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Let $X:U\subset\mathbb{R}^3\to\mathbb{R}^3$ a complete vector field of class $\mathscr{C}^k$($k\geqslant 1$), $p$ a regular point for $X$,$\varphi(t,p)$ a solution curve that goes through $p$ and $\Sigma$ a local cross -section of $X$ that passes through $p$. Assume that $\varphi(T,p)\in\Sigma$ for a $T>0$.

Show there exists a neighborhood $V$ of $p$ and a diffeomorphism over $\pi$ such that for every $\Sigma_0=V\cap\Sigma$ is homeomorphic to a disc and contains p, and $\pi(x)\in \Sigma$ is the first return of point x to the section $\Sigma$.

I think I could use the Flow box theorem so that it would give a diffeormphism let's say $\psi:V\to(-\epsilon,\epsilon)\times B$. However I am not sure and I do not know how to formalize that proof.

I am stuck at this problem as I do not have any idea on how to solve it.

Question:

How should I solve the problem?

Thanks in advance!

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  • $\begingroup$ What is your definition for "local cross section"? And, what does the phrase "... a diffeomorphism over $\pi$..." mean? $\endgroup$ – Mnifldz Nov 19 at 23:31
  • $\begingroup$ Probably you know the implicit function theorem. What is your difficulty? $\endgroup$ – John B Nov 19 at 23:40
  • $\begingroup$ @Mnifldz It is a transversal section. $\endgroup$ – Pedro Gomes Nov 20 at 12:29
  • $\begingroup$ @JohnB Thanks for your answer. I do not know how to apply the implicit function theorem on this case. Could you please show me? I need to see an example on how to solve these sort of questions. $\endgroup$ – Pedro Gomes Nov 20 at 14:33
  • $\begingroup$ Then you can have a look at the proof of the flow box theorem. $\endgroup$ – John B Nov 20 at 15:31

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