Isometry

Definition

Symbol-free definition

A function from a metric space (often, a Riemannian manifold) to another is termed an isometry if it preserves distance between points.

Definition with symbols

Let $(M,d)$ and $(M',d')$ be metric spaces. A function $f:M\to M'$ is termed an isometry if for all $x,y\in M$:

$d(x,y)=d'(f(x),f(y))$

We are often concerned with self-isometries of a metric space, viz isometries from a metric space to itself.