Concept
In differential geometry, a discipline within mathematics, a distribution on a manifold 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 . 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, e.g. integrable systems, Poisson geometry, non-commutative geometry, sub-Riemannian geometry, differential topology, etc. Even though they share the same name, distributions presented in this article have nothing to do with distributions in the sense of analysis. Let be a smooth manifold; a (smooth) distribution assigns to any point a vector subspace in a smooth way. More precisely, consists in a collection of vector subspaces with the following property. Around any there exist a neighbourhood and a collection of vector fields such that, for any point , span The set of smooth vector fields is also called a local basis of . Note that the number may be different for different neighbourhoods. The notation is used to denote both the assignment and the subset . Given an integer , a smooth distribution on is called regular of rank if all the subspaces have the same dimension. Locally, this amounts to ask that every local basis is given by linearly independent vector fields. More compactly, a regular distribution is a vector subbundle of rank (this is actually the most commonly used definition). A rank distribution is sometimes called an -plane distribution, and when , one talks about hyperplane distributions. Unless stated otherwise, by "distribution" we mean a smooth regular distribution (in the sense explained above). Given a distribution , its sections consist of the vector fields which are tangent to , and they form a vector subspace of the space of all vector fields on . A distribution is called involutive if is also a Lie subalgebra: in other words, for any two vector fields , the Lie bracket belongs to .