Definition:Diffeomorphism

This article, or a section of it, needs explaining. In particular: No longer consistent. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

This page has been identified as a candidate for refactoring of medium complexity. In particular: Turn into exposition of similar concepts in different contexts Until this has been finished, please leave {{Refactor}} in the code. New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Refactor}} from the code.

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