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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-

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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

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

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