In differential geometry, a discipline within mathematics, a distribution on a manifold ${\displaystyle M}$ is an assignment ${\displaystyle x\mapsto \Delta _{x}\subseteq T_{x}M}$ of vector subspaces satisfying certain properties. In the most common situations, a distribution is asked to be a vector subbundle of the tangent bundle ${\displaystyle TM}$. If the length of the vector is not needed, i.e. if one is concerned only with the subspaces themselves, projectivized, then it can also be constructed as a subbundle of the contact 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, 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.

Let ${\displaystyle M}$ be a smooth manifold; a (smooth) distribution ${\displaystyle \Delta }$ assigns to any point ${\displaystyle x\in M}$ a vector subspace ${\displaystyle \Delta _{x}\subset T_{x}M}$ in a smooth way. More precisely, ${\displaystyle \Delta }$ consists of a collection ${\displaystyle \{\Delta _{x}\subset T_{x}M\}_{x\in M}}$ of vector subspaces with the following property: Around any ${\displaystyle x\in M}$ there exist a neighbourhood ${\displaystyle N_{x}\subset M}$ and a collection of vector fields ${\displaystyle X_{1},\ldots ,X_{k}}$ such that, for any point ${\displaystyle y\in N_{x}}$, span${\displaystyle \{X_{1}(y),\ldots ,X_{k}(y)\}=\Delta _{y}.}$

The set of smooth vector fields ${\displaystyle \{X_{1},\ldots ,X_{k}\}}$ is also called a local basis of ${\displaystyle \Delta }$. 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 ${\displaystyle \Delta }$ is used to denote both the assignment ${\displaystyle x\mapsto \Delta _{x}}$ and the subset ${\displaystyle \Delta =\amalg _{x\in M}\Delta _{x}\subseteq TM}$.

Given an integer ${\displaystyle n\leq m=\mathrm {dim} (M)}$, a smooth distribution ${\displaystyle \Delta }$ on ${\displaystyle M}$ is called regular of rank ${\displaystyle n}$ if all the subspaces ${\displaystyle \Delta _{x}\subset T_{x}M}$ have the same dimension ${\displaystyle n}$. Locally, this amounts to ask that every local basis is given by ${\displaystyle n}$ linearly independent vector fields.

More compactly, a regular distribution is a vector subbundle ${\displaystyle \Delta \subset TM}$ of rank ${\displaystyle n}$ (this is actually the most commonly used definition). A rank ${\displaystyle n}$ distribution is sometimes called an ${\displaystyle n}$-plane distribution, and when ${\displaystyle n=m-1}$, one talks about hyperplane distributions.

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