Frobenius’ theorem
Theorem (Frobenius).
Let M be a smooth manifold (C∞) and let Δ be a
distribution on M. Then Δ is completely integrable if and only if
Δ is involutive.
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 Δ is involutive then it is completely integrable.
Another way to the theorem is that if we have n vector fields {Xk}nk=1 on a manifold M such that they are linearly independent at every point of the manifold, and furthermore if for any k,m we have
[Xk,Xm]=∑nj=1ajXj for some C∞ functions
aj, then for any point x∈N, there exists a germ of a submanifold N⊂M, through x, such that TN is spanned
by {Xk}nk=1. Note that if we extend N to all of M, it need not be
an embedded submanifold anymore, but just an immersed one.
For n=1 above, this is just the existence and uniqueness of solution of ordinary differential equations.
References
- 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
Title | Frobenius’ theorem |
---|---|
Canonical name | FrobeniusTheorem |
Date of creation | 2013-03-22 14:52:03 |
Last modified on | 2013-03-22 14:52:03 |
Owner | jirka (4157) |
Last modified by | jirka (4157) |
Numerical id | 9 |
Author | jirka (4157) |
Entry type | Theorem |
Classification | msc 37C10 |
Classification | msc 53-00 |
Classification | msc 53B25 |
Related topic | Distribution5 |
Related topic | IntegralManifold |