Definition:Isometry (Metric Spaces)

Definition
Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces or pseudometric spaces.

Let $\phi: A_1 \to A_2$ be a bijection such that:
 * $\forall a, b \in A_1: d_1 \left({a, b}\right) = d_2 \left({\phi \left({a}\right), \phi \left({b}\right)}\right)$

Then $\phi$ is called an isometry.

That is, an isometry is a distance-preserving bijection.

Such metric spaces $M_1$ and $M_2$ are defined as being isometric.

Isometry Into
When $\phi: M_1 \to M_2$ is not actually a surjection, but satisfies the other conditions for being an isometry, then $\phi$ can be called an isometry into $M_2$.

Also defined as
Some sources do not insist that an isometry be surjective.

Make sure to know which prerequisites are used when quoting results about isometries.

Some sources do not depend on the inherent properties of bijections, but instead specify an isometry in the twofold definition:
 * $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ are isometric if there exist inverse mappings $f: A_1 \to A_2$ and $g: A_2 \to A_1$ such that:


 * $\forall x, y \in A_1: d_2 \left({f \left({x}\right), f \left({y}\right)}\right) = d_1 \left({x, y}\right)$
 * and:
 * $\forall u, v \in A_2: d_1 \left({g \left({u}\right), g \left({v}\right)}\right) = d_2 \left({u, v}\right)$

That this approach is equivalent to the main definition is demonstrated in Inverse of Isometry of Metric Spaces is Isometry.

Also known as
An isometry is also known as a metric equivalence.

Two isometric spaces can also be referred to as metrically equivalent.

Also see

 * Isometry is Homeomorphism of Induced Topologies
 * Distance-Preserving Surjection is Isometry of Metric Spaces
 * Inverse of Isometry of Metric Spaces is Isometry