I want to prove that: Given $M$ smooth $m$ dimensional manifold and $f: M \to \mathbb{R}$ smooth map such that $0$ is a regular value, then $\{x \in M : f(x) \geq 0\}$ is a smooth manifold with boundary $f^{-1}(0)$.
So, I know $f^{-1}(0)$ is a embedded submanifold of codimension $1$. If $H:=\{x \in M : f(x) \geq 0\}$ is a manifold with boundary $f^{-1}(0)$, then $H$ must have dimension $2$.
With this I have my strategy, it is to find local harts to $\mathbb{R}^2$. With $f^{-1}(0)$ being an embedded submanifold of $M$, I know that for every $p$ in $f^{-1}(0)$, there exists $(U,\phi)$ local charts of $p$ such that $\phi(U \cap f^{-1}(0))= V \times \{y\} \simeq V \subset \mathbb{R}$.
My idea is to work around this function: $x \mapsto (\phi(x),f(x))$ (This is an abuse of notation since $\phi(x)$ lies in $\mathbb{R}^m$, but I see it as a real number by composing $\phi$ with an homeomorphism $V \times \{y\} \to V$).
This is well defined in $U \cap f^{-1}(0)$ (The idea is that I can do this for any of those local charts $(U,\phi)$ and define these ''new local charts'' then try to extend them to $U\cap H$ by a map $x \mapsto (g(x),f(x))$, where $g$ is an extension of $\phi \restriction_{U\cap f^{-1}(0)}$ to $U \cap H$, so that $x\mapsto (g(x),f(x))$ is an homeomorphism. Doing this, I can assure local charts for any $p \in f^{-1}(0) \subset H$.
But I cannot seem to get anywhere with this, since I do not know how the local charts work in $H-f^{-1}(0)$.
Any help would be apreciated, thanks in advanced.
You have to show the following:
(1) Each $x \in B = f^{-1}(0)$ is contained in a boundary chart, i.e. there exists a homeomorphism $h : U \to W$ with $U \subset H$ an open neighborhood of $x$, $W \subset \mathbb{R}^n_+ = \mathbb{R}^{n-1} \times [0,\infty)$ open and $h(x) \in \mathbb{R}^{n-1} \times \{ 0 \}$.
(2) For any two boundary charts $h_i : U_i \to W_i$, the transitition $h_{12} : h_1(U_1 \cap U_2) \to h_2(U_1 \cap U_2)$ is a diffeomorphism between open subsets of subsets of $\mathbb{R}^n_+$ which means that it is the restriction of a diffeomorphism between open subsets of $\mathbb{R}^n$.
(1) Since $0$ is a regular value, each $x \in B$ has a chart $h' : U' \to W'$ in $M$ such that $h'(U' \cap B) = W' \cap \mathbb{R}^{n-1} \times \{ 0 \}$ and $f (h')^{-1}(x_1,...,x_n) = x_n$ for $(x_1,...,x_n) \in W'$. Then you see that $h : U = U' \cap H \stackrel{h'}{\rightarrow} W = W' \cap \mathbb{R}^n_+$ is a boundary chart.
(2) This is shown completely similar.