distribution

In the following we will $C^{\mathrm{\infty }}$ when we say smooth.

Definition.

Let $M$ be a smooth manifold of dimension $m$. Let $n\le m$ and for each $x\in M$, we assign an $n$-dimensional subspace ${\mathrm{\Delta }}_x\subset T_x(M)$ 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 $X_1,\mathrm{\dots },X_n$ such that for any point $y\in N_x$, $X_1(y),\mathrm{\dots },X_n(y)$ span ${\mathrm{\Delta }}_y$. We let $\mathrm{\Delta }$ refer to the collection of all the ${\mathrm{\Delta }}_x$ for all $x\in M$ and we then call $\mathrm{\Delta }$ a distribution of dimension $n$ on $M$, or sometimes a $C^{\mathrm{\infty }}$ $n$-plane distribution on $M$. The set of smooth vector fields $\{X_1,\mathrm{\dots },X_n\}$ is called a local basis of $\mathrm{\Delta }$.

Note: The naming is unfortunate here as these distributions have nothing to do with distributions in the sense of analysis (http://planetmath.org/Distribution). However the naming is in wide use.

Definition.

We say that a distribution $\mathrm{\Delta }$ on $M$ is involutive if for every point $x\in M$ there exists a local basis $\{X_1,\mathrm{\dots },X_n\}$ in a neighbourhood of $x$ such that for all $1\le i,j\le n$, $[X_i,X_j]$ (the commutator of two vector fields) is in the span of $\{X_1,\mathrm{\dots },X_n\}$. That is, if $[X_i,X_j]$ is a linear combination of $\{X_1,\mathrm{\dots },X_n\}$. Normally this is written as $[\mathrm{\Delta },\mathrm{\Delta }]\subset \mathrm{\Delta }$.

Title	distribution
Canonical name	Distribution1
Date of creation	2013-03-22 14:51:57
Last modified on	2013-03-22 14:51:57
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	6
Author	jirka (4157)
Entry type	Definition
Classification	msc 53-00
Synonym	C^∞n-plane distribution
Related topic	FrobeniussTheorem
Defines	involutive
Defines	involutive distribution
Defines	local basis