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 $TM$. Locally, for $p\in M$ there exist $r$ pointwise linearly independent vector fields $X_1,\dots ,X_r$ such that $D_p=\text{span}(\{X_{1,p},\cdots ,X_{r,p}\})$.

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

Sometimes the distribution is identified with the ${\mathcal{C}}^{\mathrm{\infty }}(M)$-module of vector fields $X$ de $M$ such that $X_p\in D_p$ for every $p\in M$, $\mathrm{\Gamma }(M,D)$. 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 $G(r,TM)$ over $M$.

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es