nLab
local diffeomorphism

Definition

A smooth function $f:X\to Y$ between two smooth manifolds is a local diffeomorphism if the following equivalent conditions hold

• $f$ is both a submersion and an immersion;

• for each point $x\in X$ the derivative $df:T_{x}X\to T_{f(x)}Y$ is an isomorphism of tangent vector spaces;

• the canonical diagram

  $$\begin{array}{ccc}TX& \stackrel{df}{\to }& TY\\ \downarrow & & \downarrow \\ X& \stackrel{f}{\to }& Y\end{array}$$\array{ T X &\stackrel{d f}{\to}& T Y \\ \downarrow && \downarrow \\ X &\stackrel{f}{\to}& Y }

  (with the differential between the tangent bundles) on top is a pullback;

• for each point $x\in X$ there exists an open subset $x\in U\subset X$ such that

  1. the image $f(U)$ is an open subset in $Y$;

  2. $f$ restricted to $U$ is a diffeomorphism onto its image

     $$f{\mid }_U:U\stackrel{\simeq }{\to }f(U)$$f|_U : U \stackrel{\simeq}{\to} f(U)

The equivalence of the conditions on tangent space with the conditions on open subsets follows by the inverse function theorem?.

Properties

General

• A local diffeomorphism is a local homeomorphism and so also an open map.

Abstract characterization

The category SmoothMfd of smooth manifolds may naturally be thought of as sitting inside the more general context of the cohesive (∞,1)-topos Smooth∞Grpd of smooth ∞-groupoids. This is canonically equipped with a notion of infinitesimal cohesion exhibited by its inclusion into SynthDiff∞Grpd. This implies that there is an intrinsic notion of formally étale morphisms of smooth $\mathrm{\infty }$-groupoids in general and of smooth manifolds in particular

Proposition

A smooth function is a formally étale morphism in this sense precisely if it is a local diffeomorphism.

See this section for more details.

Related concepts

• immersion, submersion