Isometry

Definition

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

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.