So I am studying a proof of Poincaré-Bendixon Theorem and they do so, by cases and part by part, and I have some questions about some steps they made and how is it they could do this.
(Below I am going to put a Glossary on the definitions we use.) This is how they start:
Given $z \in M$ a regular point of a flow $\phi$ associated to $X$ a $C^1$ vector field and $M$ a smooth manifold of dimension $m$, we can consider the linear sub-space $E=X(z)^{\perp}$ and given any submanifold $S$ such that $T_zS= E$. Given the regularity of $X$ in a neighborhood $U$ of $z$ we have that for every $y \in S$:
$T_yS \oplus X(z)= T_zS$ (*)
Concluding that $S$ is a tansversal section.
So my questions are
1) How do you know that there exists such $S$ such that $E=X(z)^{\perp}$?
2) How does the regularity of $X$ gives you (*)?
Glossary:
Regular Point: A point $z$ which the flow $\phi_t(z)$ is not fixed (i.e the orbit $O(z,\phi)$ has more than one point$)
Transversal section: Its the image of a diffeomorfism $s:\mathbb{R}^{m-1} \to M$ over its own image such that $Im(d_ps)\oplus X(s(p))= T_pM$ for every $p\in \mathbb{R}^{m-1}$.
