# 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 and be metric spaces. A function is termed an isometry if for all :

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