Frobenius’ theorem
Theorem (Frobenius).
Let $M$ be a smooth manifold (${C}^{\mathrm{\infty}}$) and let $\mathrm{\Delta}$ be a distribution^{} on $M$. Then $\mathrm{\Delta}$ is completely integrable if and only if $\mathrm{\Delta}$ 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 $\mathrm{\Delta}$ is involutive then it is completely integrable.
Another way to the theorem is that if we have $n$ vector fields ${\{{X}_{k}\}}_{k=1}^{n}$ 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 $[{X}_{k},{X}_{m}]={\sum}_{j=1}^{n}{a}_{j}{X}_{j}$ for some ${C}^{\mathrm{\infty}}$ functions^{} ${a}_{j}$, then for any point $x\in N$, there exists a germ of a submanifold $N\subset M$, through $x$, such that $TN$ is spanned by ${\{{X}_{k}\}}_{k=1}^{n}$. 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 |