# 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.