The Frobenius Theorem

What a joke to think that I could live without the internet.

Recall that the Frobenius Theorem is actually quite a concise statement given what we’ve defined and done. It just says that every involutive distribution is completely integrable.

Let D be involutive of dimension k and M an n-manifold and p\in M. We need a flat chart about p. We will work on a coordinate chart since it is a local property, and hence with loss of generality M=U\subset \mathbb{R}^n we replace the manifold with an open subset of Euclidean space. Suppose Y_1, \ldots , Y_k is a smooth local frame for D. We choose our coordinates such that D_p is complementary to the span of \displaystyle\left(\frac{\partial}{\partial x^{k+1}\big|_p}, \ldots , \frac{\partial}{\partial x^n}\big|_p\right).

Now let \pi:\mathbb{R}^n\to\mathbb{R}^k be the projection onto the first k coordinates. Then we get a bundle morphism d\pi:T\mathbb{R}^n\to T\mathbb{R}^k. Explicitly this is just \displaystyle d\pi\left(\sum_{i=1}^n v^i\frac{\partial}{\partial x^i}\big|_q\right)=\sum_{i=1}^k v^i\frac{\partial}{\partial x^i}\big|_{\pi(q)}.

The morphism is smooth, since it is composition of the inclusion map from the distribution followed by d\pi. Thus, by definition, the component functions in any smooth frame are smooth. In particular, we have smooth frames Y_1, \ldots, Y_k and \frac{\partial}{\partial x^j}\big|_{\pi(q)}. Thus the matrix entries of d\pi\big|_{D_q} with respect to these frames are smooth functions of q.

We also get that at any given point p, this restricted map is bijective as a map to D_p. It is certainly surjective by construction, and when restricted d\pi\big|_{D_p} is complementary to the kernel of the non-restricted bundle map by choice of coordinates. Thus it is bijective in an entire neighborhood of p. Thus we have a smooth inverse in this neighborhood (d\pi\big|_{D_q})^{-1} : T_{\pi(q)}\mathbb{R}^k\to D_q.

Define a frame for D on this neighborhood by X_i\big|_q=(d\pi\big|_{D_q})^{-1}\frac{\partial}{\partial x^i}\big|_{\pi(q)}. Note that if we can show that [X_i, X_j]=0 we will be done. This is because the frame will be a commuting frame, and the “canonical form” for a commuting frame is precisely our definition of a flat chart.

We have that \frac{\partial}{\partial x^i}\big|_{\pi(q)}=\left(d\pi\big|_{D_q}\right)X_i\big|_q=d\pi_q (X_i\big|_q). i.e. we have that X_i and \frac{\partial}{\partial x^i} are \pi-related.

But now we have it by naturality of the Lie bracket, since d\pi_q([X_i, X_j]_q)=[\frac{\partial}{\partial x^i}, \frac{\partial}{\partial x^j}]_{\pi(q)}=0. And the involutivity of D tells us that [X_i, X_j] is in D, and since d\pi on the distribution is injective, [X_i, X_j]_q=0 for every point in the neighborhood.


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