I am having some trouble with the following problem (Exercise 2.45 of the book Curves and Surfaces, by Montiel and Ros):
Let $\phi: O_1 \subset \mathbb{R}^3 \to O_2 \subset \mathbb{R}^3$ be a diffeomorphism and $S \subset O_1$ be a surface. Then $\phi:S \to \phi(S)$ is a diffeomorphism, where $O_1$ and $O_2$ are Euclidean open sets.
We know that $\phi(S)$ is another surface, so differentiability of $\phi:S \to \phi(S)$ should mean in the following sense:
Definition: Let $S$ be a surface and let $O \subset \mathbb{R}^n$ be an open set.
A: A function $f:S \to \mathbb{R}^m$ is differentiable if for any parametrization $X:U \to S$ the composition is differentiable.
C: If $S_1$ is another surface, then $f:S \to S_1$ is differentiable if the map $i \circ f:S \to \mathbb{R}^3$ is differentiable in the sense of A.
I suppose we should work with the parametrizations, but I am stuck on how to write a proper proof.
Any hints or comments will be the most appreciated. Thanks in advance.
The formal details involve taking the usual sense of "diffeomorphism" between open sets and fitting it to the more gentle definition to manifolds. In order to prove $\phi|_S$ is a diffeomorphism, we need to show the following
1) $\phi|_S$ is differentiable: Let $\psi:U\rightarrow S$ be a paremeterzation of $S$. The composition $\phi \circ \psi: U\rightarrow O_2$ is differentiable in the "usual sense" of open sets due to the chain rule, as both $\phi,\psi$ are differentiable on open sets. This now follows from definitions A,B.
2) $\phi|_S$ is invertible with differentiable inverse: We know $\phi |_{O_1}$ is bijective, which must mean $\phi|_S$ is also bijective as a restriction. We know there exists $\phi^{-1}:\phi(O_1) \rightarrow O_1$ differentiable (between open sets). Let $\varphi$ be a parameterization of $\phi(S)$. Just like in (1), $\phi^{-1}\circ \varphi$ is differentiable and therefore from defition A, $\phi^{-1}$ is differentiable in the "surface sense"
I don't know how diffeomorphisms are defined in this book, so you might have to replace every "differentiable" in my answer with $C^k$.
Another note: I should point out $\phi(O_1)$ is open because $\text{rank}D\phi = 3$ which by the open mapping theorem, means $\phi$ is an open mapping. (This can be also proved easily by the inverse function theorem though. Let me know if you need me to elaborate and fill in the details!).