I'm studying differential topology from Barrett O'Neill, Semi Riemannian Geometry, in specific, integral curves and flow. There is a lemma that I just can't figure out its proof:
If V is a vector field and p a point such that $V_p \neq 0$, then there is a coordinate system $x^1,...x^n$ at p susch that $V = \partial/\partial x^1$ on the coordinate neighborhood.
$V_p$ represents the vector field at $p$. The author takes a hypersurface $S$ such that $V_p \notin T_pS$. If $\phi$ is a local flow of V in the restriction $S \times I$, then $d\phi(\partial/\partial t)=V_p \notin T_pS$ at the point $(p,0)$. Then, it "follows" that $d\phi_{(p,0)}$ is an isomorphism. How he concluded that?
Anyone can explain me please?
By the definition of flow, we know $\phi$ is the identity when restricted to $S \times \{0\}.$ Thus its derivative $$L = d\phi_{(p,0)} : T_p S \times \mathbb R \to T_p S \times \mathbb R$$ is the identity when restricted to $T_p S \times \{0\}$. Thus we can write $L$ as a block matrix
$$\left[ \begin{matrix} I_{T_pS} & v_S \\ 0 & v_I \end{matrix} \right]$$
where the right column is $d\phi(\partial/\partial t) = V_p$. The fact that this does not lie in $T_p S$ tells us that $v_I \ne 0,$ and thus $L$ is invertible.