# 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 $TN$ 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

Title Frobenius’ theorem FrobeniusTheorem 2013-03-22 14:52:03 2013-03-22 14:52:03 jirka (4157) jirka (4157) 9 jirka (4157) Theorem msc 37C10 msc 53-00 msc 53B25 Distribution5 IntegralManifold