I am a beginner of differential geometry. I wonder if the following proposition is true:
Let $M$ be an n-dimensional manifold and
$X_1, \dots ,X_m(m \le n)$ be m commuting and linearly independent vector fields in a neighborhood of a point $p$ in M, then there is a coordinate system $(U, x_1, . . . , x_n)$ around $p$ such that
$X_1 =\frac{∂}{∂x_1},…,X_m =\frac{∂}{∂x_m}$ on $U$.
Since $X_1, \dots ,X_m(m \le n)$ are commuting, we have $[X_i,X_j]=0$ for any $i,j$.
When $m=1$, this is the proposition 1.53(Page 40) in Warner's book.
Honestly speaking, I don't even know how to prove second simplest cases when $m=2.$ If this proposition is true, then I think it should also be a theorem in some book. Every hint, solution, or reference will be appreciated!
Best Answer
Yes, your proposition is true. This can be proven using the fact that two vector fields commute iff. their flows commute: $$ [X,Y] = 0 \Leftrightarrow \phi^X_t\circ\phi_s^Y=\phi^Y_s\circ\phi^X_t. $$ First of all, the proof can be found in J. M. Lee's "Introduction to Smooth Manifolds", Thm. 9.46.
To explain the idea, I will assume that $m=n$ (for the general case, see Lee). Given a point $p\in U$ (the "origin"), the flows define maps $\phi^i|_p: I_i\rightarrow U$ onto U which are your coordinate lines through $p$, and the tangent vectors are exactly the $X_i$ by definition. The flows span a coordinate grid on $U$ and the corresponding chart is then defined by the inverse of $\Phi(t^1,\dots,t^n)=\phi^1_{t^1}\circ\dots\circ\phi^n_{t^n}(p)$. To prove that this is a chart it is crucial that the flows commute, because this guarantees that a point $q\in U$ has unique coordinates $(t^1,\dots,t^n)$ and $\Phi$ is invertible.
To see this, consider the pushforward of the cartesian vector fields $\partial_{t^i}$ at some point $t_0$: $$ (\Phi_*(\partial_{t^i}|_{t_0}))f=\partial_{t^i}|_{t_0}f(\Phi(t^1,\dots,t^n))=\partial_{t^i}|_{t_0}f(\phi^1_{t^1}\circ\dots\circ\phi^n_{t^n}(p))=\partial_{t^i}|_{t_0}f(\phi^i_{t^i}\circ\phi^1_{t^1}\circ\dots\circ\hat{\phi}^i_{t^i}\circ\dots\phi^n_{t^n}(p)=:\partial_{t^i}|_{t_0}f(\phi^i_{t^i}(q))=X_i|_{\Phi(t_0)}f. $$ Here we used the commutivity of the flows and that $t^i\mapsto\phi^i_{t^i}(q)$ is a integral curve of $X_i$. Hence $\Phi_*$ maps the $\partial_{t^i}$ onto $X_i$. In particular $\Phi_*|_0$ maps $\partial_{t^1}|_0,\dots,\partial_{t^n}|_0$ to $X_1|_p,\dots,X_n|_p$, which are both basises of $T_0\mathbb{R}^n$ and $T_pM$, respectively. Hence, $\Phi_*|_0$ is an isomorphism and by the inverse function theorem, $\Phi$ is a local diffeomorphism, thus $\Phi$ defines a chart on some neighbourhood of $0\in\mathbb{R}^n$.