In differential geometry, a discipline within mathematics, a distribution on a manifold M is an assignment of vector subspaces satisfying certain properties. In the most common situations, a distribution is asked to be a vector subbundle of the tangent bundle TM. Distributions satisfying a further integrability condition give rise to foliations, i.e. partitions of the manifold into smaller submanifolds. These notions have several applications in many fields of mathematics, including integrable systems, Poisson geometry, non-commutative geometry, sub-Riemannian geometry, differential topology. Even though they share the same name, distributions presented in this article have nothing to do with distributions in the sense of analysis.

Definition

Let M be a smooth manifold; a (smooth) distribution \Delta assigns to any point x \in M a vector subspace in a smooth way. More precisely, \Delta consists of a collection of vector subspaces with the following property: Around any x \in M there exist a neighbourhood and a collection of vector fields such that, for any point y \in N_x, span The set of smooth vector fields is also called a local basis of \Delta. These need not be linearly independent at every point, and so aren't formally a vector space basis at every point; thus, the term local generating set can be more appropriate. The notation \Delta is used to denote both the assignment and the subset.

Regular distributions

Given an integer, a smooth distribution \Delta on M is called regular of rank n if all the subspaces have the same dimension n. Locally, this amounts to ask that every local basis is given by n linearly independent vector fields. More compactly, a regular distribution is a vector subbundle of rank n (this is actually the most commonly used definition). A rank n distribution is sometimes called an n-plane distribution, and when n = m-1, one talks about hyperplane distributions.

Special classes of distributions

Unless stated otherwise, by "distribution" we mean a smooth regular distribution (in the sense explained above).

Involutive distributions

Given a distribution \Delta, its sections consist of vector fields on M, forming a vector subspace of the space of all vector fields on M. (Notation: \Gamma(TM) is the space of sections of TM.) A distribution \Delta is called involutive if is also a Lie subalgebra: in other words, for any two vector fields, the Lie bracket [X,Y] belongs to. Locally, this condition means that for every point x \in M there exists a local basis of the distribution in a neighbourhood of x such that, for all, the Lie bracket [X_i,X_j] is in the span of , i.e. [X_i,X_j] is a linear combination of Involutive distributions are a fundamental ingredient in the study of integrable systems. A related idea occurs in Hamiltonian mechanics: two functions f and g on a symplectic manifold are said to be in mutual involution if their Poisson bracket vanishes.

Integrable distributions and foliations

An integral manifold for a rank n distribution \Delta is a submanifold N \subset M of dimension n such that for every x \in N. A distribution is called integrable if through any point x \in M there is an integral manifold. The base spaces of the bundle are thus disjoint, maximal, connected integral manifolds, also called leaves; that is, \Delta defines an n-dimensional foliation of M. Locally, integrability means that for every point x \in M there exists a local chart such that, for every y \in U, the space \Delta_y is spanned by the coordinate vectors. In other words, every point admits a foliation chart, i.e. the distribution \Delta is tangent to the leaves of a foliation. Moreover, this local characterisation coincides with the definition of integrability for a G-structures, when G is the group of real invertible upper-triangular block matrices (with and (m-n,m-n)-blocks). It is easy to see that any integrable distribution is automatically involutive. The converse is less trivial but holds by Frobenius theorem.

Weakly regular distributions

Given any distribution, the associated Lie flag is a grading, defined as where, and. In other words, denotes the set of vector fields spanned by the i-iterated Lie brackets of elements in. Some authors use a negative decreasing grading for the definition. Then \Delta is called weakly regular (or just regular by some authors) if there exists a sequence G-structure on a manifold of nested vector subbundles such that G-structure on a manifold (hence ). Note that, in such case, the associated Lie flag stabilises at a certain point, since the ranks of T^iM are bounded from above by G-structure on a manifold. The string of integers is then called the grow vector of \Delta. Any weakly regular distribution has an associated graded vector bundleMoreover, the Lie bracket of vector fields descends, for any, to a -linear bundle morphism , called the (i,j)-curvature. In particular, the (0,0)-curvature vanishes identically if and only if the distribution is involutive. Patching together the curvatures, one obtains a morphism, also called the Levi bracket, which makes into a bundle of nilpotent Lie algebras; for this reason, is also called the nilpotentisation of \Delta. The bundle, however, is in general not locally trivial, since the Lie algebras are not isomorphic when varying the point x \in M. If this happens, the weakly regular distribution \Delta is also called regular (or strongly regular by some authors). Note that the names (strongly, weakly) regular used here are completely unrelated with the notion of regularity discussed above (which is always assumed), i.e. the dimension of the spaces \Delta_x being constant.

Bracket-generating distributions

A distribution is called bracket-generating (or non-holonomic, or it is said to satisfy the Hörmander condition) if taking a finite number of Lie brackets of elements in is enough to generate the entire space of vector fields on M. With the notation introduced above, such condition can be written as for certain ; then one says also that \Delta is bracket-generating in m+1 steps, or has depth m+1. Clearly, the associated Lie flag of a bracket-generating distribution stabilises at the point m. Even though being weakly regular and being bracket-generating are two independent properties (see the examples below), when a distribution satisfies both of them, the integer m from the two definitions is the same. Thanks to the Chow-Rashevskii theorem, given a bracket-generating distribution on a connected manifold, any two points in M can be joined by a path tangent to the distribution.

Examples of regular distributions

Integrable distributions

Non-integrable distributions

Singular distributions

A singular distribution, generalised distribution, or Stefan-Sussmann distribution, is a smooth distribution which is not regular. This means that the subspaces may have different dimensions, and therefore the subset is no longer a smooth subbundle. In particular, the number of elements in a local basis spanning \Delta_x will change with x, and those vector fields will no longer be linearly independent everywhere. It is not hard to see that the dimension of \Delta_x is lower semicontinuous, so that at special points the dimension is lower than at nearby points.

Integrability and singular foliations

The definitions of integral manifolds and of integrability given above applies also to the singular case (removing the requirement of the fixed dimension). However, Frobenius theorem does not hold in this context, and involutivity is in general not sufficient for integrability (counterexamples in low dimensions exist). After several partial results, the integrability problem for singular distributions was fully solved by a theorem independently proved by Stefan and Sussmann. It states that a singular distribution \Delta is integrable if and only if the following two properties hold: Similarly to the regular case, an integrable singular distribution defines a singular foliation, which intuitively consists in a partition of M into submanifolds (the maximal integral manifolds of \Delta) of different dimensions. The definition of singular foliation can be made precise in several equivalent ways. Actually, in the literature there is a plethora of variations, reformulations and generalisations of the Stefan-Sussman theorem, using different notion of singular foliations according to which applications one has in mind, e.g. Poisson geometry or non-commutative geometry.

Examples

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