Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable.

Given two differentiable manifolds ${\displaystyle M}$ and ${\displaystyle N}$, a continuously differentiable map ${\displaystyle f\colon M\rightarrow N}$ is a diffeomorphism if it is a bijection and its inverse ${\displaystyle f^{-1}\colon N\rightarrow M}$ is differentiable as well. If these functions are ${\displaystyle r}$ times continuously differentiable, ${\displaystyle f}$ is called a ${\displaystyle C^{r}}$-diffeomorphism.

Two manifolds ${\displaystyle M}$ and ${\displaystyle N}$ are diffeomorphic (usually denoted ${\displaystyle M\simeq N}$) if there is a diffeomorphism ${\displaystyle f}$ from ${\displaystyle M}$ to ${\displaystyle N}$. Two ${\displaystyle C^{r}}$-differentiable manifolds are ${\displaystyle C^{r}}$-diffeomorphic if there is an ${\displaystyle r}$ times continuously differentiable bijective map between them whose inverse is also ${\displaystyle r}$ times continuously differentiable. A ${\displaystyle C^{1}}$-diffeomorphism is simply a diffeomorphism, and a ${\displaystyle C^{0}}$-diffeomorphism is a homeomorphism.

Given a subset ${\displaystyle X}$ of a manifold ${\displaystyle M}$ and a subset ${\displaystyle Y}$ of a manifold ${\displaystyle N}$, a function ${\displaystyle f:X\to Y}$ is said to be smooth if for all ${\displaystyle p}$ in ${\displaystyle X}$ there is a neighborhood ${\displaystyle U\subset M}$ of ${\displaystyle p}$ and a smooth function ${\displaystyle g:U\to N}$ such that the restrictions agree: ${\displaystyle g_{|U\cap X}=f_{|U\cap X}}$ (note that ${\displaystyle g}$ is an extension of ${\displaystyle f}$). The function ${\displaystyle f}$ is said to be a diffeomorphism if it is bijective, smooth and its inverse is smooth.

Testing whether a differentiable map is a diffeomorphism can be made locally under some mild restrictions. This is the Hadamard-Caccioppoli theorem:

If ${\displaystyle U}$, ${\displaystyle V}$ are connected open subsets of ${\displaystyle \mathbb {R} ^{n}}$ such that ${\displaystyle V}$ is simply connected, a differentiable map ${\displaystyle f:U\to V}$ is a diffeomorphism if it is proper and if the differential ${\displaystyle Df_{x}:\mathbb {R} ^{n}\to \mathbb {R} ^{n}}$ is bijective (and hence a linear isomorphism) at each point ${\displaystyle x}$ in ${\displaystyle U}$.

Some remarks:

It is essential for ${\displaystyle V}$ to be simply connected for the function ${\displaystyle f}$ to be globally invertible (under the sole condition that its derivative be a bijective map at each point). For example, consider the "realification" of the complex square function