# 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 ${\displaystyle M}$ 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 ${\displaystyle \Delta .}$

## Involutive distributions

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

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 ${\displaystyle \Delta _{x}\subset T_{x}M}$ are not required to all be of the same dimension. The definition requires that the ${\displaystyle \Delta _{x}}$ 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 ${\displaystyle \Delta _{x}}$ 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.

## References

• William M. Boothby. Section IV. 8. Frobenius's Theorem in An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, San Diego, California, 2003.
• P. Stefan, Accessible sets, orbits and foliations with singularities. Proc. London Math. Soc. 29 (1974), 699-713.
• H.J. Sussmann, Orbits of families of vector fields and integrability of distributions. Trans. Amer. Math. Soc. 180 (1973), 171-188.