distribution of subspaces

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

distribution.

A **real distribution** on a real smooth manifold $M$ is a vector subbundle of the tangent bundle $T M$. A **complex distribution** is a complex vector subbundle of the complexified tangent space $T_{\mathbf{C}}M$ 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 $T_{\mathbf{C}} M$.

One class of examples come 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 Frobenius theorem*

- N. M. J. Woodhouse,
*Geometric quantization*

Last revised on August 14, 2017 at 02:15:00. See the history of this page for a list of all contributions to it.