# Diffeomorphism

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

## Contents

### Definition

Given two manifolds M and N, a bijective map f from M to N is called a diffeomorphism if both

and its inverse

are differentiable (if these functions are r times continuously differentiable, f is called a Cr-diffeomorphism).

Two manifolds M and N are diffeomorphic (symbol usually being $\simeq$) if there is a smooth bijective map f from M to N with a smooth inverse. They are Cr diffeomorphic if there is an r times continuously differentiable bijective map between them whose inverse is also r times continuously differentiable.

### Diffeomorphisms of subsets of manifolds

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