TITLE: Inverse Function Theorem for Manifolds with Boundary as the Domain
QUESTION [6 upvotes]: In Lee's Smooth Manifolds, it is written that the inverse function theorem can fail for manifolds with boundary. As hint, it is given the inclusion of half space into euclidean space $\iota:\mathbb{H}^n\hookrightarrow\mathbb{R}^n$ My guess is that this is merely a matter of being open rather than really deep problems, or?

REPLY [10 votes]: Yes, your guess is right -- it's just that you can't find (open) neighborhoods of the given points that are diffeomorphic.
The theorem you're looking at in my book (Thm. 4.5) says

Suppose $M$ and $N$ are smooth manifolds, and $F:M\to N$ is a smooth
  map. If $p\in M$ is a point such that $dF_p$ is invertible, then there
  are connected neighborhoods $U_0$ of $p$ and $V_0$ of $F(p)$ such that
  $F|_{U_0}: U_0\to V_0$ is a diffeomorphism.

What fails in an example like $\iota:\mathbb H^n\to \mathbb R^n$ is that you can't find any neighborhood of $0$ in $\mathbb H^n$ that's diffeomorphic to a neighborhood of $0$ in $\mathbb R^n$.
It boils down to what we mean by the word "locally." To say a map $F:M\to N$ is locally invertible usually means (at least) that for each $p\in M$, there exist a neighborhood $U$ of $p$ and a neighborhood $V$ of $f(p)$ such that $f|_U:U\to V$ is bijective. If you don't require $V$ to be a neighborhood, then every smooth immersion would be locally invertible. I guess you could define the term that way, but it's not the way most people understand it.