# 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.

## Idea

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.