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

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

For example, the map $\mathbb R\to S^1$, $t\mapsto e^{it}$, is a local diffeomorphism even though it is not globally one-to-one. The inverse function theorem says that a smooth map between manifolds of equal dimension is a local diffeomorphism wherever its derivative is invertible.

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.