Editing Diffeomorphism (section)

== Definition ==

Given two differentiable manifolds <math>M</math> and <math>N</math>, a [[Differentiable manifold#Differentiability of mappings between manifolds|continuously differentiable map]] <math>f \colon M \rightarrow N </math> is a '''diffeomorphism''' if it is a [[bijection]] and its inverse <math>f^{-1} \colon N \rightarrow M</math> is differentiable as well. If these functions are <math>r</math> times continuously differentiable, <math>f</math>  is called a <math>C^r</math>-diffeomorphism.

Two manifolds <math>M</math> and <math>N</math>  are '''diffeomorphic''' (usually denoted <math>M \simeq N</math>) if there is a diffeomorphism <math>f</math>  from <math>M</math> to <math>N</math>. Two <math>C^r</math>-differentiable manifolds are <math>C^r</math>-diffeomorphic if there is an <math> r </math> times continuously differentiable bijective map between them whose inverse is also <math>r</math> times continuously differentiable.