Local Diffeomorphism

A local diffeomorphism is a smooth map between manifolds that, for every point in its domain, restricts to a diffeomorphism from some open neighborhood of that point onto an open subset of the target manifold. The inverse function theorem gives a working criterion: a smooth map is a local diffeomorphism exactly where its derivative is a linear isomorphism between the corresponding tangent spaces, which in particular forces the two manifolds to have the same dimension. Local diffeomorphisms are therefore exactly the smooth maps whose derivative is everywhere invertible.