Definition:Isometry (Metric Spaces)/Definition 1

Definition

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.

Also known as

An isometry is also known as a metric equivalence.

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

Examples

Euclidean Plane is Isometric to Complex Plane

Let $\R^2$ be the real number plane with the Euclidean metric.

Let $\C$ denote the complex plane.

Let $f: \R^2 \to \C$ be the function defined as:

$\forall \tuple {x_1, x_2} \in \R^2: \map f {x_1, x_2} = x_1 + i x_2$

Then $f$ is an isometry from $\R^2$ to $\C$.

Also see

• Equivalence of Definitions of Isometry of Metric Spaces

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

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.4$: Equivalent metrics: Definition $2.4.9$
• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces: Complete Metric Spaces
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 34$. Examples of groups: $(5)$

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• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) $\S I.5$