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!