Let $d\in\mathbb N$, $k\in\{1,\ldots,d\}$, $\mathbb H^k:=\mathbb R^{k-1}\times[0,\infty)$ and $M$ be a $k$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$ with boundary$^1$.
Assume, for simplicity, that $M$ can be described by a single chart, i.e. there is a $C^1$-diffeomorphism $\phi$ form $M$ onto an open subset of $\mathbb H^k$.
Can we show that that $$M_1:=\{x\in M:\phi(x)\not\in\partial\mathbb H^k\}$$ and $$M_2:=\{x\in M:\phi(x)\in\partial\mathbb H^k\}$$ are the manifold interior and boundary of $M$, respectively, where $\partial\mathbb H^k=\mathbb R^{k-1}\times\{0\}$?
Let $U:=\phi(M)$. For the first part, note that $$V_1:=(\mathbb H^k)^\circ\cap U,$$ where $(\mathbb H^k)^\circ=\mathbb R^{k-1}\times(0,\infty)$, is $\mathbb R^k$-open and (since $V_1\subseteq U$) $U$-open. (Since $\phi$ is continuous, $$M_1=\phi^{-1}(V)$$ is $M$-open, but I don't think this is important here, is it?). So, $M_1$ should be a $k$-dimensional embedded submanifold of $\mathbb R^d$ and $\left.\phi\right|_{M_1}$ should be a $C^1$-diffeomorphism of $M_1$ onto the open subset $V_1$ of $\mathbb R^k$, i.e. it should be a (global) chart of $M_1$.
For the second part, note that $$V_2:=U\cap\partial\mathbb H^k$$ is $\partial\mathbb H^k$-open. Let $\iota$ denote the canonical embedding of $\mathbb R^{k-1}$ into $\mathbb R^k$ with $\iota\mathbb R^{k-1}=\partial H^k$ and $\pi$ denote the canonical projection of $\mathbb R^k$ onto $\mathbb R^{k-1}$ with $\pi(\partial H^k)=\mathbb R^{k-1}$. Since $\iota$ is $(\mathbb R^{k-1},\mathbb R^k)$-continuous and $\iota\mathbb R^{k-1}=\partial\mathbb H^k$, $\iota$ is $(\mathbb R^{k-1},\partial\mathbb H^k)$-continuous and hence $$W:=\pi V_2=\{u\in\mathbb R^{k-1}:(u,0)\in U\}=\iota^{-1}(V_2)\tag1$$ is $\mathbb R^{k-1}$-open. Moreover, $$M_2=\phi^{-1}(V_2)\tag2.$$ Let $$\psi:=\left.\pi\circ\phi\right|_{M_2}.$$ Thus, $M_2$ should be a $(k-1)$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$ and $\psi$ should be a $C^1$-diffeomorphism from $M_2$ onto the open subset $$W=\pi V_2=\psi(M_2)\tag3$$ of $\mathbb R^{k-1}$, i.e. a (global) chart of $M_2$.
$^1$ i.e. each point of $M$ is locally $C^1$-diffeomorphic to $\mathbb H^k$.
If $E_i$ is a $\mathbb R$-Banach space and $B_i\subseteq E_i$, then $f:B_1\to E_2$ is called $C^1$-differentiable if $f=\left.\tilde f\right|_{B_1}$ for some $E_1$-open neighborhood $\Omega_1$ of $B_1$ and some $\tilde f\in C^1(\Omega_1,E_2)$ and $g:B_1\to B_2$ is called $C^1$-diffeomorphism if $g$ is a homeomorphism from $B_1$ onto $B_2$ and $g$ and $g^{-1}$ are $C^1$-differentiable.