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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.