PlanetMath: distribution

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distribution

(Definition)

In the following we will mean $C^\infty$ when we say smooth.

Definition 1 Let

be a smooth manifold of dimension

. Let $n \leq m$ and for each $x \in M$ , we assign an

-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

there exist

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 $\Delta_y$ . We let $\Delta$ refer to the collection of all the $\Delta_x$ for all $x \in M$ and we then call $\Delta$ a distribution of dimension

, or sometimes a $C^\infty$ -plane distribution on

. The set of smooth vector fields $\{ X_1,\ldots,X_n \}$ is called a local basis of $\Delta$ .

Note: 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.

Definition 2 We say that a distribution $\Delta$ on

is involutive if for every point $x \in M$ there exists a local basis $\{ X_1,\ldots,X_n \}$ in a neighbourhood of

such that for all $1 \leq i, j \leq n$ ,

(the commutator of two vector fields) is in the span of $\{ X_1,\ldots,X_n \}$ . That is, if

is a linear combination of $\{ X_1,\ldots,X_n \}$ . Normally this is written as $[ \Delta , \Delta ] \subset \Delta$ .

Bibliography

1: William M. Boothby. An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, San Diego, California, 2003.

"distribution" is owned by jirka.

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