Diffeomorphism - Diffeomorphisms of Subsets of Manifolds

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→ Y is said to be smooth if for all p in X there is a neighborhood of p and a smooth function g: U → N such that the restrictions agree (note that g is an extension of f). We say that f is a diffeomorphism if it is bijective, smooth, and if its inverse is smooth.