nLab distribution of subspaces

This entry is about the concept in differential geometry and Lie theory. For the concept in functional analysis see at distribution.

Definition

A real distribution on a real smooth manifold $M$ is a real vector subbundle of the tangent bundle $T M$.

A complex distribution is a complex vector subbundle of the complexified tangent bundle $T_{\mathbb{C}}M$ of $M$.

A distribution of hyperplanes is a distribution of codimension $1$ in $T M$; a distribution of complex hyperplanes is a distribution of complex codimension $1$ in $T_{\mathbb{C}} M$.

Examples

One class of examples comes from smooth foliations by submanifolds of constant dimension $m\lt n$. Then the tangent vectors at all points to the submanifolds forming the foliation form a distribution of subspaces of dimension $m$. The distributions of that form are said to be integrable.

… say something about the Frobenius theorem …

References

Discussion in the context of geometric quantization:

• N. M. J. Woodhouse, Geometric quantization