I wonder that whether there exists a version of the inverse function theorem for smooth maps from a smooth manifolds with boundary to a smooth manifold without boundary? More precisely, whether the following assertion is true?
Let $M$ be a smooth manifold with boundary $\partial M$ and $N$ be a smooth manifold without boundary whose dimension $d$ is equal to the dimension of $M$. Let $f$ be a smooth map from $M$ to $N$. Assume that there exists a point $x\in \partial M$ such that the rank of $f_M$
at $x$ is $d$ and the rank of $f_{\partial M}$ on $x$ is $d-1$. Then there exists a open neignborhood $U$ of $x$ on $M$ such that $f_{U}$ is a diffeomorphism from $U$ onto $f(U)$.
Any references or comments are well appreciated. Thanks a lot!
Best Answer
Note that, the question being local you can work in local charts. Also, recall that, by definition of manifold with boundary, and by definition of smooth maps between manifolds with boundary, you can assume w.l.o.g. that $f$ is the restriction to $U:=V\cap H$ of a $C^1$ map $\tilde f$ defined on a nbd $V$ of $x:=0\in\mathbb{R}^d$, where $H$ is a closed half-space. So $\tilde f$ is locally invertible by the usual inverse mapping theorem on open sets of $\mathbb{R}^d$, and such is f by restriction. Note that you don't have to assume anything on the invertibility of $f_{|\partial M}$. The same argument works for Banach manifolds.