# Definition:Diffeomorphism

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## Open sets in $\R^n$

Let $n$ and $k$ be natural numbers.

Let $U,V\subset \R^n$ be open sets.

Let $f : U \to V$ be a mapping.

Then $f$ is a **$C^k$-diffeomorphism** if and only if $f$ is a bijection of class $C^k$ with an inverse of class $C^k$.

## Differentiable Manifolds

Let $m, n \ge 0$ and $k$ be natural numbers with $1 \le k \le \min \set {m, n}$.

Let $M$ and $N$ be differentiable manifolds of dimensions $m$ and $n$.

Let $f: M \to N$ be a mapping.

Then $f$ is a **$C^k$-diffeomorphism** if and only if $f$ is a bijection of class $C^k$ with an inverse of class $C^k$.

### Smooth Diffeomorphism

A **smooth diffeomorphism** is a bijection which is smooth and whose inverse is smooth.

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**diffeomorphism**