Local diffeomorphism

A local diffeomorphism is a smooth map $f:M\to N$ between smooth manifolds such that every $x\in M$ has an open neighborhood $U$ for which $f(U)$ is open in $N$ and

f|_U:U\longrightarrow f(U)

is a diffeomorphism. Equivalently, $f$ has a smooth local inverse around every point of its source.

Examples

The map $\mathbb R\to S^1$ , $t\mapsto e^{it}$ , is a local diffeomorphism even though it is not globally one-to-one.

Characterization

The inverse function theorem says that a smooth map between manifolds of equal dimension is a local diffeomorphism exactly when its derivative is invertible at every point.

Analogy with algebraic geometry

An étale morphism is often called the algebraic-geometric analogue of a local diffeomorphism: both notions say that the map has no infinitesimal branching and looks locally like an isomorphism. The analogy is structural, not an identification of schemes with smooth manifolds.