Showing that these vector fields commute on the image This was a problem from my final exam I took a few days ago, and after having it graded my solution did not receive full points. The professor seems to be busy grading for other classes he is teaching and I am eager to know why I did not get full credit for it, so I was hoping someone here could help me determine what my solution is missing. 
Suppose $M$ is a smooth $n$-manifold and $F: \mathbf{R}^2 \to M$ is a smooth map. Let $X, Y \in \mathscr{X}(M)$ (smooth vector fields on $M$), and suppose that for every $s \in \mathbf{R}$, $t \mapsto F(t, s)$ and $t \mapsto F(s,t)$ are integral curves for $X, Y$ respectively. Show that $[X, Y]=0$ on $F(\mathbf{R}^2)$.
My solution:
By the fundamental theorem on flows (Lee Theorem 9.12) there are a unique smooth maximal flows with infinitesimal generators $X$ and $Y$. Since $t \mapsto F(t, s)$ and $t \mapsto F(s, t)$ are maximal integral curves for $X, Y$, they must be equal to the integral curves for $X, Y$ defined by the flow on all of $M$. Let $\theta$ be the flow of $X$ and $\Psi$ be the flow of $Y$ restricted to $F(\mathbf{R}^2)$. Now given a specific point $p \in F(\mathbf{R}^2)$ there exists $(t_p, s_p) \in \mathbf{R}^2$ such that $F(t_p, s_p) = p$. Let $t_1, t_2 \in \mathbf{R}$. Then $\theta_{t_1}(p) = F(t_p+t_1, s_p)$ and applying $\Psi_{t_2}(\theta_{t_1}(p))$ we obtain $F(t_p+t_1, s_p +t_2)$, and the same argument can be given to show this equals $\theta_{t_1}(\Psi_{t_2}(p))$.
Now by theorem 9.44 of Lee, we have that smooth vector fields commute iff their flows commute (where the flows are defined), and thus we must have that $X, Y$ commute on $F(\mathbf{R}^2)$ and therefore $[X, Y]=0$ on $F(\mathbf{R}^2)$ as desired.
 A: There are a couple of problems with your solution. Let me try to explain what they are. 

By the fundamental theorem on flows (Lee Theorem 9.12) there are a
  unique smooth maximal flows with infinitesimal generators $X$ and $Y$.
  Since $t \mapsto F(t, s)$ and $t \mapsto F(s, t)$ are maximal integral
  curves for $X, Y$, they must be equal to the integral curves for $X,Y$ defined by the flow on all of $M$. 
  Let $\theta$ be the flow of $X$
  and $\Psi$ be the flow of $Y$ restricted to $F(\mathbf{R}^2)$. Now
  given a specific point $p \in F(\mathbf{R}^2)$ there exists $(t_p,s_p) \in \mathbf{R}^2$ such that $F(t_p, s_p) = p$. Let $t_1, t_2 \in \mathbf{R}$. Then $\theta_{t_1}(p) = F(t_p+t_1, s_p)$ and applying
  $\Psi_{t_2}(\theta_{t_1}(p))$ we obtain $F(t_p+t_1, s_p +t_2)$, and
  the same argument can be given to show this equals
  $\theta_{t_1}(\Psi_{t_2}(p))$.

Nothing in the problem statement guarantees that $t\mapsto F(t,s)$ and $t\mapsto F(s,t)$ are maximal integral curves. But they are integral curves. Note that the first paragraph of the proof of Theorem 9.12 in my book shows that any two integral curves for the same vector field with the same initial condition must agree on their common domain (I don't know why I didn't state that as a separate theorem!), and therefore your conclusions in the last sentence above are true. 

Now by theorem 9.44 of Lee, we have that smooth vector fields commute
  iff their flows commute (where the flows are defined), and thus we
  must have that $X, Y$ commute on $F(\mathbf{R}^2)$ and therefore $[X,Y]=0$ on $F(\mathbf{R}^2)$ as desired.

Theorem 9.44 doesn't apply in this situation. The context of that theorem (stated at the beginning of the "Commuting Vector Fields" section on page 231) is that we are talking about smooth vector fields defined on a smooth manifold. Because $F(\mathbb R^2)$ is typically not going to be a smooth submanifold of $M$, we can't conclude from Theorem 9.44 that $X$ and $Y$ commute on $F(\mathbb R^2)$ if and only if their flows commute on $F(\mathbb R^2)$. (That approach could be made to work in the special case that $F$ is a smooth embedding, once you show that the vector fields $X$ and $Y$ are tangent to the submanifold $N = F(\mathbb R^2)$ and therefore restrict to vector fields on $N$.)
A better approach to this problem is based on the notion of $\boldsymbol F$-related vector fields. Take a look at Propositions 8.30 (naturality of Lie brackets) and 9.6 (naturality of integral curves), and think about applying them to the vector fields $\partial/\partial x$ and $\partial/\partial y$ on $\mathbb R^2$.
