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From the Introduction of Smooth Manifolds (Lee) - page 232, 2nd edition.

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Can anyone please explain why the problem mentioned at the end of the paragraph is indeed a problem?

Thank you in advance.

P.S.: I hope there is no problem posting this photo with the paragraph of the book.

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    It means that as functions of $(t, s) \in \Bbb R^2$, $\theta_t \circ \psi_s$ and $\psi_s \circ \theta_t$ will have different domains, which further complicates the idea that if $V$ and $W$ commute, then it should be that $\theta_t \circ \psi_s = \psi_s \circ \theta_t$. – 2017-02-13
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    Indeed, but we could hope that they commute in the intersection of these different domains, when both sides are defined -as the author says. Where is the problem with that? Thanks for your comment. – 2017-02-13
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    Sometimes it's best to try to think up examples. Try $M=\Bbb R^2-\{0\}$, $p=(-1,0)$, $V=\partial/\partial x$, and $W=\partial/\partial y$. – 2017-02-13
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    Possibly I am confused with something obvious, I am sorry for that. I can see that there might be the case that if $\theta_{t_0}\circ \psi_{s_0}(p)$ is defined then $\theta_{t}\circ \psi_{s}(p)$ might not be defined in any $I \times J$ where $I,J$ open intervals of $\mathbb{R}$ and $[0, t_0] \subset I$ and $[0, s_0] \subset J$. I assume that you propose this example in order to illustrate this fact. But my problem is not here, my problem is why should that be the case. 1/3 – 2017-02-14
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    Which properties of flow domains dictate that there should exist a $I \times J$ st $\theta_{t}\circ \psi_{s}(p)$ is defined on that? From the definition, $\theta^{q}_t=\theta^{q}(t)$ (as a function of $t$) is defined on an open interval containing $0$. So $\theta_{t}\circ \psi_{s_0}(p)=\theta^{(\psi_{s_0}(p))}(t)$ should be defined on an open interval $I$ containing $t_0$ and $0$. Also, obviously $\psi^{(p)}(s)=\psi_s(p)$ is defined on an open interval $J$ containing $s_0$ and $0$. 2/3 – 2017-02-14
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    But if we take an $s_1\in J$ with $s_1\neq s_0$ then why should $\theta_{t}\circ \psi_{s_1}(p)$ be defined, for $t$ in $I$ (or even $t_0$)? Thanks again and sorry for my silly questions. 3/3 – 2017-02-14

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