Distribution

(See [Lychagin_2007] page 3)
A distribution $D$ of rank $r$ on a $n$ -dimensional manifold $M$ is a subbundle of the tangent bundle $T M$ . Locally, for $p \in M$ there exist $r$ pointwise linearly independent vector fields $X_{1}, \dots, X_{r}$ such that $D_{p} = span ({X_{1, p}, \dots, X_{r, p}})$ .

(We call it rank instead of dimension since as a manifold already have a dimension, $n + r$ )

Sometimes the distribution is identified with the $C^{\infty} (M)$ -module of vector fields $X$ de $M$ such that $X_{p} \in D_{p}$ for every $p \in M$ , $Γ (M, D)$ . This is justified by the vector bundle-module of section identification.

Distributions can be involutive or not.

They can have symmetry of a distribution or cinf-symmetry of distribution (the latter only the involutive distribution).

Distributions have associated a structure 1-form of a distribution, which takes value in a kind of "vertical bundle".

Distributions have a notion of curvature of a distribution.

There is a dual description of the distribution by means of a Pfaffian system.

A distribution of constant rank $r$ on $M$ can also be seen like a smooth section of the Grassmannian bundle $G (r, T M)$ over $M$ .