# Isometry

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## 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.