Diffeomorphism

A diffeomorphism between open sets $U,V \subseteq \mathbb{R}^n$ is a map $f:U\to V$ such that

• $f$ is bijective ,
• $f$ is continuously differentiable on $U$ (i.e., $f\in C^1(U,V)$), and
• the inverse map $f^{-1}:V\to U$ is also continuously differentiable (i.e., $f^{-1}\in C^1(V,U)$).

Diffeomorphisms are the “smooth isomorphisms” of Euclidean spaces: they preserve the differentiable structure and are the natural maps appearing in the inverse function theorem and change-of-variables formula .

Examples:

• The translation $f(x)=x+a$ on $\mathbb{R}^n$ is a diffeomorphism with $f^{-1}(y)=y-a$.
• Any invertible linear map $f(x)=Ax$ with $A\in GL(n,\mathbb{R})$ is a diffeomorphism with $f^{-1}(y)=A^{-1}y$.
• The map $f(x)=x^3$ is a $C^1$ bijection $\mathbb{R}\to\mathbb{R}$, but it is not a diffeomorphism since $(f^{-1})(y)=y^{1/3}$ is not $C^1$ at $0$.