Distribution (differential geometry)

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.