"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
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TABLE OF CONTENTS
PREFACE . . . . . . . . . . . . . . . . . . . . . . . . CHAPTER I. MANIFOLDS AND LIE GROUPS . . . . . . . . . . . . 1. Differentiable manifolds . . . . . . . . . . . . . . . . . 2. Submersions and immersions . . . . . . . . . . . . . . . 3. Vector fields and flows . . . . . . . . . . . . . . . . . . 4. Lie groups . . . . . . . . . . . . . . . . . . . . . . . 5. Lie subgroups and homogeneous spaces . . . . . . . . . . CHAPTER II. DIFFERENTIAL FORMS . . . . . . . . . . . . . . . . 6. Vector bundles . . . . . . . . . . . . . . . . . . . . . 7. Differential forms . . . . . . . . . . . . . . . . . . . . 8. Derivations on the algebra of differential forms and the Fr¨licher-Nijenhuis bracket . . . . . . . . . . . . o CHAPTER III. BUNDLES AND CONNECTIONS . . . . . . . . . . . . 9. General fiber bundles and connections . . . . . . . . . . . 10. Principal fiber bundles and G-bundles . . . . . . . . . . . 11. Principal and induced connections . . . . . . . . . . . . CHAPTER IV. JETS AND NATURAL BUNDLES . . . . . . . . . . . . 12. Jets . . . . . . . . . . . . . . . . . . . . . . . . . . 13. Jet groups . . . . . . . . . . . . . . . . . . . . . . . 14. Natural bundles and operators . . . . . . . . . . . . . . 15. Prolongations of principal fiber bundles . . . . . . . . . . 16. Canonical differential forms . . . . . . . . . . . . . . . 17. Connections and the absolute differentiation . . . . . . . . CHAPTER V. FINITE ORDER THEOREMS . . . . . . . . . . . . . . 18. Bundle functors and natural operators . . . . . . . . . . . 19. Peetre-like theorems . . . . . . . . . . . . . . . . . . . 20. The regularity of bundle functors . . . . . . . . . . . . . 21. Actions of jet groups . . . . . . . . . . . . . . . . . . . 22. The order of bundle functors . . . . . . . . . . . . . . . 23. The order of natural operators . . . . . . . . . . . . . . CHAPTER VI. METHODS FOR FINDING NATURAL OPERATORS . . . 24. Polynomial GL(V )-equivariant maps . . . . . . . . . . . 25. Natural operators on linear connections, the exterior differential 26. The tensor evaluation theorem . . . . . . . . . . . . . . 27. Generalized invariant tensors . . . . . . . . . . . . . . . 28. The orbit reduction . . . . . . . . . . . . . . . . . . . 29. The method of differential equations . . . . . . . . . . .
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
. . . . 1 . . . . . . . . . . . . . . 4 . . 4 . 11 . 16 . 30 . 41 49 49 61 67 76 76 86 99 116 117 128 138 149 154 158 168 169 176 185 192 202 205 212 213 220 223 230 233 245
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CHAPTER VII. FURTHER APPLICATIONS . . . . . . . . . . . . . . . . . 30. The Fr¨licher-Nijenhuis bracket . . . . . . . . . . . . . . . . o 31. Two problems on general connections . . . . . . . . . . . . . 32. Jet functors . . . . . . . . . . . . . . . . . . . . . . . . . 33. Topics from Riemannian geometry . . . . . . . . . . . . . . . 34. Multilinear natural operators . . . . . . . . . . . . . . . . . CHAPTER VIII. PRODUCT PRESERVING FUNCTORS . . . . . . . . . . . 35. Weil algebras and Weil functors . . . . . . . . . . . . . . . . 36. Product preserving functors . . . . . . . . . . . . . . . . . 37. Examples and applications . . . . . . . . . . . . . . . . . . CHAPTER IX. BUNDLE FUNCTORS ON MANIFOLDS . . . . . . . . . . . 38. The point property . . . . . . . . . . . . . . . . . . . . . 39. The flow-natural transformation . . . . . . . . . . . . . . . 40. Natural transformations . . . . . . . . . . . . . . . . . . . 41. Star bundle functors . . . . . . . . . . . . . . . . . . . . CHAPTER X. PROLONGATION OF VECTOR FIELDS AND CONNECTIONS 42. Prolongations of vector fields to Weil bundles . . . . . . . . . . 43. The case of the second order tangent vectors . . . . . . . . . . 44. Induced vector fields on jet bundles . . . . . . . . . . . . . . 45. Prolongations of connections to F Y → M . . . . . . . . . . . 46. The cases F Y → F M and F Y → Y . . . . . . . . . . . . . . CHAPTER XI. GENERAL THEORY OF LIE DERIVATIVES . . . . . . . . . 47. The general geometric approach . . . . . . . . . . . . . . . 48. Commuting with natural operators . . . . . . . . . . . . . . 49. Lie derivatives of morphisms of fibered manifolds . . . . . . . . 50. The general bracket formula . . . . . . . . . . . . . . . . . CHAPTER XII. GAUGE NATURAL BUNDLES AND OPERATORS . . . . . . 51. Gauge natural bundles . . . . . . . . . . . . . . . . . . . 52. The Utiyama theorem . . . . . . . . . . . . . . . . . . . . 53. Base extending gauge natural operators . . . . . . . . . . . . 54. Induced linear connections on the total space of vector and principal bundles . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . Author index . . . . . . . . . . . . . . . . . . . . . . . . . Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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249 250 255 259 265 280 296 297 308 318 329 329 336 341 345 350 351 357 360 363 369 376 376 381 387 390 394 394 399 405 409 417 428 429 431
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
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PREFACE
The aim of this work 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 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 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 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 diffeomorphisms 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 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 Fr¨lichero Nijenhuis bracket (a natural extension of the Lie bracket from vector fields to
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
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Preface
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 F M → M over M and every local diffeomorphism f : M → N into a fibered 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) natural bundles on the category Mfm 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 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 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 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-productpreserving 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 secElectronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
Preface
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tions 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. The material for chapters VI, X and sections 12, 30–32, 47, 49, 50, 52–54 was prepared by the first author (I.K.), for chapters I, II, III, VIII by the second author (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 first two authors and has been based on the common cultural heritage of Middle Europe. The authors will be pleased if the reader realizes a reflection of those traditions in the book.
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
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CHAPTER I. MANIFOLDS AND LIE GROUPS
In this chapter we present an introduction to the basic structures of differential 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 - Hausdorff formula for Lie groups. We also prove that closed subgroups of Lie groups are Lie subgroups.
1. Differentiable manifolds
1.1. A topological manifold is a separable Hausdorff 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 prefix pure. A family (Uα , uα )α∈A of charts on M such that the Uα form a cover of M is called an atlas. The mappings uαβ := uα ◦ u−1 : uβ (Uαβ ) → uα (Uαβ ) are called β the chart changings for the atlas (Uα ), where Uαβ := Uα ∩ Uβ . An atlas (Uα , uα )α∈A for a manifold M is said to be a C k -atlas, if all chart changings uαβ : uβ (Uαβ ) → uα (Uαβ ) are differentiable 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 differential 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 differentiable structures. For example, every 4-dimensional manifold is smooth off 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 structures. The spheres from dimension 7 on have finitely many, see [Milnor, 56]. But the most surprising result is that on R4 there are uncountably many pairwise inequivalent (exotic) differentiable structures. This follows from the results
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
1. Differentiable manifolds
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of [Donaldson, 83] and [Freedman, 82], see [Gompf, 83] or [Freedman-Feng Luo, 89] for an overview. Note that for a Hausdorff 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 Hausdorff, separable, finite dimensional, to state it precisely. Note finally that any manifold M admits a finite atlas consisting of dim M +1 (not connected) ch..."
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