Definition:Isometry (Metric Spaces)
This page is about Isometry in the context of Metric Space. 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.
Texts which approach the subject from the direction of applied mathematics and physics refer to such a mapping as a rigid motion.
Also see
- Equivalence of Definitions of Isometry of Metric Spaces
- Isometry is Homeomorphism of Induced Topologies
- Distance-Preserving Surjection is Isometry of Metric Spaces
- Results about isometries 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