In the following we will when we say smooth.
Let be a smooth manifold of dimension and let be a distribution of dimension on . Suppose that is a connected submanifold of such that for every we have that (the tangent space of at ) is contained in (the distribution at ). We can abbreviate this by saying that . We then say that is an integral manifold of .
Do note that could be of lower dimension then and is not required to be a regular submanifold of .
We say that a distribution of dimension on is completely integrable if for each point there exists an integral manifold of passing through such that the dimension of is equal to the dimension of .
An example of an integral manifold is the integral curve of a non-vanishing vector field and then of course the span of the vector field is a completely integrable distribution.
- 1 William M. Boothby. , Academic Press, San Diego, California, 2003.
|Date of creation||2013-03-22 14:52:00|
|Last modified on||2013-03-22 14:52:00|
|Last modified by||jirka (4157)|
|Defines||completely integrable distribution|