If $dF_p$ is nonsingular, then $F(p)\in$ Int$N$

Question

Here is the problem 4-2 in John Lee's introduction to smooth manifolds:

Suppose $M$ is a smooth manifold (without boundary), $N$ is a smooth manifold with boundary, and $F: M \to N$ is smooth. Show that if $p \in M$ is a point such that $dF_p$ is non-singular, then $F(p)\in$ Int$N$.

To approach this problem, let's assume conversely that $F(p)\in$ $\partial N$, then there are two charts $(U,\varphi)$ and $(V, \psi)$(a boundary chart) centered at $p$ and $F(p)$ respectively. I can't see how the non-singularity of $dF_p$ contradicts to the fact that $F(p)$ is in the boundary chart. Maybe the proof by contradiction is not the right way? Thanks in advance!

Answer 1

$dF_p$ is nonsingular then there exists open set $U\subset M$ s.t. $F:U\rightarrow F(U)$ is a diffeomorphism Hence $F(p)$ is inerior point in $F(U)$ where $F(U)$ is open in $N$.

(Reference : differential forms and applications - do Carmo 60p.

If $H =\{ x|x_n\leq 0\}$ assume that $V$ is open in $H$. And $f : V\subset H\rightarrow {\bf R}$ is differentiable if there exists open $U \supset V$ and differentiable function $\overline{f}$ on $U$ s.t. $\overline{f}|_V=f$ Here $df_p$ is defined as $d\overline{f}_p$

That is, if $p=(0,\cdots,0),\ c(t)=(0,\cdots,0, t),\ -\epsilon \leq t\leq 0 $, then $$ df_p e_n:= \frac{d}{dt} f\circ c $$

And assume $F(p)$ is in the boundary That is if $\phi_M$ is chart for $M$, then we have a chart for $N$ $$f:=F\circ \phi_M : U:=B_\epsilon (0) \subset {\bf R}^n \rightarrow N $$ where $\phi_M(0)=p$ And there exists a chart $$ f_2:=\phi_N : U_2:=B_\delta (0)\cap H \rightarrow N,\ \phi_N(0)=p $$

Let $W:=F\circ \phi_M (U)\cap \phi_N( U_2) $ Then $$ f_2^{-1}\circ f : f^{-1}( W) \rightarrow H $$ is differentiable map Here $d (f\circ f_2^{-1} )_0 \neq 0$ so that inverse function theorem, there exists a neighborhood $V$ at $0$ in $ f^{-1}( W)$ s.t. $ f_2^{-1}\circ f$ is diffeomorphic on $V$ That is there exists a curve $c$ in $V$ s.t. $$\frac{d}{dt} f_2^{-1}\circ f \circ c = e_n \in T_0 H $$ That is $n$-th coordinate of $f_2^{-1}\circ f \circ c (t)$ is positive for small $0< t$ That is it is outside of $H$.)