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 MM is a vector subbundle of the tangent bundle TMT M. A complex distribution is a complex vector subbundle of the complexified tangent space T CMT_{\mathbf{C}}M of MM. A distribution of hyperplanes is a distribution of codimension 11 in TMT M; a distribution of complex hyperplanes is a distribution of complex codimension 11 in T CMT_{\mathbf{C}} M.

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

say something about Frobenius theorem

  • N. M. J. Woodhouse, Geometric quantization

Revised on August 14, 2017 02:15:00 by Urs Schreiber (94.220.75.2)