Definition:Isometry (Metric Spaces)

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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:

$\forall a, b \in A_1: d_1 \left({a, b}\right) = d_2 \left({\phi \left({a}\right), \phi \left({b}\right)}\right)$


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

  • Results about isometries can be found here.


Sources