Frobenius’ theorem

Theorem (Frobenius).

Let $M$ be a smooth manifold ($C^{\mathrm{\infty }}$) and let $\mathrm{\Delta }$ be adistribution 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 spaceof an integral manifold is involutive, so sometimes the theorem is onlystated in one direction, that is: If $\mathrm{\Delta }$ is involutive then it iscompletely 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 spannedby ${\{X_k\}}_{k=1}^n$. Note that if we extend $N$ to all of $M$, it need not bean 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.