ńņš. 1 |

OPERATIONS

IN

DIFFERENTIAL

GEOMETRY

Ivan KolĀ“Ė

ar

Peter W. Michor

Jan SlovĀ“k

a

Mailing address: Peter W. Michor,

Institut fĀØr Mathematik der UniversitĀØt Wien,

u a

Strudlhofgasse 4, A-1090 Wien, Austria.

Ivan KolĀ“Ė, Jan SlovĀ“k,

ar a

Department of Algebra and Geometry

Faculty of Science, Masaryk University

JanĀ“Ėkovo nĀ“m 2a, CS-662 95 Brno, Czechoslovakia

ac a

Electronic edition. Originally published by Springer-Verlag, Berlin Heidelberg

1993, ISBN 3-540-56235-4, ISBN 0-387-56235-4.

Typeset by AMS-TEX

v

TABLE OF CONTENTS

PREFACE ........................ ....1

CHAPTER I.

MANIFOLDS AND LIE GROUPS . . . . . . . . . . . . . . ..4

1. Diļ¬erentiable manifolds . . . . . . . . . . . . . . . . . . . ..4

2. Submersions and immersions . . . . . . . . . . . . . . . . . . 11

3. Vector ļ¬elds and ļ¬‚ows . . . . . . . . . . . . . . . . . . . . . 16

4. Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5. Lie subgroups and homogeneous spaces . . . . . . . . . . . . . 41

CHAPTER II.

DIFFERENTIAL FORMS . . . . . . . . . . . . . . . . ... 49

6. Vector bundles . . . . . . . . . . . . . . . . . . . . . ... 49

7. Diļ¬erential forms . . . . . . . . . . . . . . . . . . . . ... 61

8. Derivations on the algebra of diļ¬erential forms

and the FrĀØlicher-Nijenhuis bracket . . . . . . . . . . . .

o ... 67

CHAPTER III.

BUNDLES AND CONNECTIONS . . . . . . . . . . . . . . . 76

9. General ļ¬ber bundles and connections . . . . . . . . . . . . . . 76

10. Principal ļ¬ber bundles and G-bundles . . . . . . . . . . . . . . 86

11. Principal and induced connections ............ . . . 99

CHAPTER IV.

JETS AND NATURAL BUNDLES . . . . . . . . . . . . . . . 116

12. Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

13. Jet groups . . . . . . . . . . . . . . . . . . . . . . . . . . 128

14. Natural bundles and operators . . . . . . . . . . . . . . . . . 138

15. Prolongations of principal ļ¬ber bundles . . . . . . . . . . . . . 149

16. Canonical diļ¬erential forms ............... . . . 154

17. Connections and the absolute diļ¬erentiation . . . . . . . . . . . 158

CHAPTER V.

FINITE ORDER THEOREMS . . . . . . . . . . . . . . . . . 168

18. Bundle functors and natural operators . . . . . . . . . . . . . . 169

19. Peetre-like theorems . . . . . . . . . . . . . . . . . . . . . . 176

20. The regularity of bundle functors . . . . . . . . . . . . . . . . 185

21. Actions of jet groups . . . . . . . . . . . . . . . . . . . . . . 192

22. The order of bundle functors . . . . . . . . . . . . . . . . . . 202

23. The order of natural operators . . . . . . . . . . . . . . . . . 205

CHAPTER VI.

METHODS FOR FINDING NATURAL OPERATORS . . . ... 212

24. Polynomial GL(V )-equivariant maps ........... ... 213

25. Natural operators on linear connections, the exterior diļ¬erential .. 220

26. The tensor evaluation theorem . . . . . . . . . . . . . . ... 223

27. Generalized invariant tensors . . . . . . . . . . . . . . . ... 230

28. The orbit reduction . . . . . . . . . . . . . . . . . . . ... 233

29. The method of diļ¬erential equations ........... ... 245

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

vi

CHAPTER VII.

FURTHER APPLICATIONS . . . . . . . . . . . . . . . . . . 249

30. The FrĀØlicher-Nijenhuis bracket . . . . . . . . . . . . . . . .

o . 250

31. Two problems on general connections . . . . . . . . . . . . . . 255

32. Jet functors . . . . . . . . . . . . . . . . . . . . . . . . . . 259

33. Topics from Riemannian geometry . . . . . . . . . . . . . . . . 265

34. Multilinear natural operators . . . . . . . . . . . . . . . . . . 280

CHAPTER VIII.

PRODUCT PRESERVING FUNCTORS ........... . 296

35. Weil algebras and Weil functors . . . . . . . . . . . . . . . . . 297

36. Product preserving functors . . . . . . . . . . . . . . . . . . 308

37. Examples and applications . . . . . . . . . . . . . . . . . . . 318

CHAPTER IX.

BUNDLE FUNCTORS ON MANIFOLDS . . . . . . . . . . . . 329

38. The point property . . . . . . . . . . . . . . . . . . . . . . 329

39. The ļ¬‚ow-natural transformation ............... . 336

40. Natural transformations . . . . . . . . . . . . . . . . . . . . 341

41. Star bundle functors .................... . 345

CHAPTER X.

PROLONGATION OF VECTOR FIELDS AND CONNECTIONS . 350

42. Prolongations of vector ļ¬elds to Weil bundles . . . . . . . . . . . 351

43. The case of the second order tangent vectors . . . . . . . . . . . 357

44. Induced vector ļ¬elds on jet bundles . . . . . . . . . . . . . . . 360

45. Prolongations of connections to F Y ā’ M . . . . . . . . . . . . 363

46. The cases F Y ā’ F M and F Y ā’ Y . . . . . . . . . . . . . . . 369

CHAPTER XI.

GENERAL THEORY OF LIE DERIVATIVES . . . . . . . . . . 376

47. The general geometric approach ............... . 376

48. Commuting with natural operators . . . . . . . . . . . . . . . 381

49. Lie derivatives of morphisms of ļ¬bered manifolds . . . . . . . . . 387

50. The general bracket formula . . . . . . . . . . . . . . . . . . 390

CHAPTER XII.

GAUGE NATURAL BUNDLES AND OPERATORS . . . . . . . 394

51. Gauge natural bundles ................... . 394

52. The Utiyama theorem . . . . . . . . . . . . . . . . . . . . . 399

53. Base extending gauge natural operators . . . . . . . . . . . . . 405

54. Induced linear connections on the total space

of vector and principal bundles . . . . . . . . . . . . . . . . . 409

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417

List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 428

Author index ......................... . 429

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

1

PREFACE

The aim of this work is threefold:

First it should be a monographical work on natural bundles and natural op-

erators in diļ¬erential geometry. This is a ļ¬eld which every diļ¬erential geometer

has met several times, but which is not treated in detail in one place. Let us

explain a little, what we mean by naturality.

Exterior derivative commutes with the pullback of diļ¬erential forms. In the

background of this statement are the following general concepts. The vector

bundle Īk T ā— M is in fact the value of a functor, which associates a bundle over

M to each manifold M and a vector bundle homomorphism over f to each local

diļ¬eomorphism f between manifolds of the same dimension. This is a simple

example of the concept of a natural bundle. The fact that the exterior derivative

d transforms sections of Īk T ā— M into sections of Īk+1 T ā— M for every manifold M

can be expressed by saying that d is an operator from Īk T ā— M into Īk+1 T ā— M .

That the exterior derivative d commutes with local diļ¬eomorphisms now means,

that d is a natural operator from the functor Īk T ā— into functor Īk+1 T ā— . If k > 0,

one can show that d is the unique natural operator between these two natural

bundles up to a constant. So even linearity is a consequence of naturality. This

result is archetypical for the ļ¬eld we are discussing here. A systematic treatment

of naturality in diļ¬erential geometry requires to describe all natural bundles, and

this is also one of the undertakings of this book.

Second this book tries to be a rather comprehensive textbook on all basic

structures from the theory of jets which appear in diļ¬erent branches of dif-

ferential geometry. Even though Ehresmann in his original papers from 1951

underlined the conceptual meaning of the notion of an r-jet for diļ¬erential ge-

ometry, jets have been mostly used as a purely technical tool in certain problems

in the theory of systems of partial diļ¬erential equations, in singularity theory,

in variational calculus and in higher order mechanics. But the theory of nat-

ural bundles and natural operators clariļ¬es once again that jets are one of the

fundamental concepts in diļ¬erential geometry, so that a thorough treatment of

their basic properties plays an important role in this book. We also demonstrate

that the central concepts from the theory of connections can very conveniently

be formulated in terms of jets, and that this formulation gives a very clear and

geometric picture of their properties.

This book also intends to serve as a self-contained introduction to the theory

of Weil bundles. These were introduced under the name ā˜les espaces des points

prochesā™ by A. Weil in 1953 and the interest in them has been renewed by the

recent description of all product preserving functors on manifolds in terms of

products of Weil bundles. And it seems that this technique can lead to further

interesting results as well.

Third in the beginning of this book we try to give an introduction to the

fundamentals of diļ¬erential geometry (manifolds, ļ¬‚ows, Lie groups, diļ¬erential

forms, bundles and connections) which stresses naturality and functoriality from

the beginning and is as coordinate free as possible. Here we present the FrĀØlicher-

o

Nijenhuis bracket (a natural extension of the Lie bracket from vector ļ¬elds to

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

2 Preface

vector valued diļ¬erential forms) as one of the basic structures of diļ¬erential

geometry, and we base nearly all treatment of curvature and Bianchi identities

on it. This allows us to present the concept of a connection ļ¬rst on general

ļ¬ber bundles (without structure group), with curvature, parallel transport and

Bianchi identity, and only then add G-equivariance as a further property for

principal ļ¬ber bundles. We think, that in this way the underlying geometric

ideas are more easily understood by the novice than in the traditional approach,

where too much structure at the same time is rather confusing. This approach

was tested in lecture courses in Brno and Vienna with success.

A speciļ¬c feature of the book is that the authors are interested in general

points of view towards diļ¬erent structures in diļ¬erential geometry. The modern

development of global diļ¬erential geometry clariļ¬ed that diļ¬erential geomet-

ric objects form ļ¬ber bundles over manifolds as a rule. Nijenhuis revisited the

classical theory of geometric objects from this point of view. Each type of geo-

metric objects can be interpreted as a rule F transforming every m-dimensional

manifold M into a ļ¬bered manifold F M ā’ M over M and every local diļ¬eo-

morphism f : M ā’ N into a ļ¬bered manifold morphism F f : F M ā’ F N over

f . The geometric character of F is then expressed by the functoriality condition

F (g ā—¦ f ) = F g ā—¦ F f . Hence the classical bundles of geometric objects are now

studied in the form of the so called lifting functors or (which is the same) natu-

ral bundles on the category Mfm of all m-dimensional manifolds and their local

diļ¬eomorphisms. An important result by Palais and Terng, completed by Ep-

stein and Thurston, reads that every lifting functor has ļ¬nite order. This gives

a full description of all natural bundles as the ļ¬ber bundles associated with the

r-th order frame bundles, which is useful in many problems. However in several

cases it is not suļ¬cient to study the bundle functors deļ¬ned on the category

Mfm . For example, if we have a Lie group G, its multiplication is a smooth

map Āµ : G Ć— G ā’ G. To construct an induced map F Āµ : F (G Ć— G) ā’ F G,

we need a functor F deļ¬ned on the whole category Mf of all manifolds and

all smooth maps. In particular, if F preserves products, then it is easy to see

that F Āµ endows F G with the structure of a Lie group. A fundamental result

in the theory of the bundle functors on Mf is the complete description of all

product preserving functors in terms of the Weil bundles. This was deduced by

Kainz and Michor, and independently by Eck and Luciano, and it is presented in

chapter VIII of this book. At several other places we then compare and contrast

the properties of the product preserving bundle functors and the non-product-

preserving ones, which leads us to interesting geometric results. Further, some

functors of modern diļ¬erential geometry are deļ¬ned on the category of ļ¬bered

manifolds and their local isomorphisms, the bundle of general connections be-

ing the simplest example. Last but not least we remark that Eck has recently

introduced the general concepts of gauge natural bundles and gauge natural op-

erators. Taking into account the present role of gauge theories in theoretical

physics and mathematics, we devote the last chapter of the book to this subject.

If we interpret geometric objects as bundle functors deļ¬ned on a suitable cat-

egory over manifolds, then some geometric constructions have the role of natural

transformations. Several others represent natural operators, i.e. they map sec-

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Preface 3

tions of certain ļ¬ber bundles to sections of other ones and commute with the

action of local isomorphisms. So geometric means natural in such situations.

That is why we develop a rather general theory of bundle functors and natural

operators in this book. The principal advantage of interpreting geometric as nat-

ural is that we obtain a well-deļ¬ned concept. Then we can pose, and sometimes

even solve, the problem of determining all natural operators of a prescribed type.

This gives us the complete list of all possible geometric constructions of the type

in question. In some cases we even discover new geometric operators in this way.

Our practical experience taught us that the most eļ¬ective way how to treat

natural operators is to reduce the question to a ļ¬nite order problem, in which

the corresponding jet spaces are ļ¬nite dimensional. Since the ļ¬nite order natural

operators are in a simple bijection with the equivariant maps between the corre-

sponding standard ļ¬bers, we can apply then several powerful tools from classical

algebra and analysis, which can lead rather quickly to a complete solution of the

problem. Such a passing to a ļ¬nite order situation has been of great proļ¬t in

other branches of mathematics as well. Historically, the starting point for the

reduction to the jet spaces is the famous Peetre theorem saying that every linear

support non-increasing operator has locally ļ¬nite order. We develop an essential

generalization of this technique and we present a uniļ¬ed approach to the ļ¬nite

order results for both natural bundles and natural operators in chapter V.

The primary purpose of chapter VI is to explain some general procedures,

which can help us in ļ¬nding all the equivariant maps, i.e. all natural operators of

a given type. Nevertheless, the greater part of the geometric results is original.

Chapter VII is devoted to some further examples and applications, including

Gilkeyā™s theorem that all diļ¬erential forms depending naturally on Riemannian

metrics and satisfying certain homogeneity conditions are in fact Pontryagin

forms. This is essential in the recent heat kernel proofs of the Atiyah Singer

Index theorem. We also characterize the Chern forms as the only natural forms

on linear symmetric connections. In a special section we comment on the results

of Kirillov and his colleagues who investigated multilinear natural operators with

the help of representation theory of inļ¬nite dimensional Lie algebras of vector

ļ¬elds. In chapter X we study systematically the natural operators on vector ļ¬elds

and connections. Chapter XI is devoted to a general theory of Lie derivatives,

in which the geometric approach clariļ¬es, among other things, the relations to

natural operators.

The material for chapters VI, X and sections 12, 30ā“32, 47, 49, 50, 52ā“54 was

prepared by the ļ¬rst author (I.K.), for chapters I, II, III, VIII by the second au-

thor (P.M.) and for chapters V, IX and sections 13ā“17, 33, 34, 48, 51 by the third

author (J.S.). The authors acknowledge A. Cap, M. Doupovec, and J. JanyĖka, s

for reading the manuscript and for several critical remarks and comments and

A. A. Kirillov for commenting section 34.

The joint work of the authors on the book has originated in the seminar of

the ļ¬rst two authors and has been based on the common cultural heritage of

Middle Europe. The authors will be pleased if the reader realizes a reļ¬‚ection of

those traditions in the book.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

4

CHAPTER I.

MANIFOLDS AND LIE GROUPS

In this chapter we present an introduction to the basic structures of diļ¬erential

geometry which stresses global structures and categorical thinking. The material

presented is standard - but some parts are not so easily found in text books:

we treat initial submanifolds and the Frobenius theorem for distributions of non

constant rank, and we give a very quick proof of the Campbell - Baker - Hausdorļ¬

formula for Lie groups. We also prove that closed subgroups of Lie groups are

Lie subgroups.

1. Diļ¬erentiable manifolds

1.1. A topological manifold is a separable Hausdorļ¬ space M which is locally

homeomorphic to Rn . So for any x ā M there is some homeomorphism u : U ā’

u(U ) ā Rn , where U is an open neighborhood of x in M and u(U ) is an open

subset in Rn . The pair (U, u) is called a chart on M .

From topology it follows that the number n is locally constant on M ; if n is

constant, M is sometimes called a pure manifold. We will only consider pure

manifolds and consequently we will omit the preļ¬x pure.

A family (UĪ± , uĪ± )Ī±āA of charts on M such that the UĪ± form a cover of M is

called an atlas. The mappings uĪ±Ī² := uĪ± ā—¦ uā’1 : uĪ² (UĪ±Ī² ) ā’ uĪ± (UĪ±Ī² ) are called

Ī²

the chart changings for the atlas (UĪ± ), where UĪ±Ī² := UĪ± ā© UĪ² .

An atlas (UĪ± , uĪ± )Ī±āA for a manifold M is said to be a C k -atlas, if all chart

changings uĪ±Ī² : uĪ² (UĪ±Ī² ) ā’ uĪ± (UĪ±Ī² ) are diļ¬erentiable of class C k . Two C k -

atlases are called C k -equivalent, if their union is again a C k -atlas for M . An

equivalence class of C k -atlases is called a C k -structure on M . From diļ¬erential

topology we know that if M has a C 1 -structure, then it also has a C 1 -equivalent

C ā -structure and even a C 1 -equivalent C Ļ -structure, where C Ļ is shorthand

for real analytic. By a C k -manifold M we mean a topological manifold together

with a C k -structure and a chart on M will be a chart belonging to some atlas

of the C k -structure.

But there are topological manifolds which do not admit diļ¬erentiable struc-

tures. For example, every 4-dimensional manifold is smooth oļ¬ some point, but

there are such which are not smooth, see [Quinn, 82], [Freedman, 82]. There

are also topological manifolds which admit several inequivalent smooth struc-

tures. The spheres from dimension 7 on have ļ¬nitely many, see [Milnor, 56].

But the most surprising result is that on R4 there are uncountably many pair-

wise inequivalent (exotic) diļ¬erentiable structures. This follows from the results

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

1. Diļ¬erentiable manifolds 5

of [Donaldson, 83] and [Freedman, 82], see [Gompf, 83] or [Freedman-Feng Luo,

89] for an overview.

Note that for a Hausdorļ¬ C ā -manifold in a more general sense the following

properties are equivalent:

(1) It is paracompact.

(2) It is metrizable.

(3) It admits a Riemannian metric.

(4) Each connected component is separable.

In this book a manifold will usually mean a C ā -manifold, and smooth is

used synonymously for C ā , it will be Hausdorļ¬, separable, ļ¬nite dimensional,

to state it precisely.

Note ļ¬nally that any manifold M admits a ļ¬nite atlas consisting of dim M +1

(not connected) charts. This is a consequence of topological dimension theory

[Nagata, 65], a proof for manifolds may be found in [Greub-Halperin-Vanstone,

Vol. I, 72].

1.2. A mapping f : M ā’ N between manifolds is said to be C k if for each

x ā M and each chart (V, v) on N with f (x) ā V there is a chart (U, u) on M

with x ā U , f (U ) ā V , and v ā—¦ f ā—¦ uā’1 is C k . We will denote by C k (M, N ) the

space of all C k -mappings from M to N .

A C k -mapping f : M ā’ N is called a C k -diļ¬eomorphism if f ā’1 : N ā’ M

exists and is also C k . Two manifolds are called diļ¬eomorphic if there exists a dif-

feomorphism between them. From diļ¬erential topology we know that if there is a

C 1 -diļ¬eomorphism between M and N , then there is also a C ā -diļ¬eomorphism.

All smooth manifolds together with the C ā -mappings form a category, which

will be denoted by Mf . One can admit non pure manifolds even in Mf , but

we will not stress this point of view.

A mapping f : M ā’ N between manifolds of the same dimension is called

a local diļ¬eomorphism, if each x ā M has an open neighborhood U such that

f |U : U ā’ f (U ) ā‚ N is a diļ¬eomorphism. Note that a local diļ¬eomorphism

need not be surjective or injective.

1.3. The set of smooth real valued functions on a manifold M will be denoted

by C ā (M, R), in order to distinguish it clearly from spaces of sections which

will appear later. C ā (M, R) is a real commutative algebra.

The support of a smooth function f is the closure of the set, where it does

not vanish, supp(f ) = {x ā M : f (x) = 0}. The zero set of f is the set where f

vanishes, Z(f ) = {x ā M : f (x) = 0}.

Any manifold admits smooth partitions of unity: Let (UĪ± )Ī±āA be an open

cover of M . Then there is a family (Ļ•Ī± )Ī±āA of smooth functions on M , such

that supp(Ļ•Ī± ) ā‚ UĪ± , (supp(Ļ•Ī± )) is a locally ļ¬nite family, and Ī± Ļ•Ī± = 1

(locally this is a ļ¬nite sum).

1.4. Germs. Let M and N be manifolds and x ā M . We consider all smooth

mappings f : Uf ā’ N , where Uf is some open neighborhood of x in M , and we

put f ā¼ g if there is some open neighborhood V of x with f |V = g|V . This is an

x

equivalence relation on the set of mappings considered. The equivalence class of

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6 Chapter I. Manifolds and Lie groups

a mapping f is called the germ of f at x, sometimes denoted by germx f . The

ā

space of all germs at x of mappings M ā’ N will be denoted by Cx (M, N ).

This construction works also for other types of mappings like real analytic or

holomorphic ones, if M and N have real analytic or complex structures.

If N = R we may add and multiply germs, so we get the real commutative

ā

algebra Cx (M, R) of germs of smooth functions at x.

Using smooth partitions of unity (see 1.3) it is easily seen that each germ of

a smooth function has a representative which is deļ¬ned on the whole of M . For

ā

germs of real analytic or holomorphic functions this is not true. So Cx (M, R)

is the quotient of the algebra C ā (M, R) by the ideal of all smooth functions

f : M ā’ R which vanish on some neighborhood (depending on f ) of x.

1.5. The tangent space of Rn . Let a ā Rn . A tangent vector with foot

point a is simply a pair (a, X) with X ā Rn , also denoted by Xa . It induces

a derivation Xa : C ā (Rn , R) ā’ R by Xa (f ) = df (a)(Xa ). The value depends

only on the germ of f at a and we have Xa (f Ā· g) = Xa (f ) Ā· g(a) + f (a) Ā· Xa (g)

(the derivation property).

If conversely D : C ā (Rn , R) ā’ R is linear and satisļ¬es D(f Ā· g) = D(f ) Ā·

g(a) + f (a) Ā· D(g) (a derivation at a), then D is given by the action of a tangent

vector with foot point a. This can be seen as follows. For f ā C ā (Rn , R) we

have

1

d

+ t(x ā’ a))dt

f (x) = f (a) + dt f (a

0

n 1

ā‚f

+ t(x ā’ a))dt (xi ā’ ai )

= f (a) + ā‚xi (a

0

i=1

n

hi (x)(xi ā’ ai ).

= f (a) +

i=1

D(1) = D(1 Ā· 1) = 2D(1), so D(constant) = 0. Thus

n

hi (x)(xi ā’ ai ))

D(f ) = D(f (a) +

i=1

n n

i i

hi (a)(D(xi ) ā’ 0)

D(hi )(a ā’ a ) +

=0+

i=1 i=1

n

ā‚f i

= ā‚xi (a)D(x ),

i=1

where xi is the i-th coordinate function on Rn . So we have the expression

n n

i

D(xi ) ā‚xi |a .

ā‚ ā‚

) ā‚xi |a (f ),

D(f ) = D(x D=

i=1 i=1

n

D(xi )ei ), where (ei ) is the

Thus D is induced by the tangent vector (a, i=1

standard basis of Rn .

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

1. Diļ¬erentiable manifolds 7

1.6. The tangent space of a manifold. Let M be a manifold and let x ā

M and dim M = n. Let Tx M be the vector space of all derivations at x of

ā

Cx (M, R), the algebra of germs of smooth functions on M at x. (Using 1.3 it

may easily be seen that a derivation of C ā (M, R) at x factors to a derivation of

ā

Cx (M, R).)

So Tx M consists of all linear mappings Xx : C ā (M, R) ā’ R satisfying Xx (f Ā·

g) = Xx (f ) Ā· g(x) + f (x) Ā· Xx (g). The space Tx M is called the tangent space of

M at x.

If (U, u) is a chart on M with x ā U , then uā— : f ā’ f ā—¦ u induces an iso-

morphism of algebras Cu(x) (Rn , R) ā¼ Cx (M, R), and thus also an isomorphism

ā

=ā

Tx u : Tx M ā’ Tu(x) Rn , given by (Tx u.Xx )(f ) = Xx (f ā—¦ u). So Tx M is an n-

dimensional vector space. The dot in Tx u.Xx means that we apply the linear

mapping Tx u to the vector Xx ā” a dot will frequently denote an application of

a linear or ļ¬ber linear mapping.

We will use the following notation: u = (u1 , . . . , un ), so ui denotes the i-th

coordinate function on U , and

:= (Tx u)ā’1 ( ā‚xi |u(x) ) = (Tx u)ā’1 (u(x), ei ).

ā‚ ā‚

ā‚ui |x

ā‚

ā‚ui |x ā Tx M is the derivation given by

So

ā‚(f ā—¦ uā’1 )

ā‚

ā‚ui |x (f ) = (u(x)).

ā‚xi

From 1.5 we have now

n

(Tx u.Xx )(xi ) ā‚xi |u(x) =

ā‚

Tx u.Xx =

i=1

n n

i

Xx (ui ) ā‚xi |u(x) .

ā‚ ā‚

Xx (x ā—¦ u) ā‚xi |u(x)

= =

i=1 i=1

1.7. The tangent bundle. For a manifold M of dimension n we put T M :=

xāM Tx M , the disjoint union of all tangent spaces. This is a family of vec-

tor spaces parameterized by M , with projection ĻM : T M ā’ M given by

ĻM (Tx M ) = x.

ā’1

For any chart (UĪ± , uĪ± ) of M consider the chart (ĻM (UĪ± ), T uĪ± ) on T M ,

ā’1

where T uĪ± : ĻM (UĪ± ) ā’ uĪ± (UĪ± ) Ć— Rn is given by the formula T uĪ± .X =

(uĪ± (ĻM (X)), TĻM (X) uĪ± .X). Then the chart changings look as follows:

ā’1

T uĪ² ā—¦ (T uĪ± )ā’1 : T uĪ± (ĻM (UĪ±Ī² )) = uĪ± (UĪ±Ī² ) Ć— Rn ā’

ā’1

ā’ uĪ² (UĪ±Ī² ) Ć— Rn = T uĪ² (ĻM (UĪ±Ī² )),

((T uĪ² ā—¦ (T uĪ± )ā’1 )(y, Y ))(f ) = ((T uĪ± )ā’1 (y, Y ))(f ā—¦ uĪ² )

= (y, Y )(f ā—¦ uĪ² ā—¦ uā’1 ) = d(f ā—¦ uĪ² ā—¦ uā’1 )(y).Y

Ī± Ī±

= df (uĪ² ā—¦ uā’1 (y)).d(uĪ² ā—¦ uā’1 )(y).Y

Ī± Ī±

= (uĪ² ā—¦ uā’1 (y), d(uĪ² ā—¦ uā’1 )(y).Y )(f ).

Ī± Ī±

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8 Chapter I. Manifolds and Lie groups

So the chart changings are smooth. We choose the topology on T M in such

a way that all T uĪ± become homeomorphisms. This is a Hausdorļ¬ topology,

since X, Y ā T M may be separated in M if Ļ(X) = Ļ(Y ), and in one chart if

Ļ(X) = Ļ(Y ). So T M is again a smooth manifold in a canonical way; the triple

(T M, ĻM , M ) is called the tangent bundle of M .

ā

1.8. Kinematic deļ¬nition of the tangent space. Consider C0 (R, M ), the

space of germs at 0 of smooth curves R ā’ M . We put the following equivalence

ā

relation on C0 (R, M ): the germ of c is equivalent to the germ of e if and only

if c(0) = e(0) and in one (equivalently each) chart (U, u) with c(0) = e(0) ā U

d d

we have dt |0 (u ā—¦ c)(t) = dt |0 (u ā—¦ e)(t). The equivalence classes are called velocity

vectors of curves in M . We have the following mappings

u

ā ā

g

e

C0 (R, M )/ ā¼ C0 (R, M )

ee

ee Ī²

ue u

ev0

Ī±

w M,

TM ĻM

ā

d

where Ī±(c)(germc(0) f ) = dt |0 f (c(t)) and Ī² : T M ā’ C0 (R, M ) is given by:

Ī²((T u)ā’1 (y, Y )) is the germ at 0 of t ā’ uā’1 (y + tY ). So T M is canonically

identiļ¬ed with the set of all possible velocity vectors of curves in M .

1.9. Let f : M ā’ N be a smooth mapping between manifolds. Then f induces a

linear mapping Tx f : Tx M ā’ Tf (x) N for each x ā M by (Tx f.Xx )(h) = Xx (hā—¦f )

for h ā Cf (x) (N, R). This mapping is linear since f ā— : Cf (x) (N, R) ā’ Cx (M, R),

ā ā ā

given by h ā’ h ā—¦ f , is linear, and Tx f is its adjoint, restricted to the subspace

of derivations.

If (U, u) is a chart around x and (V, v) is one around f (x), then

uā’1 ),

(Tx f. ā‚ui |x )(v j ) = j j

ā‚ ā‚ ā‚

ā‚ui |x (v ā—¦ f) = ā‚xi (v ā—¦ f ā—¦

ā‚ ā‚ jā‚

Tx f. ā‚ui |x = j (Tx f. ā‚ui |x )(v ) ā‚v j |f (x) by 1.7

ā‚(v j ā—¦f ā—¦uā’1 ) ā‚

(u(x)) ā‚vj |f (x) .

= ā‚xi

j

ā‚ ā‚

So the matrix of Tx f : Tx M ā’ Tf (x) N in the bases ( ā‚ui |x ) and ( ā‚vj |f (x) ) is just

the Jacobi matrix d(v ā—¦ f ā—¦ uā’1 )(u(x)) of the mapping v ā—¦ f ā—¦ uā’1 at u(x), so

Tf (x) v ā—¦ Tx f ā—¦ (Tx u)ā’1 = d(v ā—¦ f ā—¦ uā’1 )(u(x)).

Let us denote by T f : T M ā’ T N the total mapping, given by T f |Tx M :=

Tx f . Then the composition T v ā—¦ T f ā—¦ (T u)ā’1 : u(U ) Ć— Rm ā’ v(V ) Ć— Rn is given

by (y, Y ) ā’ ((v ā—¦ f ā—¦ uā’1 )(y), d(v ā—¦ f ā—¦ uā’1 )(y)Y ), and thus T f : T M ā’ T N is

again smooth.

If f : M ā’ N and g : N ā’ P are smooth mappings, then we have T (g ā—¦ f ) =

T g ā—¦ T f . This is a direct consequence of (g ā—¦ f )ā— = f ā— ā—¦ g ā— , and it is the global

version of the chain rule. Furthermore we have T (IdM ) = IdT M .

If f ā C ā (M, R), then T f : T M ā’ T R = R Ć— R. We then deļ¬ne the

diļ¬erential of f by df := pr2 ā—¦ T f : T M ā’ R. Let t denote the identity function

on R, then (T f.Xx )(t) = Xx (t ā—¦ f ) = Xx (f ), so we have df (Xx ) = Xx (f ).

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

1. Diļ¬erentiable manifolds 9

1.10. Submanifolds. A subset N of a manifold M is called a submanifold, if for

each x ā N there is a chart (U, u) of M such that u(U ā© N ) = u(U ) ā© (Rk Ć— 0),

where Rk Ć— 0 ā’ Rk Ć— Rnā’k = Rn . Then clearly N is itself a manifold with

(U ā© N, u|U ā© N ) as charts, where (U, u) runs through all submanifold charts as

above and the injection i : N ā’ M is an embedding in the following sense:

An embedding f : N ā’ M from a manifold N into another one is an injective

smooth mapping such that f (N ) is a submanifold of M and the (co)restricted

mapping N ā’ f (N ) is a diļ¬eomorphism.

If f : Rn ā’ Rq is smooth and the rank of f (more exactly: the rank of its

derivative) is q at each point of f ā’1 (0), say, then f ā’1 (0) is a submanifold of Rn

of dimension n ā’ q or empty. This is an immediate consequence of the implicit

function theorem.

The following theorem needs three applications of the implicit function theo-

rem for its proof, which can be found in [DieudonnĀ“, I, 60, 10.3.1].

e

Theorem. Let f : W ā’ Rq be a smooth mapping, where W is an open subset

of Rn . If the derivative df (x) has constant rank k for each x ā W , then for each

a ā W there are charts (U, u) of W centered at a and (V, v) of Rq centered at

f (a) such that v ā—¦ f ā—¦ uā’1 : u(U ) ā’ v(V ) has the following form:

(x1 , . . . , xn ) ā’ (x1 , . . . , xk , 0, . . . , 0).

So f ā’1 (b) is a submanifold of W of dimension n ā’ k for each b ā f (W ).

1.11. Example: Spheres. We consider the space Rn+1 , equipped with the

xi y i . The n-sphere S n is then the subset

standard inner product x, y =

{x ā Rn+1 : x, x = 1}. Since f (x) = x, x , f : Rn+1 ā’ R, satisļ¬es df (x)y =

2 x, y , it is of rank 1 oļ¬ 0 and by 1.10 the sphere S n is a submanifold of Rn+1 .

In order to get some feeling for the sphere we will describe an explicit atlas

for S n , the stereographic atlas. Choose a ā S n (ā˜south poleā™). Let

xā’ x,a a

u+ : U+ ā’ {a}ā„ ,

U+ := S n \ {a}, u+ (x) = 1ā’ x,a ,

xā’ x,a a

uā’ : Uā’ ā’ {a}ā„ ,

Uā’ := S n \ {ā’a}, uā’ (x) = 1+ x,a .

From an obvious drawing in the 2-plane through 0, x, and a it is easily seen that

u+ is the usual stereographic projection. We also get

|y|2 ā’1

uā’1 (y) = for y ā {a}ā„

2

|y|2 +1 a + |y|2 +1 y

+

and (uā’ ā—¦ uā’1 )(y) = y

|y|2 . The latter equation can directly be seen from a

+

drawing.

1.12. Products. Let M and N be smooth manifolds described by smooth at-

lases (UĪ± , uĪ± )Ī±āA and (VĪ² , vĪ² )Ī²āB , respectively. Then the family (UĪ± Ć— VĪ² , uĪ± Ć—

vĪ² : UĪ± Ć— VĪ² ā’ Rm Ć— Rn )(Ī±,Ī²)āAĆ—B is a smooth atlas for the cartesian product

M Ć— N . Clearly the projections

pr1 pr2

M āā’ M Ć— N ā’ā’ N

ā’ ā’

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

10 Chapter I. Manifolds and Lie groups

are also smooth. The product (M Ć— N, pr1 , pr2 ) has the following universal

property:

For any smooth manifold P and smooth mappings f : P ā’ M and g : P ā’ N

the mapping (f, g) : P ā’ M Ć— N , (f, g)(x) = (f (x), g(x)), is the unique smooth

mapping with pr1 ā—¦ (f, g) = f , pr2 ā—¦ (f, g) = g.

From the construction of the tangent bundle in 1.7 it is immediately clear

that

T (pr1 ) T (pr2 )

T M ā ā’ ā’ T (M Ć— N ) ā’ ā’ ā’ T N

ā’ā’ ā’ā’

is again a product, so that T (M Ć— N ) = T M Ć— T N in a canonical way.

Clearly we can form products of ļ¬nitely many manifolds.

1.13. Theorem. Let M be a connected manifold and suppose that f : M ā’ M

is smooth with f ā—¦ f = f . Then the image f (M ) of f is a submanifold of M .

This result can also be expressed as: ā˜smooth retractsā™ of manifolds are man-

ifolds. If we do not suppose that M is connected, then f (M ) will not be a

pure manifold in general, it will have diļ¬erent dimension in diļ¬erent connected

components.

Proof. We claim that there is an open neighborhood U of f (M ) in M such that

the rank of Ty f is constant for y ā U . Then by theorem 1.10 the result follows.

For x ā f (M ) we have Tx f ā—¦ Tx f = Tx f , thus im Tx f = ker(Id ā’Tx f ) and

rank Tx f + rank(Id ā’Tx f ) = dim M . Since rank Tx f and rank(Id ā’Tx f ) can-

not fall locally, rank Tx f is locally constant for x ā f (M ), and since f (M ) is

connected, rank Tx f = r for all x ā f (M ).

But then for each x ā f (M ) there is an open neighborhood Ux in M with

rank Ty f ā„ r for all y ā Ux . On the other hand rank Ty f = rank Ty (f ā—¦ f ) =

rank Tf (y) f ā—¦ Ty f ā¤ rank Tf (y) f = r. So the neighborhood we need is given by

U = xāf (M ) Ux .

1.14. Corollary. 1. The (separable) connected smooth manifolds are exactly

the smooth retracts of connected open subsets of Rn ā™s.

2. f : M ā’ N is an embedding of a submanifold if and only if there is an

open neighborhood U of f (M ) in N and a smooth mapping r : U ā’ M with

r ā—¦ f = IdM .

Proof. Any manifold M may be embedded into some Rn , see 1.15 below. Then

there exists a tubular neighborhood of M in Rn (see [Hirsch, 76, pp. 109ā“118]),

and M is clearly a retract of such a tubular neighborhood. The converse follows

from 1.13.

For the second assertion repeat the argument for N instead of Rn .

1.15. Embeddings into Rn ā™s. Let M be a smooth manifold of dimension m.

Then M can be embedded into Rn , if

(1) n = 2m + 1 (see [Hirsch, 76, p 55] or [BrĀØcker-JĀØnich, 73, p 73]),

o a

(2) n = 2m (see [Whitney, 44]).

(3) Conjecture (still unproved): The minimal n is n = 2m ā’ Ī±(m) + 1, where

Ī±(m) is the number of 1ā™s in the dyadic expansion of m.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

2. Submersions and immersions 11

There exists an immersion (see section 2) M ā’ Rn , if

(1) n = 2m (see [Hirsch, 76]),

(2) n = 2m ā’ Ī±(m) (see [Cohen, 82]).

2. Submersions and immersions

2.1. Deļ¬nition. A mapping f : M ā’ N between manifolds is called a sub-

mersion at x ā M , if the rank of Tx f : Tx M ā’ Tf (x) N equals dim N . Since the

rank cannot fall locally (the determinant of a submatrix of the Jacobi matrix is

not 0), f is then a submersion in a whole neighborhood of x. The mapping f is

said to be a submersion, if it is a submersion at each x ā M .

2.2. Lemma. If f : M ā’ N is a submersion at x ā M , then for any chart

(V, v) centered at f (x) on N there is chart (U, u) centered at x on M such that

v ā—¦ f ā—¦ uā’1 looks as follows:

(y 1 , . . . , y n , y n+1 , . . . , y m ) ā’ (y 1 , . . . , y n )

Proof. Use the inverse function theorem.

2.3. Corollary. Any submersion f : M ā’ N is open: for each open U ā‚ M

the set f (U ) is open in N .

2.4. Deļ¬nition. A triple (M, p, N ), where p : M ā’ N is a surjective submer-

sion, is called a ļ¬bered manifold. M is called the total space, N is called the

base.

A ļ¬bered manifold admits local sections: For each x ā M there is an open

neighborhood U of p(x) in N and a smooth mapping s : U ā’ M with pā—¦s = IdU

and s(p(x)) = x.

The existence of local sections in turn implies the following universal property:

RR

M

RT

R

u

p

w

f

N P

If (M, p, N ) is a ļ¬bered manifold and f : N ā’ P is a mapping into some further

manifold, such that f ā—¦ p : M ā’ P is smooth, then f is smooth.

2.5. Deļ¬nition. A smooth mapping f : M ā’ N is called an immersion at

x ā M if the rank of Tx f : Tx M ā’ Tf (x) N equals dim M . Since the rank is

maximal at x and cannot fall locally, f is an immersion on a whole neighborhood

of x. f is called an immersion if it is so at every x ā M .

2.6. Lemma. If f : M ā’ N is an immersion, then for any chart (U, u) centered

at x ā M there is a chart (V, v) centered at f (x) on N such that v ā—¦ f ā—¦ uā’1 has

the form:

(y 1 , . . . , y m ) ā’ (y 1 , . . . , y m , 0, . . . , 0)

Proof. Use the inverse function theorem.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

12 Chapter I. Manifolds and Lie groups

2.7 Corollary. If f : M ā’ N is an immersion, then for any x ā M there is

an open neighborhood U of x ā M such that f (U ) is a submanifold of N and

f |U : U ā’ f (U ) is a diļ¬eomorphism.

2.8. Deļ¬nition. If i : M ā’ N is an injective immersion, then (M, i) is called

an immersed submanifold of N .

A submanifold is an immersed submanifold, but the converse is wrong in gen-

eral. The structure of an immersed submanifold (M, i) is in general not deter-

mined by the subset i(M ) ā‚ N . All this is illustrated by the following example.

Consider the curve Ī³(t) = (sin3 t, sin t. cos t) in R2 . Then ((ā’Ļ, Ļ), Ī³|(ā’Ļ, Ļ))

and ((0, 2Ļ), Ī³|(0, 2Ļ)) are two diļ¬erent immersed submanifolds, but the image

of the embedding is in both cases just the ļ¬gure eight.

2.9. Let M be a submanifold of N . Then the embedding i : M ā’ N is an

injective immersion with the following property:

(1) For any manifold Z a mapping f : Z ā’ M is smooth if and only if

i ā—¦ f : Z ā’ N is smooth.

The example in 2.8 shows that there are injective immersions without property

(1).

2.10. We want to determine all injective immersions i : M ā’ N with property

2.9.1. To require that i is a homeomorphism onto its image is too strong as 2.11

and 2.12 below show. To look for all smooth mappings i : M ā’ N with property

2.9.1 (initial mappings in categorical terms) is too diļ¬cult as remark 2.13 below

shows.

2.11. Lemma. If an injective immersion i : M ā’ N is a homeomorphism onto

its image, then i(M ) is a submanifold of N .

Proof. Use 2.7.

2.12. Example. We consider the 2-dimensional torus T2 = R2 /Z2 . Then the

quotient mapping Ļ : R2 ā’ T2 is a covering map, so locally a diļ¬eomorphism.

Let us also consider the mapping f : R ā’ R2 , f (t) = (t, Ī±.t), where Ī± is

irrational. Then Ļ ā—¦ f : R ā’ T2 is an injective immersion with dense image, and

it is obviously not a homeomorphism onto its image. But Ļ ā—¦ f has property

2.9.1, which follows from the fact that Ļ is a covering map.

2.13. Remark. If f : R ā’ R is a function such that f p and f q are smooth for

some p, q which are relatively prime in N, then f itself turns out to be smooth,

p

see [Joris, 82]. So the mapping i : t ā’ tq , R ā’ R2 , has property 2.9.1, but i is

t

not an immersion at 0.

2.14. Deļ¬nition. For an arbitrary subset A of a manifold N and x0 ā A let

Cx0 (A) denote the set of all x ā A which can be joined to x0 by a smooth curve

in N lying in A.

A subset M in a manifold N is called initial submanifold of dimension m, if

the following property is true:

(1) For each x ā M there exists a chart (U, u) centered at x on N such that

u(Cx (U ā© M )) = u(U ) ā© (Rm Ć— 0).

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

2. Submersions and immersions 13

The following three lemmas explain the name initial submanifold.

2.15. Lemma. Let f : M ā’ N be an injective immersion between manifolds

with property 2.9.1. Then f (M ) is an initial submanifold of N .

Proof. Let x ā M . By 2.6 we may choose a chart (V, v) centered at f (x) on N

and another chart (W, w) centered at x on M such that (vā—¦f ā—¦wā’1 )(y 1 , . . . , y m ) =

(y 1 , . . . , y m , 0, . . . , 0). Let r > 0 be so small that {y ā Rm : |y| < r} ā‚ w(W )

and {z ā Rn : |z| < 2r} ā‚ v(V ). Put

U : = v ā’1 ({z ā Rn : |z| < r}) ā‚ N,

W1 : = wā’1 ({y ā Rm : |y| < r}) ā‚ M.

We claim that (U, u = v|U ) satisļ¬es the condition of 2.14.1.

uā’1 (u(U ) ā© (Rm Ć— 0)) = uā’1 ({(y 1 , . . . , y m , 0 . . . , 0) : |y| < r}) =

= f ā—¦ wā’1 ā—¦ (u ā—¦ f ā—¦ wā’1 )ā’1 ({(y 1 , . . . , y m , 0 . . . , 0) : |y| < r}) =

= f ā—¦ wā’1 ({y ā Rm : |y| < r}) = f (W1 ) ā Cf (x) (U ā© f (M )),

since f (W1 ) ā U ā© f (M ) and f (W1 ) is C ā -contractible.

Now let conversely z ā Cf (x) (U ā©f (M )). Then by deļ¬nition there is a smooth

curve c : [0, 1] ā’ N with c(0) = f (x), c(1) = z, and c([0, 1]) ā U ā© f (M ). By

property 2.9.1 the unique curve c : [0, 1] ā’ M with f ā—¦ c = c, is smooth.

ĀÆ ĀÆ

We claim that c([0, 1]) ā W1 . If not then there is some t ā [0, 1] with c(t) ā

ĀÆ ĀÆ

ā’1 m

w ({y ā R : r ā¤ |y| < 2r}) since c is smooth and thus continuous. But then

ĀÆ

we have

(v ā—¦ f )(ĀÆ(t)) ā (v ā—¦ f ā—¦ wā’1 )({y ā Rm : r ā¤ |y| < 2r}) =

c

= {(y, 0) ā Rm Ć— 0 : r ā¤ |y| < 2r} ā {z ā Rn : r ā¤ |z| < 2r}.

This means (v ā—¦ f ā—¦ c)(t) = (v ā—¦ c)(t) ā {z ā Rn : r ā¤ |z| < 2r}, so c(t) ā U , a

ĀÆ /

contradiction.

So c([0, 1]) ā W1 , thus c(1) = f ā’1 (z) ā W1 and z ā f (W1 ). Consequently we

ĀÆ ĀÆ

have Cf (x) (U ā© f (M )) = f (W1 ) and ļ¬nally f (W1 ) = uā’1 (u(U ) ā© (Rm Ć— 0)) by

the ļ¬rst part of the proof.

2.16. Lemma. Let M be an initial submanifold of a manifold N . Then there

is a unique C ā -manifold structure on M such that the injection i : M ā’ N

is an injective immersion. The connected components of M are separable (but

there may be uncountably many of them).

Proof. We use the sets Cx (Ux ā© M ) as charts for M , where x ā M and (Ux , ux )

is a chart for N centered at x with the property required in 2.14.1. Then the

chart changings are smooth since they are just restrictions of the chart changings

on N . But the sets Cx (Ux ā© M ) are not open in the induced topology on M

in general. So the identiļ¬cation topology with respect to the charts (Cx (Ux ā©

M ), ux )xāM yields a topology on M which is ļ¬ner than the induced topology, so

it is Hausdorļ¬. Clearly i : M ā’ N is then an injective immersion. Uniqueness of

the smooth structure follows from the universal property of lemma 2.17 below.

Finally note that N admits a Riemannian metric since it is separable, which can

be induced on M , so each connected component of M is separable.

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14 Chapter I. Manifolds and Lie groups

2.17. Lemma. Any initial submanifold M of a manifold N with injective

immersion i : M ā’ N has the universal property 2.9.1:

For any manifold Z a mapping f : Z ā’ M is smooth if and only if i ā—¦ f : Z ā’

N is smooth.

Proof. We have to prove only one direction and we will suppress the embedding i.

For z ā Z we choose a chart (U, u) on N , centered at f (z), such that u(Cf (z) (U ā©

M )) = u(U ) ā© (Rm Ć— 0). Then f ā’1 (U ) is open in Z and contains a chart (V, v)

centered at z on Z with v(V ) a ball. Then f (V ) is C ā -contractible in U ā© M , so

f (V ) ā Cf (z) (U ā©M ), and (u|Cf (z) (U ā©M ))ā—¦f ā—¦v ā’1 = u ā—¦f ā—¦v ā’1 is smooth.

2.18. Transversal mappings. Let M1 , M2 , and N be manifolds and let

fi : Mi ā’ N be smooth mappings for i = 1, 2. We say that f1 and f2 are

transversal at y ā N , if

im Tx1 f1 + im Tx2 f2 = Ty N whenever f1 (x1 ) = f2 (x2 ) = y.

Note that they are transversal at any y which is not in f1 (M1 ) or not in f2 (M2 ).

The mappings f1 and f2 are simply said to be transversal, if they are transversal

at every y ā N .

If P is an initial submanifold of N with injective immersion i : P ā’ N , then

f : M ā’ N is said to be transversal to P , if i and f are transversal.

Lemma. In this case f ā’1 (P ) is an initial submanifold of M with the same

codimension in M as P has in N , or the empty set. If P is a submanifold, then

also f ā’1 (P ) is a submanifold.

Proof. Let x ā f ā’1 (P ) and let (U, u) be an initial submanifold chart for P

centered at f (x) on N , i.e. u(Cx (U ā© P )) = u(U ) ā© (Rp Ć— 0). Then the mapping

f pr2

u

M ā f ā’1 (U ) ā’ U ā’ u(U ) ā Rp Ć— Rnā’p ā’ ā’ Rnā’p

ā’ ā’ ā’

is a submersion at x since f is transversal to P . So by lemma 2.2 there is a chart

(V, v) on M centered at x such that we have

(pr2 ā—¦ u ā—¦ f ā—¦ v ā’1 )(y 1 , . . . , y nā’p , . . . , y m ) = (y 1 , . . . , y nā’p ).

But then z ā Cx (f ā’1 (P ) ā© V ) if and only if v(z) ā v(V ) ā© (0 Ć— Rmā’n+p ), so

v(Cx (f ā’1 (P ) ā© V )) = v(V ) ā© (0 Ć— Rmā’n+p ).

2.19. Corollary. If f1 : M1 ā’ N and f2 : M2 ā’ N are smooth and transver-

sal, then the topological pullback

Ć— M2 = M1 Ć—N M2 := {(x1 , x2 ) ā M1 Ć— M2 : f1 (x1 ) = f2 (x2 )}

M1

(f1 ,N,f2 )

is a submanifold of M1 Ć— M2 , and it has the following universal property.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

2. Submersions and immersions 15

For any smooth mappings g1 : P ā’ M1 and g2 : P ā’ M2 with f1 ā—¦g1 = f2 ā—¦g2

there is a unique smooth mapping (g1 , g2 ) : P ā’ M1 Ć—N M2 with pr1 ā—¦ (g1 , g2 ) =

g1 and pr2 ā—¦ (g1 , g2 ) = g2 .

RR g2

P

R ,g )

T

R

(g 1 2

u

wM

M1 Ć—N M2

g1 2

pr2

u u

pr1 f2

wM w

N 1

f1

This is also called the pullback property in the category Mf of smooth man-

ifolds and smooth mappings. So one may say, that transversal pullbacks exist

in the category Mf .

Proof. M1 Ć—N M2 = (f1 Ć— f2 )ā’1 (ā), where f1 Ć— f2 : M1 Ć— M2 ā’ N Ć— N and

where ā is the diagonal of N Ć— N , and f1 Ć— f2 is transversal to ā if and only if

f1 and f2 are transversal.

2.20. The category of ļ¬bered manifolds. Consider a ļ¬bered manifold

(M, p, N ) from 2.4 and a point x ā N . Since p is a surjective submersion, the

injection ix : x ā’ N of x into N and p : M ā’ N are transversal. By 2.19, pā’1 (x)

is a submanifold of M , which is called the ļ¬ber over x ā N .

ĀÆĀÆĀÆ ĀÆĀÆĀÆ

Given another ļ¬bered manifold (M , p, N ), a morphism (M, p, N ) ā’ (M , p, N )

means a smooth map f : M ā’ N transforming each ļ¬ber of M into a ļ¬ber of

ĀÆ ĀÆĀÆ ĀÆ

M . The relation f (Mx ) ā‚ Mx deļ¬nes a map f : N ā’ N , which is characterized

by the property p ā—¦ f = f ā—¦ p. Since p ā—¦ f is a smooth map, f is also smooth by

ĀÆ ĀÆ

2.4. Clearly, all ļ¬bered manifolds and their morphisms form a category, which

will be denoted by FM. Transforming every ļ¬bered manifold (M, p, N ) into its

ĀÆĀÆĀÆ

base N and every ļ¬bered manifold morphism f : (M, p, N ) ā’ (M , p, N ) into the

ĀÆ

induced map f : N ā’ N deļ¬nes the base functor B : FM ā’ Mf .

ĀÆĀÆ

If (M, p, N ) and (M , p, N ) are two ļ¬bered manifolds over the same base N ,

ĀÆ ĀÆ

then the pullback M Ć—(p,N,p) M = M Ć—N M is called the ļ¬bered product of M

ĀÆ

ĀÆ ĀÆ

and M . If p, p and N are clear from the context, then M Ć—N M is also denoted

ĀÆ

ĀÆ ĀÆĀÆĀÆ

by M ā• M . Moreover, if f1 : (M1 , p1 , N ) ā’ (M1 , p1 , N ) and f2 : (M2 , p2 , N ) ā’

ĀÆĀÆĀÆ ĀÆ

(M2 , p2 , N ) are two FM-morphisms over the same base map f0 : N ā’ N , then

ĀÆ ĀÆĀÆ

the values of the restriction f1 Ć— f2 |M1 Ć—N M2 lie in M1 Ć—N M2 . The restricted

ĀÆ ĀÆĀÆ

map will be denoted by f1 Ć—f0 f2 : M1 Ć—N M2 ā’ M1 Ć—N M2 or f1 ā•f2 : M1 ā•M2 ā’

ĀÆ ĀÆ

M1 ā• M2 and will be called the ļ¬bered product of f1 and f2 .

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

16 Chapter I. Manifolds and Lie groups

3. Vector ļ¬elds and ļ¬‚ows

3.1. Deļ¬nition. A vector ļ¬eld X on a manifold M is a smooth section of

the tangent bundle; so X : M ā’ T M is smooth and ĻM ā—¦ X = IdM . A local

vector ļ¬eld is a smooth section, which is deļ¬ned on an open subset only. We

denote the set of all vector ļ¬elds by X(M ). With point wise addition and scalar

multiplication X(M ) becomes a vector space.

ā‚ ā‚

Example. Let (U, u) be a chart on M . Then the ā‚ui : U ā’ T M |U , x ā’ ā‚ui |x ,

described in 1.6, are local vector ļ¬elds deļ¬ned on U .

Lemma. If X is a vector ļ¬eld on M and (U, u) is a chart on M and x ā U , then

m m

ā‚ ā‚

we have X(x) = i=1 X(x)(ui ) ā‚ui |x . We write X|U = i=1 X(ui ) ā‚ui .

ā‚

3.2. The vector ļ¬elds ( ā‚ui )m on U , where (U, u) is a chart on M , form a

i=1

holonomic frame ļ¬eld. By a frame ļ¬eld on some open set V ā‚ M we mean

m = dim M vector ļ¬elds si ā X(V ) such that s1 (x), . . . , sm (x) is a linear basis

of Tx M for each x ā V . In general, a frame ļ¬eld on V is said to be holonomic, if

ā‚

V can be covered by an atlas (UĪ± , uĪ± )Ī±āA such that si |UĪ± = ā‚ui for all Ī± ā A.

Ī±

In the opposite case, the frame ļ¬eld is called anholonomic.

With the help of partitions of unity and holonomic frame ļ¬elds one may

construct ā˜manyā™ vector ļ¬elds on M . In particular the values of a vector ļ¬eld

can be arbitrarily preassigned on a discrete set {xi } ā‚ M .

3.3. Lemma. The space X(M ) of vector ļ¬elds on M coincides canonically with

the space of all derivations of the algebra C ā (M, R) of smooth functions, i.e.

those R-linear operators D : C ā (M, R) ā’ C ā (M, R) with D(f g) = D(f )g +

f D(g).

Proof. Clearly each vector ļ¬eld X ā X(M ) deļ¬nes a derivation (again called

X, later sometimes called LX ) of the algebra C ā (M, R) by the prescription

X(f )(x) := X(x)(f ) = df (X(x)).

If conversely a derivation D of C ā (M, R) is given, for any x ā M we consider

Dx : C ā (M, R) ā’ R, Dx (f ) = D(f )(x). Then Dx is a derivation at x of

C ā (M, R) in the sense of 1.5, so Dx = Xx for some Xx ā Tx M . In this

way we get a section X : M ā’ T M . If (U, u) is a chart on M , we have

m iā‚

i=1 X(x)(u ) ā‚ui |x by 1.6. Choose V open in M , V ā‚ V ā‚ U , and

Dx =

Ļ• ā C ā (M, R) such that supp(Ļ•) ā‚ U and Ļ•|V = 1. Then Ļ• Ā· ui ā C ā (M, R)

and (Ļ•ui )|V = ui |V . So D(Ļ•ui )(x) = X(x)(Ļ•ui ) = X(x)(ui ) and X|V =

m ā‚

i

i=1 D(Ļ•u )|V Ā· ā‚ui |V is smooth.

3.4. The Lie bracket. By lemma 3.3 we can identify X(M ) with the vector

space of all derivations of the algebra C ā (M, R), which we will do without any

notational change in the following.

If X, Y are two vector ļ¬elds on M , then the mapping f ā’ X(Y (f ))ā’Y (X(f ))

is again a derivation of C ā (M, R), as a simple computation shows. Thus there is

a unique vector ļ¬eld [X, Y ] ā X(M ) such that [X, Y ](f ) = X(Y (f )) ā’ Y (X(f ))

holds for all f ā C ā (M, R).

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

3. Vector ļ¬elds and ļ¬‚ows 17

ā‚

X i ā‚ui

In a local chart (U, u) on M one immediately veriļ¬es that for X|U =

ā‚

and Y |U = Y i ā‚ui we have

X i ā‚ui , Y j ā‚uj = X i ( ā‚ui Y j ) ā’ Y i ( ā‚ui X j )

ā‚ ā‚ ā‚ ā‚ ā‚

ā‚uj ,

i j i,j

since second partial derivatives commute. The R-bilinear mapping

] : X(M ) Ć— X(M ) ā’ X(M )

[,

is called the Lie bracket. Note also that X(M ) is a module over the algebra

C ā (M, R) by point wise multiplication (f, X) ā’ f X.

] : X(M ) Ć— X(M ) ā’ X(M ) has the following

Theorem. The Lie bracket [ ,

properties:

[X, Y ] = ā’[Y, X],

[X, [Y, Z]] = [[X, Y ], Z] + [Y, [X, Z]], the Jacobi identity,

[f X, Y ] = f [X, Y ] ā’ (Y f )X,

[X, f Y ] = f [X, Y ] + (Xf )Y.

The form of the Jacobi identity we have chosen says that ad(X) = [X, ] is

a derivation for the Lie algebra (X(M ), [ , ]).

The pair (X(M ), [ , ]) is the prototype of a Lie algebra. The concept of a

Lie algebra is one of the most important notions of modern mathematics.

Proof. All these properties can be checked easily for the commutator [X, Y ] =

X ā—¦ Y ā’ Y ā—¦ X in the space of derivations of the algebra C ā (M, R).

3.5. Integral curves. Let c : J ā’ M be a smooth curve in a manifold M

deļ¬ned on an interval J. We will use the following notations: c (t) = c(t) =

Ė™

d

dt c(t) := Tt c.1. Clearly c : J ā’ T M is smooth. We call c a vector ļ¬eld along

c since we have ĻM ā—¦ c = c.

A smooth curve c : J ā’ M will be called an integral curve or ļ¬‚ow line of a

vector ļ¬eld X ā X(M ) if c (t) = X(c(t)) holds for all t ā J.

3.6. Lemma. Let X be a vector ļ¬eld on M . Then for any x ā M there is

an open interval Jx containing 0 and an integral curve cx : Jx ā’ M for X (i.e.

cx = X ā—¦ cx ) with cx (0) = x. If Jx is maximal, then cx is unique.

Proof. In a chart (U, u) on M with x ā U the equation c (t) = X(c(t)) is an

ordinary diļ¬erential equation with initial condition c(0) = x. Since X is smooth

there is a unique local solution by the theorem of Picard-LindelĀØf, which even

o

depends smoothly on the initial values, [DieudonnĀ“ I, 69, 10.7.4]. So on M there

e

are always local integral curves. If Jx = (a, b) and limtā’bā’ cx (t) =: cx (b) exists

in M , there is a unique local solution c1 deļ¬ned in an open interval containing

b with c1 (b) = cx (b). By uniqueness of the solution on the intersection of the

two intervals, c1 prolongs cx to a larger interval. This may be repeated (also on

the left hand side of Jx ) as long as the limit exists. So if we suppose Jx to be

maximal, Jx either equals R or the integral curve leaves the manifold in ļ¬nite

(parameter-) time in the past or future or both.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

18 Chapter I. Manifolds and Lie groups

3.7. The ļ¬‚ow of a vector ļ¬eld. Let X ā X(M ) be a vector ļ¬eld. Let us

write FlX (x) = FlX (t, x) := cx (t), where cx : Jx ā’ M is the maximally deļ¬ned

t

integral curve of X with cx (0) = x, constructed in lemma 3.6. The mapping FlX

is called the ļ¬‚ow of the vector ļ¬eld X.

Theorem. For each vector ļ¬eld X on M , the mapping FlX : D(X) ā’ M is

smooth, where D(X) = xāM Jx Ć— {x} is an open neighborhood of 0 Ć— M in

R Ć— M . We have

FlX (t + s, x) = FlX (t, FlX (s, x))

in the following sense. If the right hand side exists, then the left hand side exists

and we have equality. If both t, s ā„ 0 or both are ā¤ 0, and if the left hand side

exists, then also the right hand side exists and we have equality.

Proof. As mentioned in the proof of 3.6, FlX (t, x) is smooth in (t, x) for small

t, and if it is deļ¬ned for (t, x), then it is also deļ¬ned for (s, y) nearby. These are

local properties which follow from the theory of ordinary diļ¬erential equations.

Now let us treat the equation FlX (t + s, x) = FlX (t, FlX (s, x)). If the right

hand side exists, then we consider the equation

FlX (t + s, x) = FlX (u, x)|u=t+s = X(FlX (t + s, x)),

d d

dt du

FlX (t + s, x)|t=0 = FlX (s, x).

But the unique solution of this is FlX (t, FlX (s, x)). So the left hand side exists

and equals the right hand side.

If the left hand side exists, let us suppose that t, s ā„ 0. We put

FlX (u, x) if u ā¤ s

cx (u) =

FlX (u ā’ s, FlX (s, x)) if u ā„ s.

FlX (u, x) = X(FlX (u, x))

d

for u ā¤ s

du

d

du cx (u) = =

FlX (u ā’ s, FlX (s, x)) = X(FlX (u ā’ s, FlX (s, x)))

d

du

for 0 ā¤ u ā¤ t + s.

= X(cx (u))

Also cx (0) = x and on the overlap both deļ¬nitions coincide by the ļ¬rst part of

the proof, thus we conclude that cx (u) = FlX (u, x) for 0 ā¤ u ā¤ t + s and we

have FlX (t, FlX (s, x)) = cx (t + s) = FlX (t + s, x).

Now we show that D(X) is open and FlX is smooth on D(X). We know

already that D(X) is a neighborhood of 0 Ć— M in R Ć— M and that FlX is smooth

near 0 Ć— M .

For x ā M let Jx be the set of all t ā R such that FlX is deļ¬ned and smooth

on an open neighborhood of [0, t] Ć— {x} (respectively on [t, 0] Ć— {x} for t < 0)

in R Ć— M . We claim that Jx = Jx , which ļ¬nishes the proof. It suļ¬ces to show

that Jx is not empty, open and closed in Jx . It is open by construction, and

not empty, since 0 ā Jx . If Jx is not closed in Jx , let t0 ā Jx ā© (Jx \ Jx ) and

suppose that t0 > 0, say. By the local existence and smoothness FlX exists and is

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

3. Vector ļ¬elds and ļ¬‚ows 19

smooth near [ā’Īµ, Īµ] Ć— {y := FlX (t0 , x)} for some Īµ > 0, and by construction FlX

exists and is smooth near [0, t0 ā’ Īµ] Ć— {x}. Since FlX (ā’Īµ, y) = FlX (t0 ā’ Īµ, x) we

conclude for t near [0, t0 ā’ Īµ], x near x, and t near [ā’Īµ, Īµ], that FlX (t + t , x ) =

FlX (t , FlX (t, x )) exists and is smooth. So t0 ā Jx , a contradiction.

3.8. Let X ā X(M ) be a vector ļ¬eld. Its ļ¬‚ow FlX is called global or complete,

if its domain of deļ¬nition D(X) equals R Ć— M . Then the vector ļ¬eld X itself

will be called a complete vector ļ¬eld. In this case FlX is also sometimes called

t

exp tX; it is a diļ¬eomorphism of M .

The support supp(X) of a vector ļ¬eld X is the closure of the set {x ā M :

X(x) = 0}.

Lemma. Every vector ļ¬eld with compact support on M is complete.

Proof. Let K = supp(X) be compact. Then the compact set 0 Ć— K has positive

distance to the disjoint closed set (RĆ—M )\D(X) (if it is not empty), so [ā’Īµ, Īµ]Ć—

K ā‚ D(X) for some Īµ > 0. If x ā K then X(x) = 0, so FlX (t, x) = x for all t

/

and R Ć— {x} ā‚ D(X). So we have [ā’Īµ, Īµ] Ć— M ā‚ D(X). Since FlX (t + Īµ, x) =

FlX (t, FlX (Īµ, x)) exists for |t| ā¤ Īµ by theorem 3.7, we have [ā’2Īµ, 2Īµ]Ć—M ā‚ D(X)

and by repeating this argument we get R Ć— M = D(X).

So on a compact manifold M each vector ļ¬eld is complete. If M is not

compact and of dimension ā„ 2, then in general the set of complete vector ļ¬elds

on M is neither a vector space nor is it closed under the Lie bracket, as the

2

following example on R2 shows: X = y ā‚x and Y = x ā‚y are complete, but

ā‚ ā‚

2

neither X + Y nor [X, Y ] is complete.

3.9. f -related vector ļ¬elds. If f : M ā’ M is a diļ¬eomorphism, then for any

vector ļ¬eld X ā X(M ) the mapping T f ā’1 ā—¦ X ā—¦ f is also a vector ļ¬eld, which

we will denote f ā— X. Analogously we put fā— X := T f ā—¦ X ā—¦ f ā’1 = (f ā’1 )ā— X.

But if f : M ā’ N is a smooth mapping and Y ā X(N ) is a vector ļ¬eld there

may or may not exist a vector ļ¬eld X ā X(M ) such that the following diagram

commutes:

wu

u

Tf

TM TN

(1) X Y

w N.

f

M

Deļ¬nition. Let f : M ā’ N be a smooth mapping. Two vector ļ¬elds X ā

X(M ) and Y ā X(N ) are called f -related, if T f ā—¦ X = Y ā—¦ f holds, i.e. if diagram

(1) commutes.

Example. If X ā X(M ) and Y ā X(N ) and X Ć— Y ā X(M Ć— N ) is given by

(X Ć— Y )(x, y) = (X(x), Y (y)), then we have:

(2) X Ć— Y and X are pr1 -related.

(3) X Ć— Y and Y are pr2 -related.

(4) X and X Ć— Y are ins(y)-related if and only if Y (y) = 0, where

ins(y)(x) = (x, y), ins(y) : M ā’ M Ć— N .

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

20 Chapter I. Manifolds and Lie groups

3.10. Lemma. Consider vector ļ¬elds Xi ā X(M ) and Yi ā X(N ) for i = 1, 2,

and a smooth mapping f : M ā’ N . If Xi and Yi are f -related for i = 1, 2, then

also Ī»1 X1 + Ī»2 X2 and Ī»1 Y1 + Ī»2 Y2 are f -related, and also [X1 , X2 ] and [Y1 , Y2 ]

are f -related.

Proof. The ļ¬rst assertion is immediate. To show the second let h ā C ā (N, R).

Then by assumption we have T f ā—¦ Xi = Yi ā—¦ f , thus:

(Xi (h ā—¦ f ))(x) = Xi (x)(h ā—¦ f ) = (Tx f.Xi (x))(h) =

= (T f ā—¦ Xi )(x)(h) = (Yi ā—¦ f )(x)(h) = Yi (f (x))(h) = (Yi (h))(f (x)),

so Xi (h ā—¦ f ) = (Yi (h)) ā—¦ f , and we may continue:

[X1 , X2 ](h ā—¦ f ) = X1 (X2 (h ā—¦ f )) ā’ X2 (X1 (h ā—¦ f )) =

= X1 (Y2 (h) ā—¦ f ) ā’ X2 (Y1 (h) ā—¦ f ) =

= Y1 (Y2 (h)) ā—¦ f ā’ Y2 (Y1 (h)) ā—¦ f = [Y1 , Y2 ](h) ā—¦ f.

But this means T f ā—¦ [X1 , X2 ] = [Y1 , Y2 ] ā—¦ f .

3.11. Corollary. If f : M ā’ N is a local diļ¬eomorphism (so (Tx f )ā’1 makes

sense for each x ā M ), then for Y ā X(N ) a vector ļ¬eld f ā— Y ā X(M ) is deļ¬ned

by (f ā— Y )(x) = (Tx f )ā’1 .Y (f (x)). The linear mapping f ā— : X(N ) ā’ X(M ) is

then a Lie algebra homomorphism, i.e. f ā— [Y1 , Y2 ] = [f ā— Y1 , f ā— Y2 ].

3.12. The Lie derivative of functions. For a vector ļ¬eld X ā X(M ) and

f ā C ā (M, R) we deļ¬ne LX f ā C ā (M, R) by

X

d

LX f (x) := dt |0 f (Fl (t, x)) or

Xā—

ā—¦ FlX ).

d d

LX f := dt |0 (Flt ) f = dt |0 (f t

Since FlX (t, x) is deļ¬ned for small t, for any x ā M , the expressions above make

sense.

Lemma. dt (FlX )ā— f = (FlX )ā— X(f ), in particular for t = 0 we have LX f =

d

t t

X(f ) = df (X).

3.13. The Lie derivative for vector ļ¬elds. For X, Y ā X(M ) we deļ¬ne

LX Y ā X(M ) by

Xā— X

ā—¦ Y ā—¦ FlX ),

d d

LX Y := dt |0 (Flt ) Y dt |0 (T (Flā’t )

= t

and call it the Lie derivative of Y along X.

Xā—

= (FlX )ā— LX Y = (FlX )ā— [X, Y ].

d

Lemma. LX Y = [X, Y ] and dt (Flt ) Y t t

Proof. Let f ā C ā (M, R) be a function and consider the mapping Ī±(t, s) :=

Y (FlX (t, x))(f ā—¦ FlX ), which is locally deļ¬ned near 0. It satisļ¬es

s

Ī±(t, 0) = Y (FlX (t, x))(f ),

Ī±(0, s) = Y (x)(f ā—¦ FlX ),

s

X X

ā‚ ā‚ ā‚

ā‚t Ī±(0, 0) = ā‚t 0 Y (Fl (t, x))(f ) = ā‚t 0 (Y f )(Fl (t, x)) = X(x)(Y f ),

X X

ā‚ ā‚ ā‚

ā‚s |0 Y (x)(f ā—¦ Fls ) = Y (x) ā‚s |0 (f ā—¦ Fls ) = Y (x)(Xf ).

ā‚s Ī±(0, 0) =

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

3. Vector ļ¬elds and ļ¬‚ows 21

But on the other hand we have

(FlX (u, x))(f ā—¦ FlX ) =

ā‚ ā‚

ā‚u |0 Ī±(u, ā’u) ā‚u |0 Y

= ā’u

T (FlX ) ā—¦ Y ā—¦ FlX

ā‚

ā‚u |0

= (f ) = (LX Y )x (f ),

ā’u u

x

so the ļ¬rst assertion follows. For the second claim we compute as follows:

Xā—

T (FlX ) ā—¦ T (FlX ) ā—¦ Y ā—¦ FlX ā—¦ FlX

ā‚ ā‚

ā‚s |0

ā‚t (Flt ) Y = ā’t ā’s s t

= T (FlX ) ā—¦ T (FlX ) ā—¦ Y ā—¦ FlX ā—¦ FlX

ā‚

ā‚s |0

ā’t ā’s s t

= T (FlX ) ā—¦ [X, Y ] ā—¦ FlX = (FlX )ā— [X, Y ].

ā’t t t

3.14. Lemma. Let X ā X(M ) and Y ā X(N ) be f -related vector ļ¬elds for

a smooth mapping f : M ā’ N . Then we have f ā—¦ FlX = FlY ā—¦f , whenever

t t ā—

both sides are deļ¬ned. In particular, if f is a diļ¬eomorphism we have Flf Y =

t

f ā’1 ā—¦ FlY ā—¦f .

t

Proof. We have dt (f ā—¦ FlX ) = T f ā—¦ dt FlX = T f ā—¦ X ā—¦ FlX = Y ā—¦ f ā—¦ F lt

d d X

t t t

and f (FlX (0, x)) = f (x). So t ā’ f (FlX (t, x)) is an integral curve of the vector

ļ¬eld Y on N with initial value f (x), so we have f (FlX (t, x)) = FlY (t, f (x)) or

f ā—¦ FlX = FlY ā—¦f .

t t

3.15. Corollary. Let X, Y ā X(M ). Then the following assertions are equiva-

lent

(1) LX Y = [X, Y ] = 0.

(2) (FlX )ā— Y = Y , wherever deļ¬ned.

t

(3) Flt ā—¦ FlY = FlY ā—¦ FlX , wherever deļ¬ned.

X

s s t

Proof. (1) ā” (2) is immediate from lemma 3.13. To see (2) ā” (3) we note

Xā—

that FlX ā—¦ FlY = FlY ā—¦ FlX if and only if FlY = FlX ā—¦ FlY ā—¦ FlX = Fl(Flt ) Y by

ā’t

t s s t s s t s

lemma 3.14; and this in turn is equivalent to Y = (FlX )ā— Y .

t

3.16. Theorem. Let M be a manifold, let Ļ•i : R Ć— M ā UĻ•i ā’ M be smooth

mappings for i = 1, . . . , k where each UĻ•i is an open neighborhood of {0} Ć— M

in R Ć— M , such that each Ļ•i is a diļ¬eomorphism on its domain, Ļ•i = IdM , and

t 0

j j ā’1

ā—¦ (Ļ•t ) ā—¦ Ļ•j ā—¦ Ļ•i .

i ā’1

ā‚ i i j i

ā‚t 0 Ļ•t = Xi ā X(M ). We put [Ļ• , Ļ• ]t = [Ļ•t , Ļ•t ] := (Ļ•t ) t t

Then for each formal bracket expression P of lenght k we have

|0 P (Ļ•1 , . . . , Ļ•k )

ā‚

for 1 ā¤ < k,

0= t t

ā‚t

1 ā‚k 1 k

k! ā‚tk |0 P (Ļ•t , . . . , Ļ•t ) ā X(M )

P (X1 , . . . , Xk ) =

in the sense explained in step 2 of the proof. In particular we have for vector

ļ¬elds X, Y ā X(M )

Y X Y X

ā‚

ā‚t 0 (Flā’t ā—¦ Flā’t ā—¦ Flt ā—¦ Flt ),

0=

1 ā‚2 Y X Y X

2 ā‚t2 |0 (Flā’t ā—¦ Flā’t ā—¦ Flt ā—¦ Flt ).

[X, Y ] =

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

22 Chapter I. Manifolds and Lie groups

Proof. Step 1. Let c : R ā’ M be a smooth curve. If c(0) = x ā M , c (0) =

0, . . . , c(kā’1) (0) = 0, then c(k) (0) is a well deļ¬ned tangent vector in Tx M which

is given by the derivation f ā’ (f ā—¦ c)(k) (0) at x.

For we have

k

k

(k) (k)

(f ā—¦ c)(j) (0)(g ā—¦ c)(kā’j) (0)

((f.g) ā—¦ c) (0) = ((f ā—¦ c).(g ā—¦ c)) (0) = j

j=0

= (f ā—¦ c)(k) (0)g(x) + f (x)(g ā—¦ c)(k) (0),

since all other summands vanish: (f ā—¦ c)(j) (0) = 0 for 1 ā¤ j < k.

Step 2. Let Ļ• : R Ć— M ā UĻ• ā’ M be a smooth mapping where UĻ• is an open

neighborhood of {0} Ć— M in R Ć— M , such that each Ļ•t is a diļ¬eomorphism on

its domain and Ļ•0 = IdM . We say that Ļ•t is a curve of local diļ¬eomorphisms

though IdM .

ā‚j 1 ā‚k

From step 1 we see that if ā‚tj |0 Ļ•t = 0 for all 1 ā¤ j < k, then X := k! ā‚tk |0 Ļ•t

is a well deļ¬ned vector ļ¬eld on M . We say that X is the ļ¬rst non-vanishing

derivative at 0 of the curve Ļ•t of local diļ¬eomorphisms. We may paraphrase this

as (ā‚t |0 Ļ•ā— )f = k!LX f .

k

t

Claim 3. Let Ļ•t , Ļt be curves of local diļ¬eomorphisms through IdM and let

f ā C ā (M, R). Then we have

k

j kā’j

ā— ā—

Ļ•ā— )f (ā‚t |0 Ļt )(ā‚t |0 Ļ•ā— )f.

ā—

k

k k

ā‚t |0 (Ļ•t ā—¦ Ļt ) f = ā‚t |0 (Ļt ā—¦ =

t t

j

j=0

Also the multinomial version of this formula holds:

k! j j

ā‚t |0 (Ļ•1 ā—¦ . . . ā—¦ Ļ•t )ā— f = (ā‚t 1 |0 (Ļ•t )ā— ) . . . (ā‚t 1 |0 (Ļ•1 )ā— )f.

k

t t

j1 ! . . . j !

j1 +Ā·Ā·Ā·+j =k

We only show the binomial version. For a function h(t, s) of two variables we

have

k

j kā’j

k

k

ā‚t h(t, t) = ā‚t ā‚s h(t, s)|s=t ,

j

j=0

since for h(t, s) = f (t)g(s) this is just a consequence of the Leibnitz rule, and

linear combinations of such decomposable tensors are dense in the space of all

functions of two variables in the compact C ā -topology, so that by continuity

the formula holds for all functions. In the following form it implies the claim:

k

j kā’j

k

k

ā‚t |0 f (Ļ•(t, Ļ(t, x))) = ā‚t ā‚s f (Ļ•(t, Ļ(s, x)))|t=s=0 .

j

j=0

Claim 4. Let Ļ•t be a curve of local diļ¬eomorphisms through IdM with ļ¬rst

k

non-vanishing derivative k!X = ā‚t |0 Ļ•t . Then the inverse curve of local diļ¬eo-

morphisms Ļ•ā’1 has ļ¬rst non-vanishing derivative ā’k!X = ā‚t |0 Ļ•ā’1 .

k

t t

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

3. Vector ļ¬elds and ļ¬‚ows 23

For we have Ļ•ā’1 ā—¦ Ļ•t = Id, so by claim 3 we get for 1 ā¤ j ā¤ k

t

j

j jā’i

ā‚t |0 (Ļ•ā’1 (ā‚t |0 Ļ•ā— )(ā‚t (Ļ•ā’1 )ā— )f =

ā— j i

ā—¦ Ļ•t ) f =

0= t t

t

i

i=0

j j

= ā‚t |0 Ļ•ā— (Ļ•ā’1 )ā— f + Ļ•ā— ā‚t |0 (Ļ•ā’1 )ā— f,

t

t 0

0

j j

ńņš. 1 |