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**.

*say something about Frobenius theorem*

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

Created on October 20, 2011 00:18:58
by Zoran Škoda
(161.53.130.104)