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Let $M$ and $N$ be surfaces, and $f: M\rightarrow N$ a differentiable and bijective map

(a) Can you ensure that f is a diffeomorphism?

(b) Suppose further that the map $T_{p}f:T_{M}\rightarrow T_{f(p)}N$ is an isomorphism for each $p\in M$, can you ensure that a diffeomorphism?

in (a) i think that is false, but I try to make a example, but the differentiability of the inverse is always hard

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(a) is false for curves, just take $M$ and $N$ to be the real line. The product of this map with the identity on the real line gives you an example for surfaces. For (b) look at the inverse function theorem. – Ryan Budney Nov 27 '12 at 16:33
up vote 3 down vote accepted

For (a) let $M = N = \mathbb{R}^2$ with standard coordinates $(x,y)$ for $M$ and $(u,v)$ for $N$. Let $f$ be: $$ f(x,y) = (x^3,y) $$

Clearly $f$ is bijective and differentiable. But the inverse map $$ g(u,v) = (\sqrt[3]{u},v) $$ is not differentiable when $u = 0$.

For (b) by the inverse function theorem $f$ is a local diffeomorphism. Then since it is bijective it is a diffeomorphism.

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