# $V\cap \Sigma$ homeomorphic to a disc

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?

• What is your definition for "local cross section"? And, what does the phrase "... a diffeomorphism over $\pi$..." mean? – Mnifldz Nov 19 at 23:31