How to proof Frobenius Theorem in general?

Question

The general Frobenius Theorem stating that

Let $u_1,\dots,u_k$ be $k$ smooth linearly independent vector field on $M$. Let

$$W=\operatorname{Span}(u_1,\cdots,u_k)$$ Then $[u_i,u_j]\in W$ for any $i,j$ if and only if there exist foliation by $k$ dimension hypersurface tangent to $M$.

To my understanding,

there exist foliation by $k$ dimension hypersurface tangent to $M$.

means there is a cooridnate $(w_1,\cdots,w_{n-k},x_1,\cdots,x_k)$ such that

$$u_i=\frac{\partial}{\partial x_i}$$

I know the proof of the special case where $k=2$. To prove the general case, there is a hint saying using induction on $k$. However, I am not clear how to commit the induction. I tried to consider $[u_{k-1},u_k]$, but we only have

$$[u_{k-1},u_k]\in\operatorname{Span}(u_1,\cdots,u_k)$$ rather than $$[u_{k-1},u_k]\in\operatorname{Span}(u_{k-1},u_k)$$ So I cannot perform the simpler case. Could anyone help?

This post imported from StackExchange Mathematics at 2014-09-24 21:03 (UTC), posted by SE-user hxhxhx88

Comments

The case k=1 follows from an existence and uniqueness theorem for ODE's. The step to k=2 from the proof you know should be the same as the on from k-1 to k in the induction.

This post imported from StackExchange Mathematics at 2014-09-24 21:03 (UTC), posted by SE-user MBN

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Although I think this is a good and interesting question, I don't think it belongs on this site (math.SE would be better, clearly).

This post imported from StackExchange Mathematics at 2014-09-24 21:03 (UTC), posted by SE-user Danu

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@MBN, sorry I don't get your words. Could you explain a little more?

This post imported from StackExchange Mathematics at 2014-09-24 21:03 (UTC), posted by SE-user hxhxhx88

Answer 1

Here is a way of thinking about Frobenius theorem that might help to understand the theorem and its proof.

The condition that one has a set of $k$ linearly independent complete vector fields on $M$ closed by the Lie bracket of vector fields means that one has a Lie subalgebra of the Lie algebra of vector fields of $M$.

One can, then, invoke Picard-Lindelöf's theorem (on existence and uniqueness of solutions for ODE's) and use the flow of each vector field to provide an action of the associated Lie group on $M$. This action induces a foliation on $M$, the orbits of this action are the leaves of the foliation.

This post imported from StackExchange Mathematics at 2014-09-24 21:03 (UTC), posted by SE-user Romero Solha

Comments

Yes, I can imagine such picture, but still not sure how to prove rigorously. Anyway, thanks:)

This post imported from StackExchange Mathematics at 2014-09-24 21:03 (UTC), posted by SE-user hxhxhx88

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This is a, rigorous, proof of it. It is not hard to work the details of it.

This post imported from StackExchange Mathematics at 2014-09-24 21:03 (UTC), posted by SE-user Romero Solha