Let $S\subset M$ be compact embedded k dimensional submanifold,the embedding $\phi:B(0) \subset \Bbb{R}^{n-k}\to M$ be local transversal of $S$ at $p$(That is $\phi$ transversal to $S$ and $\phi^{-1}(S) = 0$)
Prove that there exists a local coordinate $(U,(x^1,...,x^n))$ center at $p$ such that
- $S\cap U = \{x^{k+1} = ...= x^n = 0\}$
- the map $\phi$ in this local coordiante has the form $\phi(y^1,...,y^{n-k}) = (0,...,0,y^1,...,y^{n-k})$
If we consider the condition individually, both of the result are direct consequence of constant rank theorem. The problem is how to combine them together? I know we need to apply the transversality condition , which says $$T_pS \oplus \phi_*(T_0B) = T_pM$$
My IDEA is first using the slice chart lemma find a local chart $(V,\varphi = (x_1,...,x^n))$ such that the condition 1 holds, then find some local diffeomorphism:
$$\Phi:\Bbb{R}^n \to \Bbb{R}^n \\ (x^1,..,x^n) \mapsto (y^1,...,y^n)$$ such that $\Phi\circ\varphi$ is still slice chart map, and 2 is satisfied I have no idea how to find such $\Phi$?
(This question comes from Liviu I. Nicolaescu's differential geometry book page 252. lemma 7.3.15)