Frobenius’ theorem

Theorem (Frobenius).

Let $M$ be a smooth manifold ( $C^{\infty}$ ) and let $\Delta$ be a distribution on $M$ . Then $\Delta$ is completely integrable if and only if $\Delta$ 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 $\Delta$ 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 $C^{\infty}$ functions $a_{j}$ , then for any point $x\in N$ , there exists a germ of a submanifold $N\subset M$ , through $x$ , such that $T N$ 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