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NATURAL
OPERATIONS
IN
DIFFERENTIAL
GEOMETRY



Ivan Kol´ˇ
ar
Peter W. Michor
Jan Slov´k
a




Mailing address: Peter W. Michor,
Institut f¨r Mathematik der Universit¨t Wien,
u a
Strudlhofgasse 4, A-1090 Wien, Austria.

Ivan Kol´ˇ, Jan Slov´k,
ar a
Department of Algebra and Geometry
Faculty of Science, Masaryk University
Jan´ˇkovo n´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. Di¬erentiable manifolds . . . . . . . . . . . . . . . . . . . ..4
2. Submersions and immersions . . . . . . . . . . . . . . . . . . 11
3. Vector ¬elds and ¬‚ows . . . . . . . . . . . . . . . . . . . . . 16
4. Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5. Lie subgroups and homogeneous spaces . . . . . . . . . . . . . 41
CHAPTER II.
DIFFERENTIAL FORMS . . . . . . . . . . . . . . . . ... 49
6. Vector bundles . . . . . . . . . . . . . . . . . . . . . ... 49
7. Di¬erential forms . . . . . . . . . . . . . . . . . . . . ... 61
8. Derivations on the algebra of di¬erential forms
and the Fr¨licher-Nijenhuis bracket . . . . . . . . . . . .
o ... 67
CHAPTER III.
BUNDLES AND CONNECTIONS . . . . . . . . . . . . . . . 76
9. General ¬ber bundles and connections . . . . . . . . . . . . . . 76
10. Principal ¬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 ¬ber bundles . . . . . . . . . . . . . 149
16. Canonical di¬erential forms ............... . . . 154
17. Connections and the absolute di¬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 di¬erential .. 220
26. The tensor evaluation theorem . . . . . . . . . . . . . . ... 223
27. Generalized invariant tensors . . . . . . . . . . . . . . . ... 230
28. The orbit reduction . . . . . . . . . . . . . . . . . . . ... 233
29. The method of di¬erential equations ........... ... 245

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CHAPTER VII.
FURTHER APPLICATIONS . . . . . . . . . . . . . . . . . . 249
30. The Fr¨licher-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 ¬‚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 ¬elds to Weil bundles . . . . . . . . . . . 351
43. The case of the second order tangent vectors . . . . . . . . . . . 357
44. Induced vector ¬elds on jet bundles . . . . . . . . . . . . . . . 360
45. Prolongations of connections to F Y ’ M . . . . . . . . . . . . 363
46. The cases F Y ’ F M and F Y ’ 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 ¬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 di¬erential geometry. This is a ¬eld which every di¬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 di¬erential forms. In the
background of this statement are the following general concepts. The vector
bundle Λk T — 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
di¬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 Λk T — M into sections of Λk+1 T — M for every manifold M
can be expressed by saying that d is an operator from Λk T — M into Λk+1 T — M .
That the exterior derivative d commutes with local di¬eomorphisms now means,
that d is a natural operator from the functor Λk T — into functor Λk+1 T — . 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 ¬eld we are discussing here. A systematic treatment
of naturality in di¬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 di¬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 di¬erential ge-
ometry, jets have been mostly used as a purely technical tool in certain problems
in the theory of systems of partial di¬erential equations, in singularity theory,
in variational calculus and in higher order mechanics. But the theory of nat-
ural bundles and natural operators clari¬es once again that jets are one of the
fundamental concepts in di¬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 ˜les espaces des points
proches™ 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 di¬erential geometry (manifolds, ¬‚ows, Lie groups, di¬erential
forms, bundles and connections) which stresses naturality and functoriality from
the beginning and is as coordinate free as possible. Here we present the Fr¨licher-
o
Nijenhuis bracket (a natural extension of the Lie bracket from vector ¬elds to

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


vector valued di¬erential forms) as one of the basic structures of di¬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 ¬rst on general
¬ber bundles (without structure group), with curvature, parallel transport and
Bianchi identity, and only then add G-equivariance as a further property for
principal ¬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 speci¬c feature of the book is that the authors are interested in general
points of view towards di¬erent structures in di¬erential geometry. The modern
development of global di¬erential geometry clari¬ed that di¬erential geomet-
ric objects form ¬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 ¬bered manifold F M ’ M over M and every local di¬eo-
morphism f : M ’ N into a ¬bered manifold morphism F f : F M ’ F N over
f . The geometric character of F is then expressed by the functoriality condition
F (g —¦ f ) = F g —¦ 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
di¬eomorphisms. An important result by Palais and Terng, completed by Ep-
stein and Thurston, reads that every lifting functor has ¬nite order. This gives
a full description of all natural bundles as the ¬ber bundles associated with the
r-th order frame bundles, which is useful in many problems. However in several
cases it is not su¬cient to study the bundle functors de¬ned on the category
Mfm . For example, if we have a Lie group G, its multiplication is a smooth
map µ : G — G ’ G. To construct an induced map F µ : F (G — G) ’ F G,
we need a functor F de¬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 µ 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 di¬erential geometry are de¬ned on the category of ¬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 de¬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-

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
Preface 3


tions of certain ¬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-de¬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 e¬ective way how to treat
natural operators is to reduce the question to a ¬nite order problem, in which
the corresponding jet spaces are ¬nite dimensional. Since the ¬nite order natural
operators are in a simple bijection with the equivariant maps between the corre-
sponding standard ¬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 ¬nite order situation has been of great pro¬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 ¬nite order. We develop an essential
generalization of this technique and we present a uni¬ed approach to the ¬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 ¬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
Gilkey™s theorem that all di¬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 in¬nite dimensional Lie algebras of vector
¬elds. In chapter X we study systematically the natural operators on vector ¬elds
and connections. Chapter XI is devoted to a general theory of Lie derivatives,
in which the geometric approach clari¬es, among other things, the relations to
natural operators.
The material for chapters VI, X and sections 12, 30“32, 47, 49, 50, 52“54 was
prepared by the ¬rst author (I.K.), for chapters I, II, III, VIII by the second au-
thor (P.M.) and for chapters V, IX and sections 13“17, 33, 34, 48, 51 by the third
author (J.S.). The authors acknowledge A. Cap, M. Doupovec, and J. Janyˇka, 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 ¬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 re¬‚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 di¬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 - Hausdor¬
formula for Lie groups. We also prove that closed subgroups of Lie groups are
Lie subgroups.


1. Di¬erentiable manifolds

1.1. A topological manifold is a separable Hausdor¬ space M which is locally
homeomorphic to Rn . So for any x ∈ M there is some homeomorphism u : U ’
u(U ) ⊆ 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 pre¬x pure.
A family (U± , u± )±∈A of charts on M such that the U± form a cover of M is
called an atlas. The mappings u±β := u± —¦ u’1 : uβ (U±β ) ’ u± (U±β ) are called
β
the chart changings for the atlas (U± ), where U±β := U± © Uβ .
An atlas (U± , u± )±∈A for a manifold M is said to be a C k -atlas, if all chart
changings u±β : uβ (U±β ) ’ u± (U±β ) are di¬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 di¬erential
topology we know that if M has a C 1 -structure, then it also has a C 1 -equivalent
C ∞ -structure and even a C 1 -equivalent C ω -structure, where C ω 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 di¬erentiable struc-
tures. For example, every 4-dimensional manifold is smooth o¬ 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 ¬nitely many, see [Milnor, 56].
But the most surprising result is that on R4 there are uncountably many pair-
wise inequivalent (exotic) di¬erentiable structures. This follows from the results

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1. Di¬erentiable manifolds 5


of [Donaldson, 83] and [Freedman, 82], see [Gompf, 83] or [Freedman-Feng Luo,
89] for an overview.
Note that for a Hausdor¬ C ∞ -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 ∞ -manifold, and smooth is
used synonymously for C ∞ , it will be Hausdor¬, separable, ¬nite dimensional,
to state it precisely.
Note ¬nally that any manifold M admits a ¬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 ’ N between manifolds is said to be C k if for each
x ∈ M and each chart (V, v) on N with f (x) ∈ V there is a chart (U, u) on M
with x ∈ U , f (U ) ⊆ V , and v —¦ f —¦ u’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 ’ N is called a C k -di¬eomorphism if f ’1 : N ’ M
exists and is also C k . Two manifolds are called di¬eomorphic if there exists a dif-
feomorphism between them. From di¬erential topology we know that if there is a
C 1 -di¬eomorphism between M and N , then there is also a C ∞ -di¬eomorphism.
All smooth manifolds together with the C ∞ -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 ’ N between manifolds of the same dimension is called
a local di¬eomorphism, if each x ∈ M has an open neighborhood U such that
f |U : U ’ f (U ) ‚ N is a di¬eomorphism. Note that a local di¬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 ∞ (M, R), in order to distinguish it clearly from spaces of sections which
will appear later. C ∞ (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 ∈ M : f (x) = 0}. The zero set of f is the set where f
vanishes, Z(f ) = {x ∈ M : f (x) = 0}.
Any manifold admits smooth partitions of unity: Let (U± )±∈A be an open
cover of M . Then there is a family (•± )±∈A of smooth functions on M , such
that supp(•± ) ‚ U± , (supp(•± )) is a locally ¬nite family, and ± •± = 1
(locally this is a ¬nite sum).
1.4. Germs. Let M and N be manifolds and x ∈ M . We consider all smooth
mappings f : Uf ’ N , where Uf is some open neighborhood of x in M , and we
put f ∼ 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

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
6 Chapter I. Manifolds and Lie groups


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

space of all germs at x of mappings M ’ 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

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 de¬ned on the whole of M . For

germs of real analytic or holomorphic functions this is not true. So Cx (M, R)
is the quotient of the algebra C ∞ (M, R) by the ideal of all smooth functions
f : M ’ R which vanish on some neighborhood (depending on f ) of x.
1.5. The tangent space of Rn . Let a ∈ Rn . A tangent vector with foot
point a is simply a pair (a, X) with X ∈ Rn , also denoted by Xa . It induces
a derivation Xa : C ∞ (Rn , R) ’ R by Xa (f ) = df (a)(Xa ). The value depends
only on the germ of f at a and we have Xa (f · g) = Xa (f ) · g(a) + f (a) · Xa (g)
(the derivation property).
If conversely D : C ∞ (Rn , R) ’ R is linear and satis¬es D(f · g) = D(f ) ·
g(a) + f (a) · 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 ∈ C ∞ (Rn , R) we
have
1
d
+ t(x ’ a))dt
f (x) = f (a) + dt f (a
0
n 1
‚f
+ t(x ’ a))dt (xi ’ ai )
= f (a) + ‚xi (a
0
i=1
n
hi (x)(xi ’ ai ).
= f (a) +
i=1
D(1) = D(1 · 1) = 2D(1), so D(constant) = 0. Thus
n
hi (x)(xi ’ ai ))
D(f ) = D(f (a) +
i=1
n n
i i
hi (a)(D(xi ) ’ 0)
D(hi )(a ’ a ) +
=0+
i=1 i=1
n
‚f i
= ‚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 ) ‚xi |a .
‚ ‚
) ‚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. Di¬erentiable manifolds 7


1.6. The tangent space of a manifold. Let M be a manifold and let x ∈
M and dim M = n. Let Tx M be the vector space of all derivations at x of

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 ∞ (M, R) at x factors to a derivation of

Cx (M, R).)
So Tx M consists of all linear mappings Xx : C ∞ (M, R) ’ R satisfying Xx (f ·
g) = Xx (f ) · g(x) + f (x) · 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 ∈ U , then u— : f ’ f —¦ u induces an iso-
morphism of algebras Cu(x) (Rn , R) ∼ Cx (M, R), and thus also an isomorphism

=∞
Tx u : Tx M ’ Tu(x) Rn , given by (Tx u.Xx )(f ) = Xx (f —¦ 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 ” a dot will frequently denote an application of
a linear or ¬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)’1 ( ‚xi |u(x) ) = (Tx u)’1 (u(x), ei ).
‚ ‚
‚ui |x


‚ui |x ∈ Tx M is the derivation given by
So

‚(f —¦ u’1 )

‚ui |x (f ) = (u(x)).
‚xi
From 1.5 we have now
n
(Tx u.Xx )(xi ) ‚xi |u(x) =

Tx u.Xx =
i=1
n n
i
Xx (ui ) ‚xi |u(x) .
‚ ‚
Xx (x —¦ u) ‚xi |u(x)
= =
i=1 i=1

1.7. The tangent bundle. For a manifold M of dimension n we put T M :=
x∈M Tx M , the disjoint union of all tangent spaces. This is a family of vec-
tor spaces parameterized by M , with projection πM : T M ’ M given by
πM (Tx M ) = x.
’1
For any chart (U± , u± ) of M consider the chart (πM (U± ), T u± ) on T M ,
’1
where T u± : πM (U± ) ’ u± (U± ) — Rn is given by the formula T u± .X =
(u± (πM (X)), TπM (X) u± .X). Then the chart changings look as follows:
’1
T uβ —¦ (T u± )’1 : T u± (πM (U±β )) = u± (U±β ) — Rn ’
’1
’ uβ (U±β ) — Rn = T uβ (πM (U±β )),
((T uβ —¦ (T u± )’1 )(y, Y ))(f ) = ((T u± )’1 (y, Y ))(f —¦ uβ )
= (y, Y )(f —¦ uβ —¦ u’1 ) = d(f —¦ uβ —¦ u’1 )(y).Y
± ±

= df (uβ —¦ u’1 (y)).d(uβ —¦ u’1 )(y).Y
± ±

= (uβ —¦ u’1 (y), d(uβ —¦ u’1 )(y).Y )(f ).
± ±


Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
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 u± become homeomorphisms. This is a Hausdor¬ topology,
since X, Y ∈ T M may be separated in M if π(X) = π(Y ), and in one chart if
π(X) = π(Y ). So T M is again a smooth manifold in a canonical way; the triple
(T M, πM , M ) is called the tangent bundle of M .

1.8. Kinematic de¬nition of the tangent space. Consider C0 (R, M ), the
space of germs at 0 of smooth curves R ’ M . We put the following equivalence

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) ∈ U
d d
we have dt |0 (u —¦ c)(t) = dt |0 (u —¦ e)(t). The equivalence classes are called velocity
vectors of curves in M . We have the following mappings

u
∞ ∞

g
e
C0 (R, M )/ ∼ C0 (R, M )
ee
ee β
ue u
ev0
±

w M,
TM πM

d
where ±(c)(germc(0) f ) = dt |0 f (c(t)) and β : T M ’ C0 (R, M ) is given by:
β((T u)’1 (y, Y )) is the germ at 0 of t ’ u’1 (y + tY ). So T M is canonically
identi¬ed with the set of all possible velocity vectors of curves in M .
1.9. Let f : M ’ N be a smooth mapping between manifolds. Then f induces a
linear mapping Tx f : Tx M ’ Tf (x) N for each x ∈ M by (Tx f.Xx )(h) = Xx (h—¦f )
for h ∈ Cf (x) (N, R). This mapping is linear since f — : Cf (x) (N, R) ’ Cx (M, R),
∞ ∞ ∞

given by h ’ h —¦ 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

u’1 ),
(Tx f. ‚ui |x )(v j ) = j j
‚ ‚ ‚
‚ui |x (v —¦ f) = ‚xi (v —¦ f —¦
‚ ‚ j‚
Tx f. ‚ui |x = j (Tx f. ‚ui |x )(v ) ‚v j |f (x) by 1.7
‚(v j —¦f —¦u’1 ) ‚
(u(x)) ‚vj |f (x) .
= ‚xi
j

‚ ‚
So the matrix of Tx f : Tx M ’ Tf (x) N in the bases ( ‚ui |x ) and ( ‚vj |f (x) ) is just
the Jacobi matrix d(v —¦ f —¦ u’1 )(u(x)) of the mapping v —¦ f —¦ u’1 at u(x), so
Tf (x) v —¦ Tx f —¦ (Tx u)’1 = d(v —¦ f —¦ u’1 )(u(x)).
Let us denote by T f : T M ’ T N the total mapping, given by T f |Tx M :=
Tx f . Then the composition T v —¦ T f —¦ (T u)’1 : u(U ) — Rm ’ v(V ) — Rn is given
by (y, Y ) ’ ((v —¦ f —¦ u’1 )(y), d(v —¦ f —¦ u’1 )(y)Y ), and thus T f : T M ’ T N is
again smooth.
If f : M ’ N and g : N ’ P are smooth mappings, then we have T (g —¦ f ) =
T g —¦ T f . This is a direct consequence of (g —¦ f )— = f — —¦ g — , and it is the global
version of the chain rule. Furthermore we have T (IdM ) = IdT M .
If f ∈ C ∞ (M, R), then T f : T M ’ T R = R — R. We then de¬ne the
di¬erential of f by df := pr2 —¦ T f : T M ’ R. Let t denote the identity function
on R, then (T f.Xx )(t) = Xx (t —¦ f ) = Xx (f ), so we have df (Xx ) = Xx (f ).

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
1. Di¬erentiable manifolds 9


1.10. Submanifolds. A subset N of a manifold M is called a submanifold, if for
each x ∈ N there is a chart (U, u) of M such that u(U © N ) = u(U ) © (Rk — 0),
where Rk — 0 ’ Rk — Rn’k = Rn . Then clearly N is itself a manifold with
(U © N, u|U © N ) as charts, where (U, u) runs through all submanifold charts as
above and the injection i : N ’ M is an embedding in the following sense:
An embedding f : N ’ 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 ’ f (N ) is a di¬eomorphism.
If f : Rn ’ Rq is smooth and the rank of f (more exactly: the rank of its
derivative) is q at each point of f ’1 (0), say, then f ’1 (0) is a submanifold of Rn
of dimension n ’ 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 [Dieudonn´, I, 60, 10.3.1].
e
Theorem. Let f : W ’ 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 ∈ W , then for each
a ∈ W there are charts (U, u) of W centered at a and (V, v) of Rq centered at
f (a) such that v —¦ f —¦ u’1 : u(U ) ’ v(V ) has the following form:

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

So f ’1 (b) is a submanifold of W of dimension n ’ k for each b ∈ 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 ∈ Rn+1 : x, x = 1}. Since f (x) = x, x , f : Rn+1 ’ R, satis¬es df (x)y =
2 x, y , it is of rank 1 o¬ 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 ∈ S n (˜south pole™). Let

x’ x,a a
u+ : U+ ’ {a}⊥ ,
U+ := S n \ {a}, u+ (x) = 1’ x,a ,
x’ x,a a
u’ : U’ ’ {a}⊥ ,
U’ := S n \ {’a}, u’ (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 ’1
u’1 (y) = for y ∈ {a}⊥
2
|y|2 +1 a + |y|2 +1 y
+


and (u’ —¦ u’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 (U± , u± )±∈A and (Vβ , vβ )β∈B , respectively. Then the family (U± — Vβ , u± —
vβ : U± — Vβ ’ Rm — Rn )(±,β)∈A—B is a smooth atlas for the cartesian product
M — N . Clearly the projections
pr1 pr2
M ←’ M — N ’’ N
’ ’

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
10 Chapter I. Manifolds and Lie groups


are also smooth. The product (M — N, pr1 , pr2 ) has the following universal
property:
For any smooth manifold P and smooth mappings f : P ’ M and g : P ’ N
the mapping (f, g) : P ’ M — N , (f, g)(x) = (f (x), g(x)), is the unique smooth
mapping with pr1 —¦ (f, g) = f , pr2 —¦ (f, g) = g.
From the construction of the tangent bundle in 1.7 it is immediately clear
that
T (pr1 ) T (pr2 )
T M ← ’ ’ T (M — N ) ’ ’ ’ T N
’’ ’’
is again a product, so that T (M — N ) = T M — T N in a canonical way.
Clearly we can form products of ¬nitely many manifolds.
1.13. Theorem. Let M be a connected manifold and suppose that f : M ’ M
is smooth with f —¦ f = f . Then the image f (M ) of f is a submanifold of M .
This result can also be expressed as: ˜smooth retracts™ 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 di¬erent dimension in di¬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 ∈ U . Then by theorem 1.10 the result follows.
For x ∈ f (M ) we have Tx f —¦ Tx f = Tx f , thus im Tx f = ker(Id ’Tx f ) and
rank Tx f + rank(Id ’Tx f ) = dim M . Since rank Tx f and rank(Id ’Tx f ) can-
not fall locally, rank Tx f is locally constant for x ∈ f (M ), and since f (M ) is
connected, rank Tx f = r for all x ∈ f (M ).
But then for each x ∈ f (M ) there is an open neighborhood Ux in M with
rank Ty f ≥ r for all y ∈ Ux . On the other hand rank Ty f = rank Ty (f —¦ f ) =
rank Tf (y) f —¦ Ty f ¤ rank Tf (y) f = r. So the neighborhood we need is given by
U = x∈f (M ) Ux .
1.14. Corollary. 1. The (separable) connected smooth manifolds are exactly
the smooth retracts of connected open subsets of Rn ™s.
2. f : M ’ 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 ’ M with
r —¦ 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. 109“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 ™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 [Br¨cker-J¨nich, 73, p 73]),
o a
(2) n = 2m (see [Whitney, 44]).
(3) Conjecture (still unproved): The minimal n is n = 2m ’ ±(m) + 1, where
±(m) is the number of 1™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 ’ Rn , if
(1) n = 2m (see [Hirsch, 76]),
(2) n = 2m ’ ±(m) (see [Cohen, 82]).


2. Submersions and immersions

2.1. De¬nition. A mapping f : M ’ N between manifolds is called a sub-
mersion at x ∈ M , if the rank of Tx f : Tx M ’ 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 ∈ M .
2.2. Lemma. If f : M ’ N is a submersion at x ∈ 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 —¦ f —¦ u’1 looks as follows:
(y 1 , . . . , y n , y n+1 , . . . , y m ) ’ (y 1 , . . . , y n )
Proof. Use the inverse function theorem.
2.3. Corollary. Any submersion f : M ’ N is open: for each open U ‚ M
the set f (U ) is open in N .
2.4. De¬nition. A triple (M, p, N ), where p : M ’ N is a surjective submer-
sion, is called a ¬bered manifold. M is called the total space, N is called the
base.
A ¬bered manifold admits local sections: For each x ∈ M there is an open
neighborhood U of p(x) in N and a smooth mapping s : U ’ M with p—¦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 ¬bered manifold and f : N ’ P is a mapping into some further
manifold, such that f —¦ p : M ’ P is smooth, then f is smooth.
2.5. De¬nition. A smooth mapping f : M ’ N is called an immersion at
x ∈ M if the rank of Tx f : Tx M ’ 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 ∈ M .
2.6. Lemma. If f : M ’ N is an immersion, then for any chart (U, u) centered
at x ∈ M there is a chart (V, v) centered at f (x) on N such that v —¦ f —¦ u’1 has
the form:
(y 1 , . . . , y m ) ’ (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 ’ N is an immersion, then for any x ∈ M there is
an open neighborhood U of x ∈ M such that f (U ) is a submanifold of N and
f |U : U ’ f (U ) is a di¬eomorphism.
2.8. De¬nition. If i : M ’ 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 ) ‚ N . All this is illustrated by the following example.
Consider the curve γ(t) = (sin3 t, sin t. cos t) in R2 . Then ((’π, π), γ|(’π, π))
and ((0, 2π), γ|(0, 2π)) are two di¬erent immersed submanifolds, but the image
of the embedding is in both cases just the ¬gure eight.
2.9. Let M be a submanifold of N . Then the embedding i : M ’ N is an
injective immersion with the following property:
(1) For any manifold Z a mapping f : Z ’ M is smooth if and only if
i —¦ f : Z ’ 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 ’ 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 ’ N with property
2.9.1 (initial mappings in categorical terms) is too di¬cult as remark 2.13 below
shows.
2.11. Lemma. If an injective immersion i : M ’ 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 π : R2 ’ T2 is a covering map, so locally a di¬eomorphism.
Let us also consider the mapping f : R ’ R2 , f (t) = (t, ±.t), where ± is
irrational. Then π —¦ f : R ’ T2 is an injective immersion with dense image, and
it is obviously not a homeomorphism onto its image. But π —¦ f has property
2.9.1, which follows from the fact that π is a covering map.
2.13. Remark. If f : R ’ 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 ’ tq , R ’ R2 , has property 2.9.1, but i is
t
not an immersion at 0.
2.14. De¬nition. For an arbitrary subset A of a manifold N and x0 ∈ A let
Cx0 (A) denote the set of all x ∈ 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 ∈ M there exists a chart (U, u) centered at x on N such that
u(Cx (U © M )) = u(U ) © (Rm — 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 ’ N be an injective immersion between manifolds
with property 2.9.1. Then f (M ) is an initial submanifold of N .
Proof. Let x ∈ 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 (v—¦f —¦w’1 )(y 1 , . . . , y m ) =
(y 1 , . . . , y m , 0, . . . , 0). Let r > 0 be so small that {y ∈ Rm : |y| < r} ‚ w(W )
and {z ∈ Rn : |z| < 2r} ‚ v(V ). Put
U : = v ’1 ({z ∈ Rn : |z| < r}) ‚ N,
W1 : = w’1 ({y ∈ Rm : |y| < r}) ‚ M.
We claim that (U, u = v|U ) satis¬es the condition of 2.14.1.
u’1 (u(U ) © (Rm — 0)) = u’1 ({(y 1 , . . . , y m , 0 . . . , 0) : |y| < r}) =
= f —¦ w’1 —¦ (u —¦ f —¦ w’1 )’1 ({(y 1 , . . . , y m , 0 . . . , 0) : |y| < r}) =
= f —¦ w’1 ({y ∈ Rm : |y| < r}) = f (W1 ) ⊆ Cf (x) (U © f (M )),
since f (W1 ) ⊆ U © f (M ) and f (W1 ) is C ∞ -contractible.
Now let conversely z ∈ Cf (x) (U ©f (M )). Then by de¬nition there is a smooth
curve c : [0, 1] ’ N with c(0) = f (x), c(1) = z, and c([0, 1]) ⊆ U © f (M ). By
property 2.9.1 the unique curve c : [0, 1] ’ M with f —¦ c = c, is smooth.
¯ ¯
We claim that c([0, 1]) ⊆ W1 . If not then there is some t ∈ [0, 1] with c(t) ∈
¯ ¯
’1 m
w ({y ∈ R : r ¤ |y| < 2r}) since c is smooth and thus continuous. But then
¯
we have
(v —¦ f )(¯(t)) ∈ (v —¦ f —¦ w’1 )({y ∈ Rm : r ¤ |y| < 2r}) =
c
= {(y, 0) ∈ Rm — 0 : r ¤ |y| < 2r} ⊆ {z ∈ Rn : r ¤ |z| < 2r}.
This means (v —¦ f —¦ c)(t) = (v —¦ c)(t) ∈ {z ∈ Rn : r ¤ |z| < 2r}, so c(t) ∈ U , a
¯ /
contradiction.
So c([0, 1]) ⊆ W1 , thus c(1) = f ’1 (z) ∈ W1 and z ∈ f (W1 ). Consequently we
¯ ¯
have Cf (x) (U © f (M )) = f (W1 ) and ¬nally f (W1 ) = u’1 (u(U ) © (Rm — 0)) by
the ¬rst part of the proof.
2.16. Lemma. Let M be an initial submanifold of a manifold N . Then there
is a unique C ∞ -manifold structure on M such that the injection i : M ’ 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 © M ) as charts for M , where x ∈ 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 © M ) are not open in the induced topology on M
in general. So the identi¬cation topology with respect to the charts (Cx (Ux ©
M ), ux )x∈M yields a topology on M which is ¬ner than the induced topology, so
it is Hausdor¬. Clearly i : M ’ 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.


Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
14 Chapter I. Manifolds and Lie groups


2.17. Lemma. Any initial submanifold M of a manifold N with injective
immersion i : M ’ N has the universal property 2.9.1:
For any manifold Z a mapping f : Z ’ M is smooth if and only if i —¦ f : Z ’
N is smooth.
Proof. We have to prove only one direction and we will suppress the embedding i.
For z ∈ Z we choose a chart (U, u) on N , centered at f (z), such that u(Cf (z) (U ©
M )) = u(U ) © (Rm — 0). Then f ’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 ∞ -contractible in U © M , so
f (V ) ⊆ Cf (z) (U ©M ), and (u|Cf (z) (U ©M ))—¦f —¦v ’1 = u —¦f —¦v ’1 is smooth.
2.18. Transversal mappings. Let M1 , M2 , and N be manifolds and let
fi : Mi ’ N be smooth mappings for i = 1, 2. We say that f1 and f2 are
transversal at y ∈ 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 ∈ N .
If P is an initial submanifold of N with injective immersion i : P ’ N , then
f : M ’ N is said to be transversal to P , if i and f are transversal.
Lemma. In this case f ’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 ’1 (P ) is a submanifold.
Proof. Let x ∈ f ’1 (P ) and let (U, u) be an initial submanifold chart for P
centered at f (x) on N , i.e. u(Cx (U © P )) = u(U ) © (Rp — 0). Then the mapping

f pr2
u
M ⊇ f ’1 (U ) ’ U ’ u(U ) ⊆ Rp — Rn’p ’ ’ Rn’p
’ ’ ’

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 —¦ u —¦ f —¦ v ’1 )(y 1 , . . . , y n’p , . . . , y m ) = (y 1 , . . . , y n’p ).

But then z ∈ Cx (f ’1 (P ) © V ) if and only if v(z) ∈ v(V ) © (0 — Rm’n+p ), so
v(Cx (f ’1 (P ) © V )) = v(V ) © (0 — Rm’n+p ).
2.19. Corollary. If f1 : M1 ’ N and f2 : M2 ’ N are smooth and transver-
sal, then the topological pullback

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


is a submanifold of M1 — 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 ’ M1 and g2 : P ’ M2 with f1 —¦g1 = f2 —¦g2
there is a unique smooth mapping (g1 , g2 ) : P ’ M1 —N M2 with pr1 —¦ (g1 , g2 ) =
g1 and pr2 —¦ (g1 , g2 ) = g2 .

RR g2
P
R ,g )
T
R
(g 1 2

u
wM
M1 —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 —N M2 = (f1 — f2 )’1 (∆), where f1 — f2 : M1 — M2 ’ N — N and
where ∆ is the diagonal of N — N , and f1 — f2 is transversal to ∆ if and only if
f1 and f2 are transversal.
2.20. The category of ¬bered manifolds. Consider a ¬bered manifold
(M, p, N ) from 2.4 and a point x ∈ N . Since p is a surjective submersion, the
injection ix : x ’ N of x into N and p : M ’ N are transversal. By 2.19, p’1 (x)
is a submanifold of M , which is called the ¬ber over x ∈ N .
¯¯¯ ¯¯¯
Given another ¬bered manifold (M , p, N ), a morphism (M, p, N ) ’ (M , p, N )
means a smooth map f : M ’ N transforming each ¬ber of M into a ¬ber of
¯ ¯¯ ¯
M . The relation f (Mx ) ‚ Mx de¬nes a map f : N ’ N , which is characterized
by the property p —¦ f = f —¦ p. Since p —¦ f is a smooth map, f is also smooth by
¯ ¯
2.4. Clearly, all ¬bered manifolds and their morphisms form a category, which
will be denoted by FM. Transforming every ¬bered manifold (M, p, N ) into its
¯¯¯
base N and every ¬bered manifold morphism f : (M, p, N ) ’ (M , p, N ) into the
¯
induced map f : N ’ N de¬nes the base functor B : FM ’ Mf .
¯¯
If (M, p, N ) and (M , p, N ) are two ¬bered manifolds over the same base N ,
¯ ¯
then the pullback M —(p,N,p) M = M —N M is called the ¬bered product of M
¯
¯ ¯
and M . If p, p and N are clear from the context, then M —N M is also denoted
¯
¯ ¯¯¯
by M • M . Moreover, if f1 : (M1 , p1 , N ) ’ (M1 , p1 , N ) and f2 : (M2 , p2 , N ) ’
¯¯¯ ¯
(M2 , p2 , N ) are two FM-morphisms over the same base map f0 : N ’ N , then
¯ ¯¯
the values of the restriction f1 — f2 |M1 —N M2 lie in M1 —N M2 . The restricted
¯ ¯¯
map will be denoted by f1 —f0 f2 : M1 —N M2 ’ M1 —N M2 or f1 •f2 : M1 •M2 ’
¯ ¯
M1 • M2 and will be called the ¬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 ¬elds and ¬‚ows

3.1. De¬nition. A vector ¬eld X on a manifold M is a smooth section of
the tangent bundle; so X : M ’ T M is smooth and πM —¦ X = IdM . A local
vector ¬eld is a smooth section, which is de¬ned on an open subset only. We
denote the set of all vector ¬elds by X(M ). With point wise addition and scalar
multiplication X(M ) becomes a vector space.
‚ ‚
Example. Let (U, u) be a chart on M . Then the ‚ui : U ’ T M |U , x ’ ‚ui |x ,
described in 1.6, are local vector ¬elds de¬ned on U .
Lemma. If X is a vector ¬eld on M and (U, u) is a chart on M and x ∈ U , then
m m
‚ ‚
we have X(x) = i=1 X(x)(ui ) ‚ui |x . We write X|U = i=1 X(ui ) ‚ui .

3.2. The vector ¬elds ( ‚ui )m on U , where (U, u) is a chart on M , form a
i=1
holonomic frame ¬eld. By a frame ¬eld on some open set V ‚ M we mean
m = dim M vector ¬elds si ∈ X(V ) such that s1 (x), . . . , sm (x) is a linear basis
of Tx M for each x ∈ V . In general, a frame ¬eld on V is said to be holonomic, if

V can be covered by an atlas (U± , u± )±∈A such that si |U± = ‚ui for all ± ∈ A.
±
In the opposite case, the frame ¬eld is called anholonomic.
With the help of partitions of unity and holonomic frame ¬elds one may
construct ˜many™ vector ¬elds on M . In particular the values of a vector ¬eld
can be arbitrarily preassigned on a discrete set {xi } ‚ M .
3.3. Lemma. The space X(M ) of vector ¬elds on M coincides canonically with
the space of all derivations of the algebra C ∞ (M, R) of smooth functions, i.e.
those R-linear operators D : C ∞ (M, R) ’ C ∞ (M, R) with D(f g) = D(f )g +
f D(g).
Proof. Clearly each vector ¬eld X ∈ X(M ) de¬nes a derivation (again called
X, later sometimes called LX ) of the algebra C ∞ (M, R) by the prescription
X(f )(x) := X(x)(f ) = df (X(x)).
If conversely a derivation D of C ∞ (M, R) is given, for any x ∈ M we consider
Dx : C ∞ (M, R) ’ R, Dx (f ) = D(f )(x). Then Dx is a derivation at x of
C ∞ (M, R) in the sense of 1.5, so Dx = Xx for some Xx ∈ Tx M . In this
way we get a section X : M ’ T M . If (U, u) is a chart on M , we have
m i‚
i=1 X(x)(u ) ‚ui |x by 1.6. Choose V open in M , V ‚ V ‚ U , and
Dx =
• ∈ C ∞ (M, R) such that supp(•) ‚ U and •|V = 1. Then • · ui ∈ C ∞ (M, R)
and (•ui )|V = ui |V . So D(•ui )(x) = X(x)(•ui ) = X(x)(ui ) and X|V =
m ‚
i
i=1 D(•u )|V · ‚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 ∞ (M, R), which we will do without any
notational change in the following.
If X, Y are two vector ¬elds on M , then the mapping f ’ X(Y (f ))’Y (X(f ))
is again a derivation of C ∞ (M, R), as a simple computation shows. Thus there is
a unique vector ¬eld [X, Y ] ∈ X(M ) such that [X, Y ](f ) = X(Y (f )) ’ Y (X(f ))
holds for all f ∈ C ∞ (M, R).

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
3. Vector ¬elds and ¬‚ows 17


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

and Y |U = Y i ‚ui we have

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

since second partial derivatives commute. The R-bilinear mapping
] : X(M ) — X(M ) ’ X(M )
[,
is called the Lie bracket. Note also that X(M ) is a module over the algebra
C ∞ (M, R) by point wise multiplication (f, X) ’ f X.
] : X(M ) — X(M ) ’ X(M ) has the following
Theorem. The Lie bracket [ ,
properties:
[X, Y ] = ’[Y, X],
[X, [Y, Z]] = [[X, Y ], Z] + [Y, [X, Z]], the Jacobi identity,
[f X, Y ] = f [X, Y ] ’ (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 —¦ Y ’ Y —¦ X in the space of derivations of the algebra C ∞ (M, R).
3.5. Integral curves. Let c : J ’ M be a smooth curve in a manifold M
de¬ned on an interval J. We will use the following notations: c (t) = c(t) =

d
dt c(t) := Tt c.1. Clearly c : J ’ T M is smooth. We call c a vector ¬eld along
c since we have πM —¦ c = c.
A smooth curve c : J ’ M will be called an integral curve or ¬‚ow line of a
vector ¬eld X ∈ X(M ) if c (t) = X(c(t)) holds for all t ∈ J.
3.6. Lemma. Let X be a vector ¬eld on M . Then for any x ∈ M there is
an open interval Jx containing 0 and an integral curve cx : Jx ’ M for X (i.e.
cx = X —¦ cx ) with cx (0) = x. If Jx is maximal, then cx is unique.
Proof. In a chart (U, u) on M with x ∈ U the equation c (t) = X(c(t)) is an
ordinary di¬erential equation with initial condition c(0) = x. Since X is smooth
there is a unique local solution by the theorem of Picard-Lindel¨f, which even
o
depends smoothly on the initial values, [Dieudonn´ I, 69, 10.7.4]. So on M there
e
are always local integral curves. If Jx = (a, b) and limt’b’ cx (t) =: cx (b) exists
in M , there is a unique local solution c1 de¬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 ¬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 ¬‚ow of a vector ¬eld. Let X ∈ X(M ) be a vector ¬eld. Let us
write FlX (x) = FlX (t, x) := cx (t), where cx : Jx ’ M is the maximally de¬ned
t
integral curve of X with cx (0) = x, constructed in lemma 3.6. The mapping FlX
is called the ¬‚ow of the vector ¬eld X.
Theorem. For each vector ¬eld X on M , the mapping FlX : D(X) ’ M is
smooth, where D(X) = x∈M Jx — {x} is an open neighborhood of 0 — M in
R — 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 ≥ 0 or both are ¤ 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 de¬ned for (t, x), then it is also de¬ned for (s, y) nearby. These are
local properties which follow from the theory of ordinary di¬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 ≥ 0. We put

FlX (u, x) if u ¤ s
cx (u) =
FlX (u ’ s, FlX (s, x)) if u ≥ s.
FlX (u, x) = X(FlX (u, x))
d
for u ¤ s
du
d
du cx (u) = =
FlX (u ’ s, FlX (s, x)) = X(FlX (u ’ s, FlX (s, x)))
d
du
for 0 ¤ u ¤ t + s.
= X(cx (u))

Also cx (0) = x and on the overlap both de¬nitions coincide by the ¬rst part of
the proof, thus we conclude that cx (u) = FlX (u, x) for 0 ¤ u ¤ 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 — M in R — M and that FlX is smooth
near 0 — M .
For x ∈ M let Jx be the set of all t ∈ R such that FlX is de¬ned and smooth
on an open neighborhood of [0, t] — {x} (respectively on [t, 0] — {x} for t < 0)
in R — M . We claim that Jx = Jx , which ¬nishes the proof. It su¬ces to show
that Jx is not empty, open and closed in Jx . It is open by construction, and
not empty, since 0 ∈ Jx . If Jx is not closed in Jx , let t0 ∈ Jx © (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 ¬elds and ¬‚ows 19


smooth near [’µ, µ] — {y := FlX (t0 , x)} for some µ > 0, and by construction FlX
exists and is smooth near [0, t0 ’ µ] — {x}. Since FlX (’µ, y) = FlX (t0 ’ µ, x) we
conclude for t near [0, t0 ’ µ], x near x, and t near [’µ, µ], that FlX (t + t , x ) =
FlX (t , FlX (t, x )) exists and is smooth. So t0 ∈ Jx , a contradiction.
3.8. Let X ∈ X(M ) be a vector ¬eld. Its ¬‚ow FlX is called global or complete,
if its domain of de¬nition D(X) equals R — M . Then the vector ¬eld X itself
will be called a complete vector ¬eld. In this case FlX is also sometimes called
t
exp tX; it is a di¬eomorphism of M .
The support supp(X) of a vector ¬eld X is the closure of the set {x ∈ M :
X(x) = 0}.
Lemma. Every vector ¬eld with compact support on M is complete.
Proof. Let K = supp(X) be compact. Then the compact set 0 — K has positive
distance to the disjoint closed set (R—M )\D(X) (if it is not empty), so [’µ, µ]—
K ‚ D(X) for some µ > 0. If x ∈ K then X(x) = 0, so FlX (t, x) = x for all t
/
and R — {x} ‚ D(X). So we have [’µ, µ] — M ‚ D(X). Since FlX (t + µ, x) =
FlX (t, FlX (µ, x)) exists for |t| ¤ µ by theorem 3.7, we have [’2µ, 2µ]—M ‚ D(X)
and by repeating this argument we get R — M = D(X).
So on a compact manifold M each vector ¬eld is complete. If M is not
compact and of dimension ≥ 2, then in general the set of complete vector ¬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 ‚x and Y = x ‚y are complete, but
‚ ‚
2
neither X + Y nor [X, Y ] is complete.
3.9. f -related vector ¬elds. If f : M ’ M is a di¬eomorphism, then for any
vector ¬eld X ∈ X(M ) the mapping T f ’1 —¦ X —¦ f is also a vector ¬eld, which
we will denote f — X. Analogously we put f— X := T f —¦ X —¦ f ’1 = (f ’1 )— X.
But if f : M ’ N is a smooth mapping and Y ∈ X(N ) is a vector ¬eld there
may or may not exist a vector ¬eld X ∈ X(M ) such that the following diagram
commutes:
wu
u
Tf
TM TN
(1) X Y

w N.
f
M
De¬nition. Let f : M ’ N be a smooth mapping. Two vector ¬elds X ∈
X(M ) and Y ∈ X(N ) are called f -related, if T f —¦ X = Y —¦ f holds, i.e. if diagram
(1) commutes.
Example. If X ∈ X(M ) and Y ∈ X(N ) and X — Y ∈ X(M — N ) is given by
(X — Y )(x, y) = (X(x), Y (y)), then we have:
(2) X — Y and X are pr1 -related.
(3) X — Y and Y are pr2 -related.
(4) X and X — Y are ins(y)-related if and only if Y (y) = 0, where
ins(y)(x) = (x, y), ins(y) : M ’ M — N .

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
20 Chapter I. Manifolds and Lie groups


3.10. Lemma. Consider vector ¬elds Xi ∈ X(M ) and Yi ∈ X(N ) for i = 1, 2,
and a smooth mapping f : M ’ N . If Xi and Yi are f -related for i = 1, 2, then
also »1 X1 + »2 X2 and »1 Y1 + »2 Y2 are f -related, and also [X1 , X2 ] and [Y1 , Y2 ]
are f -related.
Proof. The ¬rst assertion is immediate. To show the second let h ∈ C ∞ (N, R).
Then by assumption we have T f —¦ Xi = Yi —¦ f , thus:
(Xi (h —¦ f ))(x) = Xi (x)(h —¦ f ) = (Tx f.Xi (x))(h) =
= (T f —¦ Xi )(x)(h) = (Yi —¦ f )(x)(h) = Yi (f (x))(h) = (Yi (h))(f (x)),
so Xi (h —¦ f ) = (Yi (h)) —¦ f , and we may continue:
[X1 , X2 ](h —¦ f ) = X1 (X2 (h —¦ f )) ’ X2 (X1 (h —¦ f )) =
= X1 (Y2 (h) —¦ f ) ’ X2 (Y1 (h) —¦ f ) =
= Y1 (Y2 (h)) —¦ f ’ Y2 (Y1 (h)) —¦ f = [Y1 , Y2 ](h) —¦ f.
But this means T f —¦ [X1 , X2 ] = [Y1 , Y2 ] —¦ f .
3.11. Corollary. If f : M ’ N is a local di¬eomorphism (so (Tx f )’1 makes
sense for each x ∈ M ), then for Y ∈ X(N ) a vector ¬eld f — Y ∈ X(M ) is de¬ned
by (f — Y )(x) = (Tx f )’1 .Y (f (x)). The linear mapping f — : X(N ) ’ X(M ) is
then a Lie algebra homomorphism, i.e. f — [Y1 , Y2 ] = [f — Y1 , f — Y2 ].
3.12. The Lie derivative of functions. For a vector ¬eld X ∈ X(M ) and
f ∈ C ∞ (M, R) we de¬ne LX f ∈ C ∞ (M, R) by
X
d
LX f (x) := dt |0 f (Fl (t, x)) or
X—
—¦ FlX ).
d d
LX f := dt |0 (Flt ) f = dt |0 (f t

Since FlX (t, x) is de¬ned for small t, for any x ∈ M , the expressions above make
sense.
Lemma. dt (FlX )— f = (FlX )— 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 ¬elds. For X, Y ∈ X(M ) we de¬ne
LX Y ∈ X(M ) by
X— X
—¦ Y —¦ FlX ),
d d
LX Y := dt |0 (Flt ) Y dt |0 (T (Fl’t )
= t

and call it the Lie derivative of Y along X.
X—
= (FlX )— LX Y = (FlX )— [X, Y ].
d
Lemma. LX Y = [X, Y ] and dt (Flt ) Y t t

Proof. Let f ∈ C ∞ (M, R) be a function and consider the mapping ±(t, s) :=
Y (FlX (t, x))(f —¦ FlX ), which is locally de¬ned near 0. It satis¬es
s

±(t, 0) = Y (FlX (t, x))(f ),
±(0, s) = Y (x)(f —¦ FlX ),
s
X X
‚ ‚ ‚
‚t ±(0, 0) = ‚t 0 Y (Fl (t, x))(f ) = ‚t 0 (Y f )(Fl (t, x)) = X(x)(Y f ),
X X
‚ ‚ ‚
‚s |0 Y (x)(f —¦ Fls ) = Y (x) ‚s |0 (f —¦ Fls ) = Y (x)(Xf ).
‚s ±(0, 0) =

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
3. Vector ¬elds and ¬‚ows 21


But on the other hand we have
(FlX (u, x))(f —¦ FlX ) =
‚ ‚
‚u |0 ±(u, ’u) ‚u |0 Y
= ’u

T (FlX ) —¦ Y —¦ FlX

‚u |0
= (f ) = (LX Y )x (f ),
’u u
x

so the ¬rst assertion follows. For the second claim we compute as follows:
X—
T (FlX ) —¦ T (FlX ) —¦ Y —¦ FlX —¦ FlX
‚ ‚
‚s |0
‚t (Flt ) Y = ’t ’s s t

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

‚s |0
’t ’s s t

= T (FlX ) —¦ [X, Y ] —¦ FlX = (FlX )— [X, Y ].
’t t t


3.14. Lemma. Let X ∈ X(M ) and Y ∈ X(N ) be f -related vector ¬elds for
a smooth mapping f : M ’ N . Then we have f —¦ FlX = FlY —¦f , whenever
t t —
both sides are de¬ned. In particular, if f is a di¬eomorphism we have Flf Y =
t
f ’1 —¦ FlY —¦f .
t

Proof. We have dt (f —¦ FlX ) = T f —¦ dt FlX = T f —¦ X —¦ FlX = Y —¦ f —¦ F lt
d d X
t t t
and f (FlX (0, x)) = f (x). So t ’ f (FlX (t, x)) is an integral curve of the vector
¬eld Y on N with initial value f (x), so we have f (FlX (t, x)) = FlY (t, f (x)) or
f —¦ FlX = FlY —¦f .
t t

3.15. Corollary. Let X, Y ∈ X(M ). Then the following assertions are equiva-
lent
(1) LX Y = [X, Y ] = 0.
(2) (FlX )— Y = Y , wherever de¬ned.
t
(3) Flt —¦ FlY = FlY —¦ FlX , wherever de¬ned.
X
s s t

Proof. (1) ” (2) is immediate from lemma 3.13. To see (2) ” (3) we note
X—
that FlX —¦ FlY = FlY —¦ FlX if and only if FlY = FlX —¦ FlY —¦ FlX = Fl(Flt ) Y by
’t
t s s t s s t s
lemma 3.14; and this in turn is equivalent to Y = (FlX )— Y .
t

3.16. Theorem. Let M be a manifold, let •i : R — M ⊃ U•i ’ M be smooth
mappings for i = 1, . . . , k where each U•i is an open neighborhood of {0} — M
in R — M , such that each •i is a di¬eomorphism on its domain, •i = IdM , and
t 0
j j ’1
—¦ (•t ) —¦ •j —¦ •i .
i ’1
‚ i i j i
‚t 0 •t = Xi ∈ X(M ). We put [• , • ]t = [•t , •t ] := (•t ) t t
Then for each formal bracket expression P of lenght k we have

|0 P (•1 , . . . , •k )

for 1 ¤ < k,
0= t t
‚t
1 ‚k 1 k
k! ‚tk |0 P (•t , . . . , •t ) ∈ X(M )
P (X1 , . . . , Xk ) =
in the sense explained in step 2 of the proof. In particular we have for vector
¬elds X, Y ∈ X(M )
Y X Y X

‚t 0 (Fl’t —¦ Fl’t —¦ Flt —¦ Flt ),
0=
1 ‚2 Y X Y X
2 ‚t2 |0 (Fl’t —¦ Fl’t —¦ Flt —¦ 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 ’ M be a smooth curve. If c(0) = x ∈ M , c (0) =
0, . . . , c(k’1) (0) = 0, then c(k) (0) is a well de¬ned tangent vector in Tx M which
is given by the derivation f ’ (f —¦ c)(k) (0) at x.
For we have
k
k
(k) (k)
(f —¦ c)(j) (0)(g —¦ c)(k’j) (0)
((f.g) —¦ c) (0) = ((f —¦ c).(g —¦ c)) (0) = j
j=0

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

since all other summands vanish: (f —¦ c)(j) (0) = 0 for 1 ¤ j < k.
Step 2. Let • : R — M ⊃ U• ’ M be a smooth mapping where U• is an open
neighborhood of {0} — M in R — M , such that each •t is a di¬eomorphism on
its domain and •0 = IdM . We say that •t is a curve of local di¬eomorphisms
though IdM .
‚j 1 ‚k
From step 1 we see that if ‚tj |0 •t = 0 for all 1 ¤ j < k, then X := k! ‚tk |0 •t
is a well de¬ned vector ¬eld on M . We say that X is the ¬rst non-vanishing
derivative at 0 of the curve •t of local di¬eomorphisms. We may paraphrase this
as (‚t |0 •— )f = k!LX f .
k
t

Claim 3. Let •t , ψt be curves of local di¬eomorphisms through IdM and let
f ∈ C ∞ (M, R). Then we have
k
j k’j
— —
•— )f (‚t |0 ψt )(‚t |0 •— )f.

k
k k
‚t |0 (•t —¦ ψt ) f = ‚t |0 (ψt —¦ =
t t
j
j=0

Also the multinomial version of this formula holds:
k! j j
‚t |0 (•1 —¦ . . . —¦ •t )— f = (‚t 1 |0 (•t )— ) . . . (‚t 1 |0 (•1 )— )f.
k
t t
j1 ! . . . j !
j1 +···+j =k

We only show the binomial version. For a function h(t, s) of two variables we
have
k
j k’j
k
k
‚t h(t, t) = ‚t ‚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 ∞ -topology, so that by continuity
the formula holds for all functions. In the following form it implies the claim:
k
j k’j
k
k
‚t |0 f (•(t, ψ(t, x))) = ‚t ‚s f (•(t, ψ(s, x)))|t=s=0 .
j
j=0


Claim 4. Let •t be a curve of local di¬eomorphisms through IdM with ¬rst
k
non-vanishing derivative k!X = ‚t |0 •t . Then the inverse curve of local di¬eo-
morphisms •’1 has ¬rst non-vanishing derivative ’k!X = ‚t |0 •’1 .
k
t t


Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
3. Vector ¬elds and ¬‚ows 23


For we have •’1 —¦ •t = Id, so by claim 3 we get for 1 ¤ j ¤ k
t
j
j j’i
‚t |0 (•’1 (‚t |0 •— )(‚t (•’1 )— )f =
— j i
—¦ •t ) f =
0= t t
t
i
i=0
j j
= ‚t |0 •— (•’1 )— f + •— ‚t |0 (•’1 )— f,
t
t 0
0
j j

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