Distribution (differential geometry)

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For other meanings, see Distribution (disambiguation).

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.

[edit] 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 $y \in N_x$ , $X_1(y),\ldots,X_n(y)$ span $Δ y .$ We let $Δ$ refer to the collection of all the $Δ x$ for all $x \in M$ and we then call $Δ$ 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 $Δ.$

The naming is unfortunate here as these distributions have nothing to do with distributions in the sense of analysis. However the naming is in wide use.

[edit] Involutive distributions

We say that a distribution $Δ$ 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 derivative 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.