In the following we will when we say smooth.
Let be a smooth manifold of dimension . Let and for each , we assign an -dimensional subspace of the tangent space in such a way that for a neighbourhood of there exist linearly independent smooth vector fields such that for any point , span . We let refer to the collection of all the for all and we then call a distribution of dimension on , or sometimes a -plane distribution on . The set of smooth vector fields is called a local basis of .
Note: The naming is unfortunate here as these distributions have nothing to do with distributions in the sense of analysis (http://planetmath.org/Distribution). However the naming is in wide use.
We say that a distribution on is involutive if for every point there exists a local basis in a neighbourhood of such that for all , (the commutator of two vector fields) is in the span of . That is, if is a linear combination of . Normally this is written as .
- 1 William M. Boothby. , Academic Press, San Diego, California, 2003.
|Date of creation||2013-03-22 14:51:57|
|Last modified on||2013-03-22 14:51:57|
|Last modified by||jirka (4157)|