difeomorphism

A diffeomorphism is a bijective and differentiable mapping of a manifold to itself. In other words, it is a morphism (in the sense of category theory) between the manifold and itself, where the morphism is a smooth (differentiable) map and its inverse is also smooth. Diffeomorphisms play a significant role in topology and differential geometry.

The local expression of a diffeomorphism is a diffeomorphic map in the sense of classical differential geometry on Euclidean spaces, specifically a bijective map defined on open sets of Euclidean spaces. Globally, on a manifold, a diffeomorphism is a smooth map that has a smooth inverse, which can be composed to produce other diffeomorphisms.

The fundamental theorem of smooth manifolds establishes that smooth maps between manifolds can be characterized in a unique way up to diffeomorphism, implying that the diffeomorphisms form a group under composition. This allows the problem of finding all diffeomorphisms between different manifolds to be reduced to finding all smooth maps, provided that the target and source spaces are the same.

In practical terms, diffeomorphisms are utilized in various mathematical disciplines, including differential geometry and topology. However, several technical challenges in utilizing diffeomorphisms exist, such as reconciliation of apparent equivalence under reparameterization with failure to meet directly invertible criteria.