Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism 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.

Given two manifolds M and N, a differentiable map f : M → N is called a diffeomorphism if it is a bijection and its inverse f⁻¹ : N → M is differentiable as well. If these functions are r times continuously differentiable, f is called a C^r-diffeomorphism.

Two manifolds M and N are diffeomorphic (symbol usually being ≃) if there is a diffeomorphism f from M to N. They are C^r diffeomorphic if there is an r times continuously differentiable bijective map between them whose inverse is also r times continuously differentiable.

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

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