One direction in the proof is pretty easy since the tangent space of an integral manifold is involutive, so sometimes the theorem is only stated in one direction, that is: If is involutive then it is completely integrable.
Another way to the theorem is that if we have vector fields on a manifold such that they are linearly independent at every point of the manifold, and furthermore if for any we have for some functions , then for any point , there exists a germ of a submanifold , through , such that is spanned by . Note that if we extend to all of , it need not be an embedded submanifold anymore, but just an immersed one.
For above, this is just the existence and uniqueness of solution of ordinary differential equations.
- 1 William M. Boothby. , Academic Press, San Diego, California, 2003.
- 2 Frobenius theorem at Wikipedia: http://en.wikipedia.org/wiki/Frobenius_theoremhttp://en.wikipedia.org/wiki/Frobenius_theorem
|Date of creation||2013-03-22 14:52:03|
|Last modified on||2013-03-22 14:52:03|
|Last modified by||jirka (4157)|