Distribution (differential geometry)

In differential geometry, a discipline within mathematics, a distribution is a subset of the tangent bundle of a manifold satisfying certain properties. Distributions are used to build up notions of integrability, and specifically of a foliation of a manifold.

Even though they share the same name, distributions we discuss in this article have nothing to do with distributions in the sense of analysis.

Let ${\displaystyle M}$ be a ${\displaystyle C^{\infty }}$ manifold of dimension ${\displaystyle m}$, and let ${\displaystyle n\leq m}$. Suppose that for each ${\displaystyle x\in M}$, we assign an ${\displaystyle n}$-dimensional subspace ${\displaystyle \Delta _{x}\subset T_{x}(M)}$ of the tangent space in such a way that for a neighbourhood ${\displaystyle N_{x}\subset M}$ of ${\displaystyle x}$ there exist ${\displaystyle n}$ linearly independent smooth vector fields ${\displaystyle X_{1},\ldots ,X_{n}}$ such that for any point ${\displaystyle y\in N_{x}}$, span ${\displaystyle \{X_{1}(y),\ldots ,X_{n}(y)\}=\Delta _{y}.}$ We let ${\displaystyle \Delta }$ refer to the collection of all the ${\displaystyle \Delta _{x}}$ for all ${\displaystyle x\in M}$ and we then call ${\displaystyle \Delta }$ a distribution of dimension ${\displaystyle n}$ on ${\displaystyle M}$, or sometimes a ${\displaystyle C^{\infty }}$ ${\displaystyle n}$-plane distribution on ${\displaystyle M.}$ The set of smooth vector fields ${\displaystyle \{X_{1},\ldots ,X_{n}\}}$ is called a local basis of ${\displaystyle \Delta .}$

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