# Definition:Isometry (Metric Spaces)

*This page is about isometry in the context of metric spaces. For other uses, see isometry.*

## Definition

### Definition 1

Let $M_1 = \tuple {A_1, d_1}$ and $M_2 = \tuple {A_2, d_2}$ be metric spaces or pseudometric spaces.

Let $\phi: A_1 \to A_2$ be a bijection such that:

- $\forall a, b \in A_1: \map {d_1} {a, b} = \map {d_2} {\map \phi a, \map \phi b}$

Then $\phi$ is called an **isometry**.

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

### Definition 2

Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces or pseudometric spaces.

- $M_1$ and $M_2$ are
**isometric**if and only if there exist inverse mappings $\phi: A_1 \to A_2$ and $\phi^{-1}: A_2 \to A_1$ such that:

- $\forall a, b \in A_1: \map {d_1} {a, b} = \map {d_2} {\map \phi a, \map \phi b}$

- and:
- $\forall u, v \in A_2: \map {d_2} {u, v} = \map {d_1} {\map {\phi^{-1} } u, \map {\phi^{-1} } v}$

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

### Isometry Into

Let $\phi: A_1 \to A_2$ be an injection such that:

- $\forall a, b \in A_1: \map {d_1} {a, b} = \map {d_2} {\map \phi a, \map \phi b}$

Then $\phi$ is called an **isometry (from $M_1$) 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**.

## 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

- Equivalence of Definitions of Isometry of Metric Spaces
- Isometry is Homeomorphism of Induced Topologies
- Distance-Preserving Surjection is Isometry of Metric Spaces

- Definition:Isometry (Euclidean Geometry): its application to conventional Euclidean geometry

- Results about
**isometries**in the context of**metric spaces**can be found**here**.

## Sources

- 1963: Louis Auslander and Robert E. MacKenzie:
*Introduction to Differentiable Manifolds*... (previous) ... (next): Euclidean, Affine, and Differentiable Structure on $R^n$: $\text {1-1}$: Euclidean $n$-Space, Linear $n$-Space, and Affine $n$-Space - 2003: John H. Conway and Derek A. Smith:
*On Quaternions And Octonions*... (previous) ... (next): $\S 1$: The Complex Numbers: Introduction: $1.1$: The Algebra $\R$ of Real Numbers