# 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.

## Definition

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

## Involutive distributions

We say that a distribution $\Delta$  on $M$  is involutive if for every point $x\in M$  there exists a local basis $\{X_{1},\ldots ,X_{n}\}$  of the distribution in a neighbourhood of $x$  such that for all $1\leq i,j\leq n$ , $[X_{i},X_{j}]$  (the Lie bracket of two vector fields) is in the span of $\{X_{1},\ldots ,X_{n}\}.$  That is, if $[X_{i},X_{j}]$  is a linear combination of $\{X_{1},\ldots ,X_{n}\}.$  Normally this is written as $[\Delta ,\Delta ]\subset \Delta .$

Involutive distributions are the tangent spaces to foliations. Involutive distributions are important in that they satisfy the conditions of the Frobenius theorem, and thus lead to 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.

## Generalized distributions

A generalized distribution, or Stefan-Sussmann distribution, is similar to a distribution, but the subspaces $\Delta _{x}\subset T_{x}M$  are not required to all be of the same dimension. The definition requires that the $\Delta _{x}$  are determined locally by a set of vector fields, but these 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.

One class of examples is furnished by a non-free action of a Lie group on a manifold, the vector fields in question being the infinitesimal generators of the group action (a free action gives rise to a genuine distribution). Another arises in dynamical systems, where the set of vector fields in the definition is the set of vector fields that commute with a given one. There are also examples and applications in Control theory, where the generalized distribution represents infinitesimal constraints of the system.