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Natural operations in differential geometry. Transl. from the English. (Estestvennye operatsii v differentsial’noj geometrii.) (Russian) Zbl 1084.53001
Metody Matematicheskogo Modelirovaniya 10. Kiev: TIMPANI (ISBN 966-7649-10-5/hbk). viii, 447 p. (2001).
After eight years of the English version of this book, this first treatise on natural bundles and natural operators (see M. de Léon’s review, Zbl 0782.53013) is included as number ten of MMM monograph series, a Russian book version, managing editor being S. S. Moscaliuc.
In MMM’s conception, this treatise together with volume 11 covers the systematic presentation of theoretical categorial methods; precisely it is dedicated to the differential geometrical approach.
The principal “geometric” categories which are interesting in the authors’s points of view are the following: \({\mathcal M}f\) – the category of manifolds and smooth mappings, \({\mathcal M}f_m\) – the category of \(m\)-dimensional manifolds and local diffeomorphisms, \(\mathcal{TM}\) – the category of fibred manifolds and fiber respecting mappings, \(\mathcal{FM}_m\) – the category of fibred manifolds with \(m\)-dimensional bases and fiber respecting mappings with local diffeomorphisms as base maps, \(\mathcal{FM}_{m,n}\) – the category of fibred manifolds with \(m\)-dimensional bases and \(n\)-dimensional fibres and locally invertible fiber respecting mappings, \(\mathcal{FM}^*\) – the category of star bundles, \(\mathcal{PB}\) – the category of principal fiber bundles, \(\mathcal{PB}_m\) – the category of principal bundles over \(m\)-dimensional manifolds and of \(\mathcal{PB}\)-morphisms covering local diffeomorphisms and, finally, \(\mathcal{VB}\) – the category of vector bundles.
The main concepts of natural bundles (bundle functor) and natural operator are given in two steps. Firstly, in chapter IV, §14, the original (Nijenhuis, Terng, Palais-Terng) definition of a natural bundle as a covariant functor \(F:{\mathcal M}f_m\to\mathcal{FM}\) satisfying three conditions (prolongation, locality and regularity) is given and their first geometric properties are derived: a natural operator between two natural bundles \(F\) and \(G\) is a system of regular operators \(A_M:C^\infty(FM)\to C^\infty(GM)\), \(M\in\)ob\({\mathcal M}f_m\), satisfying some properties of naturality and locality:
1) for every section \(s\in C^\infty(FM)\) and every diffeomorphism \(f:M\to N\) holds: \[ A_N(Ff\circ s\circ f^{-1})=Gf\circ A_M s \circ f^{-1}; \]
2) \(A_U(s|_U)=(A_M s)|_U\), for every \(s\in C^\infty(FM)\) and every open submanifold \(U\subset M\).
Secondly, in chapter V, a general framework for the theory of differential geometric objects as natural bundles on categories over manifolds which include \({\mathcal M}f,{\mathcal M}f_m,\mathcal{FM},\mathcal{FM}_m\) \(\mathcal{FM}_{m,n}\) is given. So, the central result, theorem 20.5, which proves the regularity of bundle functors for a class of so-called admissible, locally flat categories is given. Moreover, the well known finite order theorems of Peetre and its nonlinear generalizations are proved.
That complex process to get in the theory of natural bundles and operators is based, at first, on a large introduction (three chapters) to the fundamentals of differential geometry (manifolds, flows, Lie groups, differential forms, bundles and connections) and, on the other hand, on a really comprehensive textbook on all basic structures from the theory of jets (chapter IV).
Nonstandard aspects in this preparatory part of the book consist in the introductory methods which stress naturality and functoriality from the beginning and are as coordinate free as possible. In fact, the derivation of Frölicher-Nijenhuis is introduced as a natural extension of the Lie bracket from vector fields to vector valued differential forms, the so-called Frölicher-Nijenhuis bracket, and is considered as one of the basic structures of differential geometry so that nearly all treatment of curvature and Bianchi identities is based on it. This point of view allows the authors to present the concept of a connection first on general fibre bundles and only then add \(G\)-equivariance as a further property for principal fibre bundles. In the same order of ideas, the authors set up the jets as one of the fundamental concepts in differential geometry so that the presentation of their basic properties plays an important role in the book’s conception.
The keystone of this conception is well resumed by the authors in the preface of the book: “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 other ones represent natural operators, i.e., they map sections of certain fiber bundles to sections of other ones and commute with the action of local isomorphism: Hence geometric means natural in such situations.”
In this idea, the main problem of this treatise, namely the problem of determining all natural operators of a prescribed type, can give the complete list of all possible geometric constructions of the type in question.
This problem is sometimes solved and this fact illustrates the efficacy and the power of the book’s procedure. For instance, the exterior derivative, a first order natural operator from the functor \(\lambda^{k}T^*\) into the functor \(\lambda^{k+1}T^*\) is an archetypical one: \(d\) is the unique natural operator between these two natural bundles up to a constant.
The answer to the problem is given by reducing the question to a finite order problem and by establishing the bijective correspondences: between the set of all \(r\)-th order natural bundles on \(m\)-dimensional manifolds and the set of smooth left actions of the jet group \(G_m^r\) on smooth manifolds; between the set of all \(k\)-th order natural operators \(A:F_m\rightsquigarrow F'\), \(F,F'\) being natural bundles over \({\mathcal M}f_m\) of finite orders \(r\) or \(r'\) with standard fibres \(S\) or \(S'\), and the set of all smooth \(G^q_m\)-equivariant maps between the left \(G_m^q\)-spaces \(T_m^k S\) and \(S'\) where \(q=\max\{r+k,r'\}\). Such correspondences are established also in the general context of natural bundles and operators on categories over manifolds.
In this context, the study of some general algebraic and analytical procedures which can be useful in finding all equivariant maps, i.e., all natural operators of given type is very important. This is the basic purpose of chapter VI.
Chapter VII is devoted to some further examples and applications. So, all bilinear operators on the Frölicher-Nijenhuis type are obtained; Gilkey’s theorem that all differential forms depending naturally on Riemannian metrics and satisfying certain homogeneity conditions are in fact Pontryagin forms is given and the Chern forms as the only natural forms on linear symmetric connections are also characterized.
The theory of those bundle functors which are determined by local algebras in A. Weil’s sense is presented in chapter VIII. The interest in them has been renewed by the relatively recent description of all product preserving functors on manifolds in terms of products of Weil bundle functors and the natural transformations between two such functors are in bijection with the homorphisms of the local algebras of Weil type. So, indeed this book serves as a self-contained introduction to the theory of Weil functors which play an important role in the rest of the book.
The basic subject of chapter IX entitled Bundle functors on manifolds is the study of the geometric properties of arbitrary bundle functors on \({\mathcal M}f\) that do not preserve products. First, the bundle functors with the so-called point property, i.e., the range of a one-point set is a one-point set, are studied. In particular, the author proves that their fibres are numerical spaces and that they preserve products if and only if the dimensions of their values behave well. Then it is proved that an arbitrary bundle functor on manifolds is, in a certain sense, a “bundle” of functors with the point property. In the last section, the basic properties of the so-called star bundle functors, which reflect some constructions of contravariant character on \({\mathcal M}f\), are given.
Chapter X is devoted to systematic investigation of the natural operators transforming vector fields into vector fields or general connections into general connections. First, all natural transforming vector fields on a manifold \(M\) into vector fields on a Weil bundle over \(M\) are determined. Then, the prolongations of vector fields to the bundle of second order tangent vector are studied. For the prolongations of projectable vector fields to the \(r\)-jet prolongation of a fibered manifold the result that the unique natural operator, up to a multiplicative constant, is the flow operator, is given. Also, all first-order natural operators transforming connections on \(Y\to M\) to connections on \(TY\to TM\) are determined. Another problem posed in this chapter consists in the study of the prolongations of connections from \(Y\to M\) to \(FY\to M\), where \(F\) is a functor defined on local isomorphisms of fibered manifolds.
The general geometric approach of generalized Lie derivative is given in chapter XI. First, is shown how this approach generalizes the classical case of Lie differentiation. Moreover, a useful comparison of the generalized Lie derivative with the absolute derivative with respect to a general connection is presented. Then, the authors prove that every linear natural operator commutes with Lie differentiation and a similar condition in the non linear case is deduced.
In the last chapter, the first and the third author generalize the description of all natural bundles \(F:{\mathcal M}f_m\to\mathcal{FM}\) derived in the previous chapters IV (§14) and V (§22) to the gauge natural case. The so-called gauge natural bundle is a functor on principal fiber bundles with structure group \(G\) and their local isomorphisms with values in fiber bundles, but with fibration over the original base manifold. A description of all gauge natural bundles is given. In particular, the results that the regularity condition is a consequence of functoriality and locality and that any gauge natural bundle is of finite order are given.
Some interesting concrete problems on finding gauge natural operators are discussed.
I warmly recommend the book to all mathematicians and physicists interested to know and apply the differential geometrical technics.

MSC:
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
53A55 Differential invariants (local theory), geometric objects
58A20 Jets in global analysis
53C05 Connections (general theory)
58A32 Natural bundles
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