nLab
isometry

An isometry is a function that preserves a metric, either in the sense of a metric space or in the sense of a Riemannian manifold.

Metric spaces

An isometry $f:(X,d)\to (X\prime ,d\prime )$ between metric spaces is a function $f:X\to X\prime$ between the underyling sets that respects the metrics in that $d=f^{*}d\prime$. More explicitly, $d\prime (f(a),f(b))=d(a,b)$ for any points $a,b$ in $X$.

The same idea holds for extended quasi-pseudo-generalisations of metric spaces.

Manifolds

An isometry $f:(X,g)\to (X\prime ,g\prime )$ between Riemannian manifolds is a morphism $f:X\to X\prime$ between the underlying manifolds that respects the metrics in that $g=f^{*}g\prime$. More explicitly, $g\prime (f_{*}v,f_{*}w)=g(v,w)$ for any tangent vectors $v,w$ on $X$.

Global isometries

Global isometries are the isomorphisms of metric spaces or Riemannian manifolds. An isometry is global if it is a bijection whose inverse is also an isometry. Bewteen metric spaces, isometries are necessarily injections and bijective isometries necessarily have isometries as inverses, so global isometries between metric spaces are also called surjective isometries; this does not work for Riemannian manifolds (where the inverse of an isometry need not be a morphism of manifolds), nor does it work for pseudometric spaces (where an isometry need not be injective).