Diffeomorphism

A diffeomorphism is a bijective map between smooth manifolds that is differentiable with a differentiable inverse. More precisely, if M and N are smooth manifolds, a function f: M → N is a diffeomorphism if it is smooth, bijective, and its inverse f^{-1}: N → M is also smooth. When M = N, f is called a self-diffeomorphism of M. The term is used for maps that are as regular as the chosen differentiability class, such as C^k-diffeomorphisms or smooth (C^∞) diffeomorphisms.

On open subsets of Euclidean space, a map f: U → V is a diffeomorphism if it is bijective and both f and f^{-1} are continuously differentiable (or smoother). Equivalently, the Jacobian of f is invertible at every point, and by the inverse function theorem this condition ensures the existence of a smooth inverse. This makes diffeomorphisms the differentiable analogue of homeomorphisms.

Examples include translations and rotations of Euclidean space, as well as any smooth bijection with a smooth inverse. In the general setting, diffeomorphisms serve as the smooth symmetries of a manifold and identify spaces that are smoothly the same.

The collection of all diffeomorphisms of a manifold M forms a group under composition, called the diffeomorphism group Diff(M). Diffeomorphisms preserve the smooth structure and, on oriented manifolds, can preserve or reverse orientation. They play a central role in classifying manifolds up to smooth equivalence and in the study of geometric and physical symmetries.