Generalised subbundles and distributions: A comprehensive review Andrew D. Lewis∗ 2011/09/01 Last updated: 2014/07/21 Abstract Distributions, i.e., subsets of tangent bundles formed by piecing together subspaces of tangent spaces, are commonly encountered in the theory and application of differential geometry. Indeed, the theory of distributions is a fundamental part of mechanics and control theory. The theory of distributions is presented in a systematic way, and self-contained proofs are given of some of the major results. Parts of the theory are presented in the context of generalised subbundles of vector bundles. Special emphasis is placed on understanding the r^oleof sheaves and understanding the distinctions between the smooth or finitely differentiable cases and the real analytic case. The Orbit Theorem and applications, including Frobenius's Theorem and theorems on the equivalence of families of vector fields, are considered in detail. Examples illustrate the phenomenon that can occur with generalised subbundles and distributions. Keywords. Generalised subbundle, sheaf theory, distribution, Orbit Theorem, Frobe- nius's Theorem, equivalence of families of vector fields. AMS Subject Classifications (2020). 18F20, 32C05, 32C35, 32L10, 55N30, 57R27, 57R25, 58A30, 58A07, 93B05. Contents 1. Introduction2 2. A wee bit of sheaf theory5 2.1 Presheaves and sheaves of sets. .5 2.2 Germs and ´etalespaces. .7 2.3 Sheaves of rings and modules. .8 2.4 Morphisms and subsheaves. .9 2.5 Coherent real analytic sheaves. 10 2.6 Real analytic spaces. 12 2.7 The beginnings of sheaf cohomology. 14 2.8 Topologies on stalks of sheaves of sections. 15 The smooth case. 15 The real analytic case. 17 ∗Professor, Department of Mathematics and Statistics, Queen's University, Kingston, ON K7L 3N6, Canada Email: [email protected], URL: http://www.mast.queensu.ca/~andrew/ 1 2 A. D. Lewis 3. Generalised subbundles of vector bundles 20 3.1 Generalised subbundles. 21 3.2 Regular and singular points of generalised subbundles. 22 3.3 Generalised subbundles and subsheaves of sections. 25 3.4 Generators for submodules and subsheaves of sections. 28 3.5 Patchy subsheaves. 30 4. Algebraic constructions associated to generalised subbundles 34 4.1 From stalks of a sheaf to fibres. 34 4.2 Subsheaves of sections of a real analytic vector bundle. 38 4.3 Locally finitely generated subsheaves of modules. 39 4.4 Relationship between various notions of \locally finitely generated". 46 5. Global generators for generalised subbundles 49 5.1 Global generators for Cr-generalised subbundles. 50 5.2 Global generators for real analytic generalised subbundles. 55 5.3 Swan's Theorem for regular generalised subbundles. 56 6. Differential constructions associated to distributions 58 6.1 Local diffeomorphisms and flows of vector fields. 59 6.2 Lie brackets, (local) diffeomorphisms, and sheaves. 59 6.3 Lie subalgebras of vector fields and subsheaves. 63 6.4 Distributions and subsheaves invariant under vector fields and diffeomorphisms. 71 7. The Orbit Theorem 76 7.1 Partially defined vector fields. 76 7.2 Orbits. 78 7.3 Fixed-time orbits. 82 7.4 The Orbit Theorem. 85 7.5 The finitely generated Orbit Theorem. 94 7.6 The fixed-time Orbit Theorem. 95 7.7 The finitely generated fixed-time Orbit Theorem. 102 8. Frobenius's Theorem for subsheaves and distributions 103 8.1 Involutive distributions and subsheaves. 103 8.2 Integral manifolds. 105 8.3 Frobenius's Theorem, examples, and counterexamples. 107 1. Introduction Distributions arise naturally in differential geometry (for example, in the characterisa- tion of a Poisson manifold as being a disjoint union of its symplectic leaves) and in many applications of differential geometry, such as mechanics (for example, as arising from non- holonomic constraints in mechanics), and control theory (for example, in characterisations of orbits of families of vector fields). As such, distributions have been widely studied and much is known about them. However, as is often the case with objects that are widely used, there has arisen certain conventions for handling distributions that are viable \a lot of the Generalised subbundles and distributions 3 time," but which, in fact, do not have a basis in the general theory. For just one example, when distributions are used, there is often an unstated assumption of the distribution and all distributions arising from it having locally constant rank. However, this assumption is not always valid, and interesting phenomenon arise when it does not hold, e.g., the gen- eral class of abnormal sub-Riemannian minimisers described by Liu and Sussmann [1994] rely on the distribution generated by certain brackets being singular. For this reason, it seems that there may be some benefit to assembling the basic theory of distributions in one place, with complete proofs of important results, and this is what we do in this paper. The objective is to present, in one place, precise and general definitions and results relating to these definitions. Many of these definitions and results are most naturally presented in the setting of generalised subbundles of general vector bundles, and not just tangent bundles. Thus a substantial part of the paper deals with this. A principal function of the paper is that of a review paper. However, there are also some other contributions which we now outline. One of the contributions of the paper is to explicate clearly the r^oleof sheaf theory in handling distributions. This contribution arises in four ways: (1) in understanding the difficulties of dealing with real analytic global sections of vector bundles, and the manner in which such global sections arise; (2) in understanding clearly the r^oleof analyticity in results which require certain modules to be finitely generated; (3) in understanding some of the algebraic properties of generalised subbundles; (4) in properly characterising some local constructions involving distributions, e.g., invariance under vector fields and flows. To carry out these objectives in full detail requires a great deal to be drawn from sheaf theory, particularly as it relates to complex differential geometry. In this paper we make these connections explicit, we believe for the first time. Apart from this explication of sheaves in the theory of generalised subbundles and dis- tributions, another contribution of the paper is to clarify certain \folklorish" parts of the theory, i.e., things which are known to be true, but for which it is difficult (and for the author, impossible) to put together complete proofs in the existing literature. Here are a few such bits of folklore: (1) the fact that the rank of a real analytic generalised sub- bundle attains its maximal rank on the complement of an analytic set (proved here as Proposition 3.7); (2) the Serre{Swan Theorem for general vector bundles (proved here as Theorem 5.3); (3) the Noetherian properties of real analytic germs (proved here as Propo- sition 4.6); (4) the r^oleof these Noetherian properties in the local finite generation of modules (proved here as Theorem 4.10); (5) all variants of the Orbit Theorem (i.e., all com- binations of fixed-time and arbitrary-time, and finitely generated and non-finitely generated versions); (6) a counterexample that shows that smooth involutive distributions need not be integrable. The proofs of many of these facts are a matter of putting together known results; nonetheless, this has not been done to the best knowledge of the author. There are also some new results in the paper. In Section 3.5 we introduce the class of \patchy" subsheaves of the sheaf of sections of a vector bundle. In we show that the subsheaf what? of sections of a generalised subbundle is patchy. We subsequently show in Corollary 4.11 that patchy real analytic subsheaves are coherent, and this provides for these subsheaves access to the machinery of the cohomology of coherent real analytic sheaves presented in Section 2.5. Other new results are presented in Section 6.4, where we consider the notions of invariance of subsheaves of vector fields under diffeomorphisms and vector fields, and discuss the relationship between invariance under a vector field and invariance under the 4 A. D. Lewis flow of the same vector field. Here the topology of stalks of sheaves discussed in Section 2.8 features prominently, and in the smooth case we reveal the r^oleof the Whitney Spectral Theorem. What follows is an outline of the paper. Many of the properties of distributions, par- ticularly their smoothness, are prescribed locally. This raises the immediate question as to whether these local constructions give rise to meaningful global objects. The best way to systematically handle such an approach is via the use of sheaves. Thus in Section2 we introduce the elements of sheaf theory that we will require. Many of the elementary con- structions and results concerning distributions are just as easily done with vector bundles, rather than specifically with tangent bundles. For this reason, we devote a significant part of the paper to definitions and results for vector bundles. In Section3 we give the basic definitions and properties of generalised subbundles, i.e., assignments of a subspace of each fibre of a vector bundle. The set of sections of a generalised subbundle is a submodule of the module of sections. This simple observation, appropriately parsed in terms of sheaf theory, gives rise to some interesting algebraic structure for generalised subbundles. In Section4 we present some of this algebraic theory. In Section5 we study the important question of global generators of generalised subbundles. This provides an instance of the importance of sheaf theory in the study of generalised subbundles. Next in the paper we turn particularly to distributions, i.e., generalised subbundles of tangent bundles. Here the additional structure of vector fields having a flow, and all the consequences of this, play a r^ole.