the interior of E =F(f1 x Lrk--0 T(n, p, k)) is empty, and we can choose A E T(n, p) not inE as close as we want to the zero matrix.1.24. Lemma. Let f be a Cl -map of M, into R" (p > n). If the rank of fis equal to n on a compact set K C 91, (11, gyp) a local chart, then there existsrl > 0 such that for any C'-map g satisfying IIJ(g)II < rl on K we have f +gof rank nonK.

Proof. We write J(g) for the Jacobian matrix of g. Let 6(x) be themaximum of the absolute values of the determinants of the n x n submatricesof J(f) at x. 6(x) is positive and continuous on K; thus there exists 6 > 0such that 6(x) > 6 for all x E K. A determinant is a continuous functionof its components; therefore there is an it > 0 such that, if IIAII < +l, thematrix J(f) + A is of rank n on K.


1.25. Proposition. Let f be a Ck-map (2 < k < oo) of the connected andCk-differeentiable manifold Mn into RP (p > 2n) and let be a continuousfunction everywhere positive on Mn. There exists a Ck-immersion g of M,into RP such that II f (P)

- g(P)II

The Whitney Theorem 33

Let {S4}, i E N, be a sequence of such compact neighbourhoods whichform a locally finite covering of M (as in Proposition 1.25), f i being ho-meomorphic to B3. Let {bi} be a sequence of points of R2"+1 which we willchoose later. Let us say already that we will choose bi satisfying

Ilbill io.

The limit of the sequence hi is a Ck-differentiable immersion h which isinjective. Indeed, by contradiction, let us suppose that there exist f1 andP2 such that h(P1) = h(P2).

Let jo be an integer such that for j > jo we have 'y, (P1) = y, (P2 )and hj(P1) = h(P1) = h(P2) = hl(P2). Now hr+1(Pi) = hr+i(P2) giveshr(Pi)+bryr(P1) = hr(P2) +brryr(P2). Since br Gr(Dr), this implies thatyr(Pi) = 7r(Pz) and hr(P1) = hr(P2). By induction, for any i, we havehi(Pi) = h,(2) and' (P1) ='yi(P2)

At least one y. (A) is not zero, say yip (Pi) 0 0. Thus yip (P2) is notzero. So Fl and P2 belong to S, where g = hl is injective, which is incontradiction with h1(Pi) 96 hi (Pi) if Pi # P2.1.28. Theorem. Every Ck-diferentiable manifold (2 < k < oo) has a Ck-imbedding into R2"+1, and its image is a closed subset of R2"+1.

Proof. According to Proposition 1.27, there is an injective immersion hof M., into R2n+'. In order that h be an imbedding, it is necessary andsufficient that for any compact set K C R2"+n, h-1(K) is compact. Indeed,since h is injective, h will be then a homeomorphism of M" onto h(M").

We will choose f and ay for that. Pick 0 1 and all the components off (P) zero except the first; we choose f1(P) _ F 1 jy j (P). We have thath(P) E K implies IIh(P)lI 5 p = sup }IxII for x E K and IIf(P)II


The Sard Theorem1.29. Definition. Let M and W p be two C differentiable manifolds ofdimension n and p respectively, and let f be a map of class Ck of M intoWp. The points of M" where rank f < p are called critical points of f .All other points of M are called regular points. A point Q E W. such thatf -1(Q) contains at least one critical point is called a critical value. All otherpoints of WP are called regular values.

When f is a smooth real valued function (WV = R), P is a critical pointoff if all the first derivatives off at P vanish, since (rank f)p = 0. Thegradient at P vanishes. A critical point P is called non-degenemte if andonly if the matrix ((8a f /8x' 8xi)) p is non-singular.

The index of f at P is the number of negative eigenvalues of((a'f/8x'8xi))p. M. Morse proved that any bounded smooth funetlonf : M R can be uniformly approximated by smooth functions whichhave no degenerate critical point (see Milnor 18]).

If n < p, all points of M,, are critical since rank f < inf(n,p) < p, andthe critical values form a set of measure zero. This is obvious according toProposition 0.28.

But the Sard Theorem asserts that this result holds in general if the mapf is Cl: the set of critical values is of measure zero. We are at the beginningof the course (in Chapter 1) and we have only seen some definitions and a fewtheorems, but nevertheless we were able to prove a very important theorem,the Whitney Theorem, which shows that a differentiable manifold, whosedefinition is abstract, is nothing else than a surface of dimension n in RPfor p large enough. And now it is possible to give a second very importanttheorem:

1.30. Theorem (The Sard theorem). Let M" and W. be two connectedCG differentiable manifolds of dimension n and p respectively, n > p, andlet f be a map of class Ck (k > 1) of M into Wp. If k > it - p + 1, thecritical values form a set of measure zero.

In case the manifolds are only C'', we have Sard's theorem for any p ifand only if r > n.

Proof. Since the manifolds are separable (they are connected), we have onlyto prove the result for a local chart (fl, So) of M and a local chart (0,,0) ofW such that f (Sl) C 9.

Thus the theorem will be proved if the result holds when M is an openset of R" and W = R. Indeed, we will only have to consider the map

0 0 -of into tp(A) C RP.

The Sari Theorem 35

Thus henceforth f is a Ck map with k > max(n - p + 1, 1) of an openset fl C R" into RP, and A is the set of the critical points of f . When n < pwe proved the result in 0.28. Therefore we assume n > p.

First step. The Sard theorem is true for functions (p = 1).As in the proof of Proposition 0.28, we only have to prove the result

when Mn is a unit closed cube C of R". The proof is by induction on thedimension n. Suppose n = 1 and f E C'. Then x E S2 C R" is a criticalpoint off if and only if f(x) = 0. Moreover, as f is C' on the compactset C, for any c > 0 there exists m E N such that Iy - zI < 1/m impliesIf (v) - f'(z)I < E.

Thus Iy - xI < 1/m implies I f'(y)I < e, and according to the mean valuetheorem If (x) - f (y) I < e/m.

Now we proceed as in the proof of Proposition 0.28. We divide C intom intervals of length 1/m. Let J be one of them which has a critical point.We have mess f (J) < s/m. The set A of critical points of f is covered bysome intervals like J. The number of these intervals is at most m, of course.So mess f (A) < me/m = E. Since E is as small as one wants, meas f (A) = 0.

The result is true for n = 1. Now we suppose, by induction, that if 9is an open set in R"-1, and g is a Ci-1 map of 9 into R, then g(B) hasmeasure zero, B being the set of critical points of g. The result is also trueif 8 is a CO separable manifold, according to the begining of the proof.

Let Ak (1 < k < n) be the set of critical points x of f such that allderivatives of f of order less than or equal to k vanish at x, with x Ak+1when k < n. If x E An, for any c > 0 there exists m E N such thatIlv - xII S nl/2/m implies that the norm of the differential D"f (y) of ordern at y is less than E. Thus Ily - xII < n1/2/m implies

11f (y)- f(x)II


Second step. Let

Aj = {x E f2 I rank D f (x) = j }, A= U Aj.0 n. If not, we have two more lemmas to prove.

The Sard Theorem 37

1.33. Lemma. Let f be a function of class Ck on a closed unit cube in Rn,and A the set of critical points of f. Then A = A,, U A with Ao countableand A such that, for any pair (x, y) in A,

1f(x) - f(y)I When kp > n, the right hand side is as small as one wants. Hence mess f (A)=0.1.35. Remarks. When k > max(n - p + 1, 1), we have k _> n/p, sincen - p + 1 > n/p if n > p. The assumption on kin Sard's theorem issharp; Whitney (see Sternberg [141) gave a counterexample in the case whenk < max(n - p + 1, 1).1.36. Corollary. The set of regular values of a C map f of Mn ontoWp is everywhere dense in Wp. If Q is a regular value, then f (Q) is asubmanifold of Mn of dimension n - p.

V = f-1(Q) is defined locally by a set of p functions (f1,f2,... fp) = f,

and we know that rank f = p; so we have to consider a local chart at Q.


Thus, according to Theorem 1.19, V is a submanifold of M,a of dimensionn - p.

Exercises and Problems1.37. Exercise. C = {z E CJIzI = 1} being the unit circle centered at 0in C, we consider the map p of R into the torus T = C x C defined byR 3 u -+ V(u) = (e2"', e2'"u) E T. What can we say about

Exercises and Problems 39

a pointaEUi,aadlet -1, 1,e+1,..., -+')Ez. Vi (i) will be,the point f o f C- of c o m p l e x Coordinates 1 , t + 1 ,

a) We define a subset M of P3(C) byM = {i E P3(C) I t2t3 = t1t4, (t2)2 = t1S3, (S )2 = ee}

where z = ((1, g2, 3 4) E i. Show that M is a compact submanifoldof dimension 2 of P3(C).

b) Consider the map 4 : P'(C) --+ P3(C) defined byP1(C)32 -41(2)=iEP1(C)

with Z = (rl1,f12) E C2 \ {0} andz =

[(t I)3, (g1)2g2,111(112)2, (172)3] E C \ {0}.Prove that -6 is a diffeomorphism of P1 (C) onto M.

1.42. Exercise. Let M,, be a COO compact manifold of dimension n.a) Exhibit a finite cover of Mn by a family of open sets 9, (i = 1, 2,..., p)

homeomorpllic to a ball of R" such that 9, C f2i with each fli homeo-morphic to a ball of R. Prove the existence of C functions f, onM" satin ying 0 < f= < 1, supp f; C 1'2; and f; (x) = 1 when x E 9s.

b) Deduce the existence of an imbedding ip of M" into R9 with q =(n + 1)p.

1.48. Eiunrcise. Show that a proper injective immersion is an imbedding.1.44. Exercise. We identify R4with the set of 2 x 2 matrices.

a) Show that the set M2 of 2 x 2 matrices whose determinant is equalto 1 is a submanifold of R4. What is its dimension?

b) Prove that the tangent space to M2 at 12 = (o ) may be identifiedwith the set of matrices of zero trace.

C) Show that the set Mn of n x n matrices whose determinant is equal to1 may be identified with a submanifold of R"2 . What is its dimension?

d) Characterize the tangent space to Mn at the unit n x n matrix In.1.45. Exercise. Let E be the set of straight lines in R3.

a) Establish a bijection between E and the quotient set of P2 x R4 byan equivalence relation (P2 the real projective space of dimension 2).

b) We endow E with the structure of a topological space (the finestpossible) such that a is continuous. Is ir open? (A map is open if theimage of any open set is an open set.)

c) Show that E is Hausdorg.


d) On E define a structure of an analytic manifold.Hint. Consider the open sets 0, = ir(fj x Rs), where the open

sets f2; (i = 1, 2, 3) cover P2 as in the course.e) Using the proof of the previous question, show that E is a vector fiber

bundle. Characterize its elements. What is its dimension?f) Let S be the unit sphere in R3. Prove that the set of tangent straight

lines to S is a compact differential submanifold of E. What is itsdimension?

Hint. Apply in $; a theorem of the course. For the compactnesslook at the proof of compactness of P2.

1.46. Exercise. Let Mp and W. be two C differentiable manifolds ofdimension p and n respectively, and let f be a CO0 map from M. into W.We say that f is a subirnnersion at x E Mp if there exist a neighbourhoodn C Mp of x, a neighbourhood 0 C W of f (x), a C manifold V, andtwo C maps g and h such that g is a submersion of fE into V and h is animmersion of V into Wn with f lo = h o g.

a) Prove that f is a subimmersion if and only if the rank of f is constantin a neighbourhood of x.

b) Shaw that the function Mp ? x --+ r(x) =rank off at x is lowersemicontinuous.

c) Prove that the subset of M. where f is a subimmersion is dense inM.

Solutions to ExercisesSolution to Exercise 1.37.Sp is everywhere of rank 1; thus p is an immersion. If cp(u) = W(v), we

would have u-v E Z in order that e2`" = 0`", but also u-v E xZ in orderthat e2i" = e2". Since 7r is irrational, this is impossible except if u = v.

Therefore cp is an injective immersion. Let P = (e2i", eu"") be a pointof V(R). If cp is an imbedding, there would exist an open set Cl of T whoseintersection with o(R) would be a given connected arc ry of w(R) throughP. We can show that, as close as one wants to P, there are points of V(R)which do not belong to 7. It is well known that for any a > 0, there areintegers n and p such that tpsr - nJ < a. Set v = u + n, e2'"a = e2i'"' andJe2'" - e j = 1e24" -1J = Je*n ) -1J < Jean -11, which is as small asone wants.

Finally, let us prove the well known fact mentioned above. For P E Nwe set xp = inf(prr - n) for n E N, n < par. So xp = par - np, and the set{xp}pEN C 10, 1 (.

Solutions to Exercises 41

Let {xp; } be a subsequence which converges in (0, 11. So we have(P.+1 - Pi) r - (rb,+, - rip,) - 0.

Solution to Exercise 1.38.If (x, y) -- i f (x, y) is everywhere of rank 1 on Wp = f V (P)],(P)], then

Wp is a submanifold imbedded in R2 according to Theorem 1.19. Now8f /8x = 3x2 + y vanishes when y = -3x2, and Of/ft = 3y2 + x is zerowhen x = -3ya. These two equations have for solution (x = 0, y = 0) and(x = - j, y = -g ). Set f1= (0,0) and Q = (-', -' ).Then f(0,0) = landf(-,-3) =1+1/27.

Let us study Wn. WO \ fl is a submanifold. We have to see what happensat Q. The equation of Wn is x3 + xy + y3 = 0. If x is small with respectto y, the equation is xy + y3 = 0. y = 0 is impossible if x:1- 0, but we havean arc where x - -y2. By symmetry we have an arc where y - -x2. f2 isa double point. At f2 there are two arcs, one with tangent y = 0, the otherwith tangent x = 0. There is nothing more in the neighbourhood of Cl, sinceif we suppose x - ay with a 96 0, we find aye = 0, which is impossible. Letus study WQ. Wq \ Q is a submanifold. In a similar way as above, we provethat Q is an isolated point. So WQ is not a submanifold.

Therefore the set of points P for which Wp is a submanifold is A =f -'((-oo,1) U (1,1 + 1/27) U (1 + 1/27, oo)).

Let D be the line of equation x + y = 1/3. f (D) = 1 + 1/27. Thecomplementary set of A is Wn U WQ. That is, {Q} U D and a curve throughCl asymptotic to D. This set is symmetric with respect to the line y = x.


Solution to Exercise 1.39.II-1(P) is compact (a closed subset of a compact set), and II-1(P) is

a set of isolated points since n is locally a diffeomorphism. Thus II-1(P)is a finite subset of W. We will show below that cardII-1(P) is locallyconstant. This will imply that c a r d II-1(P) = Constant, since V is con-nected. L e t P i (i = 1, 2, , m) be the points of II-1(P). There are disjointneighbourhoods f2 of Pi which are homeomorphic to a neighbourhood 9of P. Let {Qk} be a sequence in 9 which converges to P. Obviouslylimsupk, card II-'(Qt) > card II-1(P). Let us prove the equality by con-tradiction. If we do not have equality, there is a sequence {xk} in W suchthat f(xk)=Qkand xkVS2ifor all hand 1.

Since W is compact, a subsequence, noted always {xk}, converges to apoint x of W. x is not a point Pi, and, by the continuity of II, II(x) = P, acontradiction. II-1(9) is diffeomorphic to F x 9, F a set of m points. Fora counterexample take W = ] - oo, 0[ U ] 1, oo[ and II the identity map.

Solution to Exercise 1.40.We have 8y2r+1/&zi = 2x` and ey f 8x; = 2x1+'-' when a - n < i < a.

We verify that V is of rank n + 1 on R"+1 ` {0}. If x1 0, the first(n + 1) x (n + 1) determinant in Drp is equal to (2x1)"+1 # 0. If x1 = 0 andx2 3& 0, the first row is zero but the (n + 1) x (n + 1) determinant in Dcpwith the n + 1 following rows is equal to (2x2)"+1 76 0, and so on.

Let S be the unit sphere in Ri+1 and let do be the restriction of (pto S,,. We consider the map f of Sa into R20 defined by f(P) = Co' for1

Chapter 2

Tangent Space

In this chapter we introduce many basic notions. First we will study tangentvectors, then differential forms. We will give two different definitions of atangent vector at a point P E M, (they are dual to each other). Then,of course, we will prove that the definitions are equivalent. M,, is a C'differentiable manifold (r > 1) and (Q, W) a local chart at P; {s{} are thecorresponding coordinates.

We dune the tangent space Tp(M), P E M. It is the set of the tangentvectors X at P, which has a natural vector space structure of dimension n.The union of all tangent spaces is the tangent bundle T(M). We will showthat if r > 1, T(M) carries a structure of differentiable manifold of classC", which is a vector fiber bundle (T(M), ir, M) of fiber R" and basis M(T(M) B X -i ,r(X) = P E M if X E Tp(M)). Likewise we define thecotangent bundle T*(M). A vector field on M is a differentiable map f ofM into T (M) such that 7r o C is the identity. Thus a vector field X on M isa mapping that assigns to each point P E M a vector X(P) of Tp(M), anassignment which satisfies some regularity condition.

Likewise we define differential p-forms, exterior differential p-forms, ... .The notions of linear tangent mapping and linear cotangent map-

ping (4)*)p associated to a differentiable map 4 of one differentiable manifoldinto another are very important.

The linear cotangent mapping V allows us to transport differentiablep-forms in the direction opposite to that of the map

This chapter continues with the definition of the bracket [X, Y] of twovector fields X and Y.

43

2. Tangent Space

We will define the exterior product, the inner product and the exteriordifferential on the direct sum of exterior differential forms; our definition isan extension to exterior differential forms of the usual differential of differ-entiable functions. We proceed with the study of orientable manifolds andof manifolds with boundary, and conclude with Stokes' formula.

Tangent Vector

2.1. Definition. Consider differentiable maps 7i of a neighbourhood of0 E R into Mn such that -t(O) = P. Let (it, jo) be a local chart at P.

We say that ryl - % if V o -n and V o rya have the same differential atzero. We verify that this definition makes sense (it does not depend on thelocal chart). It is an equivalence relation R. A tangent vector X at P toM,, is an equivalence class for R.

2.2. Definition. Let us consider a differentiable real-valued function f de-fined on a neighbourhood 0 of P E Q. We say that f is flat at P if d(f oW'1)is zero at W(P).

This definition makes sense; it does not depend on the local chart. If(S2, 3) is another local chart at P, then, on n fl fl,

d(f o'P-1) = d(f o ca ) o d(V o cp 1 }.A tangent vector at P E M, is a map X : f --- X (f) E ]R defined

on the set of the differentiable functions in a neighbourhood of P, where Xsatisfies the following conditions:

a) IfA,pER, then X(Af+ug)=AX(f)+AX(g).b) X(f)=0iffisflat atP.

It follows from a) and b) that

Tangent Vector 45

c) X(f9) = f(P)X(9)+9(P)X(f)Indeed,

X(f9) = X{(f -f(P)+f(P)](9-9(P)+9(P)]},X(f9) = X V - f (P))(9 - 9(P))] + f (P)X (9) + g(P)X(f)

since X(1) = 0 (the constant function 1 is flat).Now d((figi) o w-'],(p) = d((fi o'-')(g1 o V-1)],(p) = 0 if fi and 91

are zero at P. Thus (f - f (P))(9 - 9(P)) is flat at P, and c) follows.

2.3. Definition. The tangent space Tp(M) at P E M is the set of tangentvectors at P.

Using Definition 2.2, let us show that the tangent space of Definition 2.3has a natural vector space structure of dimension n. We set

(X + Y)(f) = X(f) + Y(f) and (AX)(f) = AX(f).With this sum and this product, Tp(M) is a vector space. And now let usexhibit a basis. {x'} being the coordinate system corresponding to (f2, gyp),we define the vector (8/8x')p by

(0x;)p(f)_ I -9-T L(p)The vectors (a/ax')p (1 < i < n) are independent since (a/ax')p(x3) _

J,', and they form a basis. Indeed, as f - E 1(a f /ax') px' is flat at p,X(f)

X(xt) a') P] (f).axThe X' = X(x) are the components of X in the basis (a/ax')p. Observethat the expression of X (f) contains only the first derivatives of f.2.4. Proposition. The two definitions of a tangent vector are equivalent.

46 2. Tangent Space

Let y(t) be a map in the equivalence class 7 (7(0) = P), and f a real-valued function in a neighbourhood of P.

Considering the map X : f -- (8(f o ry)/8t]t__o, we define a map 'Pof the set of tangent vectors (Definition 2.1) to the set of tangent vectors(Definition 2.2), ' : X. Indeed, since

d(f o7) = d(f ov-1 ocpo7) = d(fif -y1 - -t2 we have

ra(fat71)lc=o = (a(J)lc=o'because by definition [d(v o yl)]t=o = [d(cp o 72)]t-o. Moreover, X is atangent vector (Definition 2.2): (a) is obvious, and if f is flat at P, than(8(f o -y)/&)t--o = 0 since (d(f o cp-1) pi = 0.

Let us show now that ' : ry --- X is one-to-one and onto. Let Xbe a tangent vector (Definition 2.2), X = E 1 X'(8/8ci)p. Consider themap ry: (-e, e) ? t -p y(t) E Mn, the point whose coordinates are {tX{}(we suppose that p(P) = 0 E ft"). Then

(8(f 0'Y)1 _ c 8(f o V-') 8(tXi)_ X(f}.

at J t_o r 8x 8tSo 'P is onto. Moreover, if yl is not equivalent to y2i then [d(+p o y1)] t=6 #[d(+p o y2)]t=o, and it is possible to exhibit a function f such that

[d(f o -tl)]t= * [d(f o -2)]t=0.2.5. Definition. The tangent bundle T(M) is UpEM Tp(M). If 7 (M) de-notes the dual space of Tp(M), the cotangent bundle T*(M) isUpEM Tp(M). If r > 1, we will show that T(M) carries a structure ofdifferentiable manifold of class C". Likewise for Tp* (M).Linear Tangent Mapping

2.6. Definition. Let 4) be a differentiable map of Mn into Wp (two differ-entiable manifolds). Let P E M,,, and set Q = 4(P). The map d induces alinear map (4*)p of the tangent bundle Tp(M) into TQ(W) defined by

[(4)*)pX1(f) = X Y o fl,here X E Tp(M), ((P+)pX E TQ(W) and f is a differentiable function in aneighbourhood 9 of Q. We call the linear tangent mapping of $ atP.

To define a vector of TQ(W) (Definition 2.2), we must specify how itacts on differentiable functions defined in a neighbourhood of Q. Obviously,f - pX ] (f) is linear. Moreover, if f is flat at Q. then f o t, which is

Linear Tangent Mapping 47

differentiable in a neighbourhood of P, is flat at P, and we have X (f o4) = 0.So ((*)p is a linear map of Tp(M) into TQ(R').

(O*)p is nothing else than (d4i)p. Indeed, consider a local chart at Pwith coordinates {x*} and a local chart at Q with coordinates {y}. 4P is de-fined in a neighbourhood of P by p real-valued functions 4'(x1, x2, ... , x"),a = 1, 2, , p. Using intrinsic notations to simplify, we get

X(fo0)=d(fo-O)poX=(df)o(d4')poX=(dj)o($*)pX.Indeed, {X1} being the components of X in the basis {(8/8x`)p}, the com-ponents of Y = (41*) pX are

n

Y I '''' X'8x*

in the basis {(8/8y)Q}. When we use intrinsic notation, we do not specifythe local charts. In the coordinate systems {x`} and {y}, the equalityabove shows that (df)p = ((09*/8x'))p = (4 )p. When we do not specifythe point P, we write L. instead of (4)p.2.7. Definition. Linear cotangent mapping ($*)p. Let P E Mn andQ = 4'(P). By duality, we define the linear cotangent mapping (V) p ofTQ(W) into 7p(M) as follows:

7Q(W) 9 w , (4)p(w) E TA(M),(($*)p(w), X) _ (w, (4*)p(X)) for allX E Tp(M).

In case w = df we saw (Definition 2.6) that d(f o P) p o X = (df) o ($) pX.Thus

(4*)p(df) = d(f o ')p2.8. Proposition. 'l o ib* = (if o 4P)*.

Let V be a third differentiable manifold and V a differentiable mappingof W into V. If f is a differentiable function in a neighbourhood of T(Q)

48 2. Thngent Space

and X E Tp(M), then['I'.(4 X)](f) = ['T (X)](f o'I') = X(f oI o$) = 01 o'P)*(X)](f)

If $ is a diffeomorphism, we infer that 4,* is bijective and (-6-1) _('F.)-1.2.9. Example. The tangent vector -2 to a differentiable curve -y(t) of M,,(y is a differentiable map of (a, b) C R into Mn). Let to E (a, b). By defini-tion ( )i is the tangent vector at 7(to) defined by (_2)t = [N(d)lt., (1)being the unit vector on R. For a differentiable function f in a neighbour-hood of ry(to), we have

dry (f) - d(f 7) = tim f [7(to + h)] - f [7(to)]h--.o hdt)w

dt ) toVector Bundles2.10. Proposition. The tangent bundle T(M) has a structure of differen-tiable manifold of class C''-1, if Mn is a differentiable manifold of class C''with r> 1.

Let (U, p) be a local chart on Mn and P E U. If {zt} are the coordinatesof Q E U and {e } the coordinates of p(Q) E Rn, then z' _ i' for 1 < i < n.This is the equality of two real numbers. If we consider the equality of twofunctions, we must write a' = ` o V. We have

V W.- ( 0 )PIndeed, (f being a differentiable function in ((a neighbourhood of Sp(P) in R',

I la) P1 1 P (f gyp) = \Lf ).{P) = (') (P) (f).Thus ,p* is a bijection of T(U) onto V(U) x R" C R.

Let (U0, be an atlas for Mn. The set of ]T(UB), isan atlas for T(V). Let us show that this atlas is of class Cr-1. Sup-pose U. n Us # 0. On Sp.(U. n U8) set 0 = vo o cp;'; then we have8=(Q, X) = [e(Q), (dO)Q(X)], where Q E p0(Ua n Us) and X E TQ(RR).Thus (dO)Q is of class C''' 1.2.11. Definition. A differentiable manifold E is a vector fiber bundle offiber the vector space F if there exist a differentiable manifold M (calledthe basis) and a differentiable map II of E an M such that, for all P E M,II'1(P) = Ep is isomorphic to F and there exist a neighbourhood U of Pin M and a diffeomorphism p of U x F onto R-I(U) whose restriction toeach Ep is linear, p satisfying 11 o p(P, z) = P for all z E F.

The Bracket [X, 11 49

2.12. Proposition. The tangent bundle T(M) is a vector bundle of fiberIIn.

M is the basis. If X E T(M) and X E Tp(M) for a unique point P E M,then the map n is x -+ II(X) = P. Thus II-1(P) = Tp(M), which is avector space of dimension n: F = R". If (U, rp) is a local chart at P, wesaw that ip is a diffeommorphism of T(U) = II-1(U) onto rp(U) x Itn. So wecan choose p = V. 1 o (V, Id), and we know that (rp- 1) p is linear. Moreover,p(P, z) E Tp(M), and thus ]a o p(P, z) = P, for all z E Rn.2.13. Definition. Likewise we can consider the fiber bundles T*(M),A" r(M),77(M):

T*(M) = U 17(M),PEM

P

A(M),A7M= UPEM

where A'77(M) is the space of skew-symmetric p-forms on Tp(M), andU 07 p(M) Tp(M),

PEM

where 0 7P(M) 4 Tp(M) is the space of tensors of type (r, s), r timescovariant, s times contravarlant, on Tp(M).2.14. Definition. A section of a vector fiber bundle (E, II, M) is a differ-entiable map { of M into E such that II o e = identity.

A vector field is a section of T(M).An (r, s)-tensor field is a section of 7, (T(M)).An exterior differential p-form is a section of AP 7' (M). In a local chart

an exterior differential p-form17 = L aj,...jdx" A dxh A ... A dTJP,1j,

50 2. Tangent Space

where f is a C2 function on M. Let us show that, in fact, the definitionis valid for CI functions and that we have defined above a new vector field[X, Y]. Using intrinsic notations,

[X,Y](f) = d[Y(f )].X - d[X(f )].Y = d[df.Y].X - d[df.X].Y= (d2f ) (X, Y) + df.dY.X - (d2f)(Y,X) - df.dX.Y= df[dY.X - dX.YJ = [dY.X - dX.Y](f ),

since d2 f is a symmetric bilinear form. From this expression, it is clear that[X, Y] satisfies conditions (a) and (b) of Definition 2.2. If the vector fieldsX and Y are C', then [X, Y] is a Cr-1 vector field. In a coordinate system{x' } corresponding to a local chart (S2, gyp),

2.16. Definition. Lie algebra. The map (X. Y) -+ [X, Y] is bilinear andantisymmetric.

a) [X, Y] = -[Y,X], and [X, Y] satisfies the Jacobi identity.13) [X, [Y, Z]] + [Z, [X, Y]] + [Y, [Z, X]] = o.

A real vector space L, endowed with a bilinear map L x L into L satisfyinga) and 3), is called a Lie algebra. So the set of CO vector fields is a Liealgebra. A straightforward computation proves,3), and a) is obvious.2.17. Definition. Projectable vector field. Let M and W be two differen-tiable manifolds and W a differentiable map of M into W. We have definedthe linear tangent mapping, but in most cases it does not allow us to asso-ciate to a vector field X on M a vector field on W. There are two reasonsfor this. First, 4r (M) may be not all W; second, if %P is not injective, we mayhave at some points of W several tangent vectors images by +Yt of vectorsof the vector field X.

This is why we say that a vector field X is projectable by AY, if for allQ E W(M) and each P E M such that P(P) = Q, we haveindependent of i. P(M) is assumed to be W.

If 41 is a diffeomorphism, any vector field X on M is projectable by 41.Let Y be a vector field on W. The vector fields X and Y are said to be

compatible by 41 if for all P E M we have Y(T(P)).2.18. Proposition. Let X1, Y1 and X2, Y2 be two pairs of vectors fieldscompatible by 40. Then (XI, X21 and [Y1,Y2] are compatible by 4 (see 2.17for the notations). Consider P E M and Q = 41(P), and let g be a C2

The Bracket [X, Y] 51

function in a neighbourhood of Q. Then[Y1,Y2](g) = Y1{Y2(9)] _ Y2[Yl(9)] = X1[X2(9o'I')] _ X2[Xi(9o'I')]

= [X1,X2](9o W).Thus *.([X1,X2]) = ['1',(X1), `l',(X2)]-

In the proof above there are some subtle points. When we write for thedefinition of the linear tangent mapping Y2(g) = X2(9 o 41), we understandthe equality of two real numbers. But here they are functions. So we mustwrite

[1`2(9)] 0'1' = X2 (9 o ').2.19. Definition. A differentiable manifold M of dimension n is paral-leliaable if on V there are n vector fields X1, X2i - , X,,, such that, at anypoint P E M. {Xl(P),X2(P),. , X"(P)} is a basis for Tp(M).2.20. Definition. The product manifold M x W of two differentiable man-ifolds M. and Wp is a differentiable manifold of dimension n + p, defined bythe atlas (U, X 0j, (apt, O?)) ((i, j) E I x J) on the topological product spaceM X W, (Uj, pt)jEt being an atlas on M" and (0j, IPf)jEJ an atlas on Wp.2.21. Proposition. A manifold M" is parallelisable if and only if its ton-gent bundle T(M) is trivial-that is to say, difeomorphic to M x R". thedfeomorphisrn p being linear on each fiber and satisfying R o p` 1(P, z) = P(see 2.12).

Let { e1, - - - , be a basis of R" and -b the diffeomorphism of T(M)onto M x R" if T(M) is trivial.

Set X; (P) = $-1(P, ei) for all P E M. and i = 1, 2,- - , n. Thenthe set{X{(P)}, i = 1, 2, - , n, is a basis of Tp(M), as otherwise a linearcombination e 0 of et would be such that -6 -1(P, e) = 0. That isimpossible, since 0-1 is a diffeomorphism and b-'(P,0) = 0. So M" isparallelisable.

Conversely, if there exist n v e c t o r fields Xi (i = 1 , 2, , n) such thatfor all P E M. the set {XX(P) } is a basis of Tp(M). let its consider the map9 of M x R" into T(M) defined by

9`P,"ate{/ .'X;(P),

where the At are n real numbers.9 is a differentiable bijection, linear for fixed P. If we prove that the

rank of 0 is 2n, then 0 is a di$eomorphism, fi = 0-1 and T(M) is trivial. Let

52 2. Tangent Space

{z'} be a coordinate system in a neighbourhood of P. {x'} and {aV} forma coordinate system in a neighbourhood of (P, R") in M x W". Moreover,

49Xi(x) _ Xi (x)k-1

and

(gy)p = \(A) (XX(P)))Here (Id) is the n x n identity matrix, (0) is the n x n zero matrix, (A) isan n x n matrix, and the n x n matrix (X (P)), with components Xjk(P),is invertible. Thus the rank of (DO)p is 2n.

Exterior Differential2.22. Definition. The algebra of exterior differential forms A(M).

A(M) = (DP"-fl AP(M), with an exterior product defined below, is analgebra. For simplicity we will say only "diferential form? instead of "ex-terior differential form" when no confusion is possible. We suppose M is aCOO manifold. The algebras A(M) = C""- (M) and AP(M) for p > 0 are de-fined in 2.14. Given q E Aq(M) and E AP(M), we define qAf E AP'-'(M),the exterior product of in and t;, by

(n A C)(X1,... , Xp+q)

I E e(a)n(Xv(1), ... , X0(q)X(Xc(1+q), ... , XQ(P+q)),p'q' VE'P

where X1, , X,+q are p + q vector fields and the sum is over the set Pof permutations a, e(a) being the signature of a. The exterior product isassociative and anticommutative: l; A q = (-1)Pgq A l;.

We also define the inner product i(X)77 of a differential form q E Aq(M)(1 < q< n) by a vector field X. i(X)v7 is a differential (q - 1)-form definedas follows: If X; (i = 1, 2, ... , q -1) are q -1 vector fields , then

[i(X)17](X1,X2,... ,Xq-1) = FAX, X1,X2,... ,Xq-1)

We verify that if t E AP(M), theni(X)(q At) _ [i(X)q] A1: + (-1)qq A [i(X).J and i(X)[i(X)q] = 0.

2.23. Proposition. Let 0 be a differentiable map of M,a into WP. To anydifferential q-form q on W, we can associate a differential q-form 0*q EAq(M), the inverse image of q by 4b, defined by

(4*q)p(X1, X2, ... , Xq) = fi(n) ('4X 1, ... ,44Xq),

Exterior Differential 53

X1, - , Xq being q vectors of Tp(M). For a function f on W (f E A(W ),we sets*f =fo4.

Here there is no difficulty such as in Definition 2.17. Indeed if P E M,then Q = $(P) is unique and (4*ri)p = (,DP)r7i. We verify that

2.24. Definition. Exterior differential.To q E Aq(M), we associate the exterior differential form dri E Aq+l(M)

defined byq+1

rAdr1(X1, ... , Xq+1) = E(-1)i-'Xi[r1(X1, .. , Xi, ...

, Xq+l)]i=1

+,(_1)i+j]7([Xi,Xj],X1,... ,X;,... ,Xj,... ,X4+1)i

54 2. Tangent space

where aj,...jq are differentiable real-valued functions.Since [8/ax=, 88x7] = 0, by definition

a a 1an (

q+1 ^

ll

k=1

5D; [77X-J"

-, ...

q+1%k-l aaj,...jk...jg},

k=1arjk

Thus, we easily verify that

dr7 = da j, .. yq A dx't A ... A dx'q.j j

Orientable Manifolds 55

On the other hand,'Vii = E (aj,...j4 o 4')d(xl' o ID) A ... A d(xj4 0 4'),

and according to 2.25d(Vt1) _ E d(aj,...3, o A d(x" o $) A ... A d(ziQ o 4)).

J1 s

56 2. Tangent Space

the transformation (x1, x2, x") -i (-xl, x2, 'X") of R". So from Awe construct an atlas A = (f-4, c )iEI

Now if f, is equivalent to f, but in the chart (f4-, Bpi), then f, = fi ifVi = Vii, fj = -fJ otherwise. So fj and f, are positive. At X E fki nf2j, denoting by Al Cthe determinant of the Jacobian of pj o Bpi 1, we havefjJAI = fi. Thus JAI > 0 and all changes of charts of A have positiveJacobian.2.29. Definition. Let M be a connected orientable manifold. On the setof nonvanishing differential n-forms, consider the following equivalence re-lation: wl - w2 if there exists f > 0 such that wl = fw2. There are twoequivalence classes. Choosing one of them defines an orientation of M; thenM is called oriented. There are two possible orientations of an orientableconnected manifold.

Some examples of orientable manifolds are the sphere, the cylinder, thetorus, real projective spaces of odd dimension, the tangent bundle of anymanifold, and complex manifolds.

Some examples of nonorientable manifolds are the Mobius band, theKlein bottle, and real projective spaces of even dimension (see 1.9).

We can see the Mobius band in R3.(1,

a rectangle ABCD in R2:[-1, 1) X ] - E, E[, c > 0, A = (1, E), B = (1, -E), C = (-1, -E), D = (-1, e),and identify the segment AB with CD. If we do this by identifying A withD and B with C, we have a cylinder. If instead we identify A with C andB with D, we get a Mobius band.

Let us consider the atlas A with two charts, (fZl,col), P2, IM, wheref1i =]-4, 4[ x cR2,

Vl = identity,f12=]1,1( X ]-E,E[U [-1,-2[ X

P2 =/identity on ] 2, 1 [ x

,p2(x, b) _ (x + 2, -y) on [-1, -' [ x - E, E[,

Orientable Manifolds 57

(x, y are coordinates on R2). Theni21f1S22-]-41[ x ]-E,E[U]2,d[ x ]-E,E[CR2.

Let (xl, yl) and (x2r y2) be the considered coordinate systems on f1l andrespectively.On I", I [ x E, e [ the change of coordinates is x2 = X1, y2 = yl, and

on ] - 4, [ x ] - E, E[ it is x2 = xl + 2, y2 = -yl. On the first open set thechange of coordinate chart has positive Jacobian, but on the second it hasnegative Jacobian.

In the proof of Theorem 2.28 we saw that, if a manifold is orientable,from an atlas A we can construct an atlas A all of whose changes of chartsare positive only by the eventual change of coordinate x2 -p -x2. Here ifwe do t

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