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I am reading Lee's book Introduction to smooth manifolds 2nd edition chapter 9 the fundamental theorem on flows.

In the proof of the fundamental theorem on flows the author defines $t_0=\inf\{t\in\mathbb{R}:(t,p_0)\notin W\}$, and argues that since $(0,p_0)\in W$, we have $t_0>0$. (Page 213 the last paragraph.)

But I am thinking can there be some $t\in\mathbb{R}$ s.t. $t<0$ and $(t,p_0)\notin W$? It is not very clear to me why this can not happen.

More context on the theorem (you may ignore this part):

In the proof, the author wants to show that for any smooth vector field $V$ on a smooth manifold $M$, there exists a local flow $\theta$ generated by $V$.

In the proof, the author defines $\mathcal{D}$ as the flow domain (we still need to prove that it is actually a flow domain), and wants to show that $\mathcal{D}$ is open and that $\theta$ as previously defined in the book is smooth on $\mathcal{D}$. The author wants to show this by contradiction.

He defines $W\subseteq\mathcal{D}$ s.t. $\theta$ is smooth on $W$. And that for each $(t,p)\in W$ there exists a product neighborhood $J\times U\subset \mathbb{R}\times M$ s.t. $(t,p)\in J\times U\subset W$. Where $J$ is an open interval in $\mathbb{R}$ containing both $0$ and $t$.

Now the author wants to show by contradiction that $W=\mathcal{D}$.

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  • $\begingroup$ You should give more context. You'll get more people willing to answer. As it is, I have no idea what $W$ is. $\endgroup$ – Lee Mosher Mar 10 '16 at 21:38
  • $\begingroup$ Thanks for the advice, I added some context. $\endgroup$ – user40683 Mar 10 '16 at 22:05
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My understanding is the same as yours: it should be $t_0=\sup\{t\in\mathbb R:(t,p_0)\in W\}$.

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  • $\begingroup$ Thanks for your opinion! I agree with you. $\endgroup$ – user40683 Mar 10 '16 at 22:07
  • $\begingroup$ You're right. I've added this to my correction list. $\endgroup$ – Jack Lee Mar 10 '16 at 22:54
  • $\begingroup$ Thanks for you comment prof. Lee! I like your books! $\endgroup$ – user40683 Mar 10 '16 at 22:57
  • $\begingroup$ You're welcome, and thank you! $\endgroup$ – Jack Lee Mar 10 '16 at 23:01

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