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 presented 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.

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.

External links

• "Involutive distribution", Encyclopedia of Mathematics, EMS Press, 2001 [1994]

This article incorporates material from Distribution on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.