Let $M$ be a manifold and $D$ be a distribution on $M$ (thus a subbundle of the tangent bundle $TM$). Denote the fibers of the subbundle with $D_p$, $p \in M$. These are subspaces of the tangent space $T_pM$.
Definition: A smooth local section on $D$ is a vector field $X$ on $M$ such that $X_p \in D_p$ for all $p$ on an open subset of $M$.
Is the following statement true?
If for every $p \in M$, there is an open neighborhood $U$ of $p$ in
$M$ and a frame $(X_1, \dots, X_k)$ of local smooth sections on $D$
with domain $U$ (as defined in the definition above) such that $[X_i,
X_j]$ is a smooth local section on $D$ with domain $U$, then $D$ is
involutive.
I think I saw a similar statement in Lee, but I'm wondering if my interpretation of it is correct.
Best Answer
Yes, it is true, it is also called the theorem of Frobenius.
https://en.wikipedia.org/wiki/Frobenius_theorem_(differential_topology)