# nLab distribution of subspaces

The entry distribution is about the notion of generalized function or generalized density. In differential geometry there is another notion of distribution which is discussed here.

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.