distribution
In the following we will when we say smooth.
Definition.
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.
Definition.
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 .
References
- 1 William M. Boothby. , Academic Press, San Diego, California, 2003.
Title | distribution |
---|---|
Canonical name | Distribution1 |
Date of creation | 2013-03-22 14:51:57 |
Last modified on | 2013-03-22 14:51:57 |
Owner | jirka (4157) |
Last modified by | jirka (4157) |
Numerical id | 6 |
Author | jirka (4157) |
Entry type | Definition |
Classification | msc 53-00 |
Synonym | C^∞n-plane distribution |
Related topic | FrobeniussTheorem |
Defines | involutive |
Defines | involutive distribution |
Defines | local basis |