Diffeomorphism - Knowlpedia

Let $M$ and $N$ be smooth manifolds. A map $f:M\to N$ is a diffeomorphism if

$f$ is bijective,
$f$ is a smooth map, and
the inverse function $f^{-1}:N\to M$ is also smooth.

If $f$ is a diffeomorphism, then for each $p\in M$ its differential $\mathrm{d}f_p$ is a linear isomorphism $T_pM\to T_{f(p)}N$ , where $T_pM$ denotes the tangent space.

Equivalent characterizations

Equivalently, $f$ is a smooth bijection whose inverse is smooth. Any diffeomorphism is, in particular, a homeomorphism of the underlying topological spaces, and it identifies the smooth structures on $M$ and $N$ .

Examples

On $\mathbb{R}^n$ , any translation $x\mapsto x+a$ and any invertible linear map $x\mapsto Ax$ (with $A\in \mathrm{GL}(n,\mathbb{R})$ ) is a diffeomorphism.
The logarithm map $\log:(0,\infty)\to\mathbb{R}$ is a diffeomorphism of $1$ -manifolds with inverse $\exp:\mathbb{R}\to(0,\infty)$ .
Stereographic projection gives a diffeomorphism between $S^n\setminus\{N\}$ and $\mathbb{R}^n$ , showing that punctured spheres are smoothly equivalent to Euclidean space.