Ivan Kolar, Jan Slovak, Peter W. Michor
Natural operations in differential geometry
The book is published:
Springer-Verlag, Berlin, Heidelberg, New York, 1993
- MSC:
- 53-02 Research exposition (monographs, survey articles)
- 53-01 Instructional exposition (textbooks, tutorial papers, etc.)
- 58-02 Research exposition (monographs, survey articles)
- 58-01 Instructional exposition (textbooks, tutorial papers, etc.)
- 53A55 Differential invariants (local theory), geometric objects
- 53C05 Connections, general theory
- 58A20 Jets
Abstract: The aim of this book is threefold:
First it should be a monographical work on natural
bundles and natural operators in
differential geometry. This is a field which every differential
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 differential
forms. In the background of this statement are the following
general concepts.
The vector bundle $\La^kT^*M$ 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 diffeomorphism $f$ 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 $\La^kT^*M$ into sections of $\La^{k+1}T^*M$ for every
manifold $M$ can be expressed by saying that $d$ is an operator
from $\La^kT^*M$ into $\La^{k+1}T^*M$.
That the exterior
derivative $d$ commutes with local diffeomorphisms
now means, that $d$ is a natural operator from the functor $\La^kT^*$
into functor $\La^{k+1}T^*$. 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 field we are
discussing here. A systematic treatment of naturality in
differential 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 different branches of differential geometry. Even
though Ehresmann in his original papers from 1951 underlined the
conceptual meaning of the notion of an $r$-jet for differential
geometry, jets have been mostly used as a purely technical tool
in certain problems in the theory of systems of partial
differential equations, in singularity theory, in
variational calculus and in higher order mechanics. But the
theory of natural bundles and natural operators clarifies once
again that jets are one of the fundamental concepts in
differential 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 differential geometry
(manifolds, flows, Lie groups, differential forms, bundles and
connections) which stresses naturality and functoriality from
the beginning and is as coordinate free as possible. Here we
present the Frölicher-Nijenhuis bracket (a natural extension
of the Lie bracket from vector fields to vector valued
differential forms) as one of
the basic structures of differential geometry, and we base
nearly all treatment of curvature and Bianchi identities on it.
This allows us to present the concept of a connection first on
general fiber bundles (without structure group), with
curvature, parallel transport and Bianchi identity, and only
then add G-equivariance as a further property for principal
fiber 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 specific feature of the book is that the authors are interested
in general points of view towards different structures in differential
geometry. The modern development of global differential
geometry clarified that differential geometric objects form fiber
bundles over manifolds as a rule. Nijenhuis revisited the
classical theory of geometric objects
from this point of view. Each type of geometric objects can
be interpreted as
a rule $F$ transforming every $m$-dimensional
manifold $M$ into a fibered manifold $FM\> M$ over $M$ and every
local diffeomorphism $f:M\>N$ into a fibered manifold morphism
$Ff:FM\to FN$ over $f$. The geometric character of $F$ is then
expressed by the functoriality condition $F(g\o f)=Fg\o Ff$.
Hence the classical bundles of geometric objects are now studied
in the form of the so called lifting functors or (which is the
same) natural bundles on the category $\Mf_m$ of all
$m$-dimensional manifolds and their local diffeomorphisms. An
important result by Palais and Terng, completed by Epstein and
Thurston, reads that every lifting functor has finite order. This
gives a full description of all natural bundles as the fiber
bundles associated with the $r$-th order frame bundles, which is
useful in many problems. However in several cases it is not
sufficient to study the bundle functors defined on the category $\Mf_m$.
For example, if we have a Lie group $G$, its multiplication is a
smooth map $\mu:G\x G\to G$. To construct an induced map $F\mu:F(G\x
G)\to FG$, we need a functor $F$ defined 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 $F\mu$ 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 differential geometry are
defined on the category of fibered manifolds and their local
isomorphisms, the bundle of general connections being the simplest example.
Last but not least we remark that Eck has recently
introduced the general concepts of gauge natural bundles and
gauge natural operators. 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 defined
on a suitable category over manifolds, then some geometric
constructions have the role of natural transformations.
Several others represent natural operators, i.e. they map
sections of certain fiber 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 natural is that we obtain a well-defined 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 effective way
how to treat natural operators is to reduce the question to a
finite order problem, in which the corresponding jet spaces are
finite dimensional. Since the finite order natural operators are
in a simple bijection with
the equivariant maps between the corresponding standard fibers,
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 finite
order situation has been of great profit 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 finite order. We develop an essential
generalization of this technique and we present a unified
approach to the finite 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 finding 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 differential
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 infinite dimensional Lie algebras of vector fields. In
chapter X we study
systematically the natural operators on vector fields and connections.
Chapter XI is devoted to a general theory of Lie derivatives, in
which the geometric approach clarifies, among other things, the
relations to natural operators.
Notes: Electronic version, corrected.
Reviews in:
Math. Rev. 94a:58004, Zentralblatt Math 782:53013,
Review in Bull. AMS 31,1 (1994),108--112
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