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 aneighbourhood $N_x\subset M$ of $x$ there exist $n$ linearly independentsmooth 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 thecollection of all the ${\mathrm{\Delta }}_x$ for all $x\in M$ and we then call $\mathrm{\Delta }$ adistribution of dimension $n$ on $M$, or sometimes a$C^{\mathrm{\infty }}$ $n$-plane distribution on $M$. The set of smoothvector 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 nothingto 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 $[\mathrm{\Delta },\mathrm{\Delta }]\subset \mathrm{\Delta }$.