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NATURAL
OPERATIONS
IN
DIFFERENTIAL
GEOMETRY
Ivan Kolar
Peter W. Michor
Jan Slovak
Mailing address: Peter W. Michor,
Institut fur Mathematik der Universitat Wien,
Strudlhofgasse 4, A-1090 Wien, Austria.
Ivan Kolar, Jan Slovak,
Department of Algebra and Geometry
Faculty of Science, Masaryk University
Janackovo nam 2a, CS-662 95 Brno, Czechoslovakia
Electronic edition. Originally published by Springer-Verlag, Berlin Heidelberg
1993, ISBN 3-540-56235-4, ISBN 0-387-56235-4.
Typeset by A M S-T E X
v
TABLE OF CONTENTS
PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
CHAPTER I.
MANIFOLDS AND LIE GROUPS . . . . . . . . . . . . . . . . 4
1 . Dierentiable 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 . Dierential forms . . . . . . . . . . . . . . . . . . . . . . . 61
8 . Derivations on the algebra of dierential forms
and the Frolicher-Nijenhuis bracket . . . . . . . . . . . . . . . 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 dierential forms . . . . . . . . . . . . . . . . . . 154
17 . Connections and the absolute dierentiation . . . . . . . . . . . 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 dierential . . 220
26 . The tensor evaluation theorem . . . . . . . . . . . . . . . . . 223
27 . Generalized invariant tensors . . . . . . . . . . . . . . . . . . 230
28 . The orbit reduction . . . . . . . . . . . . . . . . . . . . . . 233
29 . The method of dierential equations . . . . . . . . . . . . . . 245
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
vi
CHAPTER VII.
FURTHER APPLICATIONS . . . . . . . . . . . . . . . . . . 249
30 . The Frolicher-Nijenhuis bracket . . . . . . . . . . . . . . . . . 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 FY ! M . . . . . . . . . . . . 363
46 . The cases FY ! FM and FY ! 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 dierential geometry. This is a eld which every dierential 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 dierential forms. In the
background of this statement are the following general concepts. The vector
bundle k 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
dieomorphism 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 k T M into sections of k+1 T M for every manifold M
can be expressed by saying that d is an operator from k T M into k+1 T M.
That the exterior derivative d commutes with local dieomorphisms now means,
that d is a natural operator from the functor k T into functor k+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 dierential 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 dierent 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 dierential ge-
ometry, jets have been mostly used as a purely technical tool in certain problems
in the theory of systems of partial dierential equations, in singularity theory,
in variational calculus and in higher order mechanics. But the theory of nat-
ural bundles and natural operators claries once again that jets are one of the
fundamental concepts in dierential 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 dierential geometry (manifolds, ows, Lie groups, dierential
forms, bundles and connections) which stresses naturality and functoriality from
the beginning and is as coordinate free as possible. Here we present the Frolicher-
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 dierential forms) as one of the basic structures of dierential
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 specic feature of the book is that the authors are interested in general
points of view towards dierent structures in dierential geometry. The modern
development of global dierential geometry claried that dierential 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 FM ! M over M and every local dieo-
morphism f : M ! N into a bered manifold morphism Ff : FM ! FN over
f. The geometric character of F is then expressed by the functoriality condition
F(g f) = Fg Ff. 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 Mf m of all m-dimensional manifolds and their local
dieomorphisms. 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 sucient to study the bundle functors dened on the category
Mf m . For example, if we have a Lie group G, its multiplication is a smooth
map : G G ! G. To construct an induced map F : F(G G) ! FG,
we need a functor F dened 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 endows FG 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 dierential geometry are dened 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 dened 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

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