Diffeomorphism

Definition

Given two differentiable manifolds $M$ and $N$, a differentiable map $f:M\to N$ is a diffeomorphism if it is a Bijective and its inverse $f^{-1}$ is differentiable as well.

If these functions are $r$ times continuously differentiable, $f$ is called $C^r$-diffeomorphism

If there is a diffeomorphism $f:M\to N$ between two manifolds $M$ and $N$, we call these two manifolds are diffeomorphic and denote as $M\simeq N$