Does the inverse mapping theorem in the setting of subsets of $\mathbb{R^n}$ imply the inverse mapping theorem in the setting of manifolds? Does the inverse mapping theorem in the setting of subsets of $\mathbb{R^n}$ imply the inverse mapping theorem in the setting of manifolds?
Or does one have to plunge back into the original proof and make suitable adjustments there? I'm not seeing an obvious implication.
Any comments will be much appreciated. Thanks!
 A: Yes, assuming all the conditions are satisfied, the inverse mapping theorem holds for manifolds. 
The point is that both the assumptions and conclusion of the inverse mapping theorem are about some neighborhood of a given point, and any point in a manifold has a neighborhood which looks like an open subset of $\mathbb{R}^n$. Thus, upon restricting to this open subset, we can just appeal to the inverse mapping theorem for $\mathbb{R}^n$.
A shorthand way of saying this is that the inverse mapping theorem is a "local" statement.
Edit: Here is an outline of the proof.
Suppose $f:X\rightarrow Y$ is a smooth map between $n$-manifolds, $f(x)=y$, and the Jacobian of $f$ is invertible at $x$ in some choice (hence any choice) of local coordinates. 
There exists an open subset $V\ni y$ such that $V$ looks like an open subset of $\mathbb{R}^n$. Similarly, there exists an open subset $U\subset f^{-1}(V)$ such that $U\ni x$ and $U$ looks like an open subset of $\mathbb{R}^n$. Then we can apply the usual inverse mapping theorem to $f:U\rightarrow V$. 
