A local diffeomorphism can map a boundary point to an interior point I would like to find an example for a local diffeomorphism between smooth manifolds with boundary which maps some boundary point to an interior point.
I am quite sure such an example exists.
 A: To say that $f\colon M\to N$ is a local diffeomorphism means that each point of $M$ has an open neighborhood $U$ such that $f(U)$ is open in $N$ and $f|_U$ is a diffeomorphism from $U$ onto $f(U)$. 
Theorem. If $M$ and $N$ are smooth manifolds with boundary and $f\colon M\to N$ is a local diffeomorphism, then $f(\partial M)\subseteq \partial N$.
Proof. Assume for contradiction that there is a point $p\in \partial M$ such that $f(p)\in \operatorname{Int} M$. Let $U$ and $V$ be open neighborhoods of $p$ and $f(p)$ respectively, such that $f|_U\colon U\to V$ is a diffeomorphism.
Because $p\in\partial M$, there is a vector $v\in T_pM$ that is not tangent to $\partial M$. This means, in particular, that there is no smooth curve $\gamma\colon (-\varepsilon,\varepsilon)\to M$ such that $\gamma(0)=p$ and $\gamma'(0)=v$. Let $w = df_p(v) \in T_{f(p)}N$. Because $f(p)$ is an interior point of $N$, there is a smooth curve $\gamma\colon (-\varepsilon,\varepsilon)\to V$ such that $\gamma(0) = f(p)$ and $\gamma'(0) = w$. Then $\tilde\gamma = f^{-1}\circ \gamma$ is a smooth curve in $M$ such that $\tilde\gamma(0)=p$ and $\tilde\gamma'(0) = v$, which is a contradiction. $\square$
When $M$ and $N$ both have empty boundaries (and the same dimension), it's easy to show using the inverse function theorem that a smooth map $f\colon M\to N$ is a local diffeomorphism if and only if it is an immersion (a map whose differential is injective at each point). But in the case of nonempty boundaries, this isn't true. There are plenty of examples (such as those described by Andrew Hwang) of smooth immersions that take boundary points to interior points, but they're not local diffeomorphisms.
A: Let $M$ a differentiable manifold with boundary. Let $S$ a submanidold with boundary such that $\partial S\neq \partial M$, and let $i:S\rightarrow M$ the immersion of $S$ in $M$. Then some boundary points go to internal points of $M$.
