Isometries between metric spaces

Content created by malarbol, Fredrik Bakke and Louis Wasserman.

Created on 2024-09-28.
Last modified on 2025-05-05.

module metric-spaces.isometries-metric-spaces where
Imports 
open import elementary-number-theory.positive-rational-numbers

open import foundation.binary-transport
open import foundation.dependent-pair-types
open import foundation.embeddings
open import foundation.equivalences
open import foundation.function-extensionality
open import foundation.function-types
open import foundation.homotopies
open import foundation.identity-types
open import foundation.logical-equivalences
open import foundation.propositional-truncations
open import foundation.propositions
open import foundation.sequences
open import foundation.sets
open import foundation.subtypes
open import foundation.univalence
open import foundation.universe-levels

open import metric-spaces.functions-metric-spaces
open import metric-spaces.isometries-premetric-spaces
open import metric-spaces.metric-spaces
open import metric-spaces.uniformly-continuous-functions-metric-spaces

Idea

A function between metric spaces is an isometry if it is an isometry between their carrier premetric spaces.

Definitions

The property of being a isometry between metric spaces

module _
  {l1 l2 l1' l2' : Level}
  (A : Metric-Space l1 l2) (B : Metric-Space l1' l2')
  (f : map-type-Metric-Space A B)
  where

  is-isometry-prop-Metric-Space : Prop (l1 ⊔ l2 ⊔ l2')
  is-isometry-prop-Metric-Space =
    is-isometry-prop-Premetric-Space
      ( premetric-Metric-Space A)
      ( premetric-Metric-Space B)
      ( f)

  is-isometry-Metric-Space : UU (l1 ⊔ l2 ⊔ l2')
  is-isometry-Metric-Space =
    type-Prop is-isometry-prop-Metric-Space

  is-prop-is-isometry-Metric-Space :
    is-prop is-isometry-Metric-Space
  is-prop-is-isometry-Metric-Space =
    is-prop-type-Prop is-isometry-prop-Metric-Space

The set of isometries between metric spaces

module _
  {l1 l2 l1' l2' : Level}
  (A : Metric-Space l1 l2) (B : Metric-Space l1' l2')
  where

  set-isometry-Metric-Space : Set (l1 ⊔ l2 ⊔ l1' ⊔ l2')
  set-isometry-Metric-Space =
    set-subset
      ( set-map-type-Metric-Space A B)
      ( is-isometry-prop-Metric-Space A B)

  isometry-Metric-Space : UU (l1 ⊔ l2 ⊔ l1' ⊔ l2')
  isometry-Metric-Space = type-Set set-isometry-Metric-Space

  is-set-isometry-Metric-Space : is-set isometry-Metric-Space
  is-set-isometry-Metric-Space =
    is-set-type-Set set-isometry-Metric-Space

module _
  {l1 l2 l1' l2' : Level}
  (A : Metric-Space l1 l2) (B : Metric-Space l1' l2')
  (f : isometry-Metric-Space A B)
  where

  map-isometry-Metric-Space : map-type-Metric-Space A B
  map-isometry-Metric-Space =
    map-isometry-Premetric-Space
      ( premetric-Metric-Space A)
      ( premetric-Metric-Space B)
      ( f)

  is-isometry-map-isometry-Metric-Space :
    is-isometry-Metric-Space A B map-isometry-Metric-Space
  is-isometry-map-isometry-Metric-Space =
    is-isometry-map-isometry-Premetric-Space
      ( premetric-Metric-Space A)
      ( premetric-Metric-Space B)
      ( f)

Properties

The identity function on a metric space is an isometry

module _
  {l1 l2 : Level} (A : Metric-Space l1 l2)
  where

  is-isometry-id-Metric-Space :
    is-isometry-Metric-Space A A (id-Metric-Space A)
  is-isometry-id-Metric-Space d x y = id-iff

  isometry-id-Metric-Space : isometry-Metric-Space A A
  isometry-id-Metric-Space =
    id-Metric-Space A , is-isometry-id-Metric-Space

Equality of isometries in metric spaces is equivalent to homotopies between their carrier maps

module _
  {l1 l2 l1' l2' : Level}
  (A : Metric-Space l1 l2) (B : Metric-Space l1' l2')
  (f g : isometry-Metric-Space A B)
  where

  equiv-eq-htpy-map-isometry-Metric-Space :
    (f ＝ g) ≃
    (map-isometry-Metric-Space A B f ~ map-isometry-Metric-Space A B g)
  equiv-eq-htpy-map-isometry-Metric-Space =
    equiv-eq-htpy-map-isometry-Premetric-Space
      ( premetric-Metric-Space A)
      ( premetric-Metric-Space B)
      ( f)
      ( g)

  htpy-eq-map-isometry-Metric-Space :
    (f ＝ g) →
    (map-isometry-Metric-Space A B f ~ map-isometry-Metric-Space A B g)
  htpy-eq-map-isometry-Metric-Space =
    map-equiv equiv-eq-htpy-map-isometry-Metric-Space

  eq-htpy-map-isometry-Metric-Space :
    (map-isometry-Metric-Space A B f ~ map-isometry-Metric-Space A B g) →
    (f ＝ g)
  eq-htpy-map-isometry-Metric-Space =
    map-inv-equiv equiv-eq-htpy-map-isometry-Metric-Space

Composition of isometries

module _
  {l1a l2a l1b l2b l1c l2c : Level}
  (A : Metric-Space l1a l2a)
  (B : Metric-Space l1b l2b)
  (C : Metric-Space l1c l2c)
  (g : isometry-Metric-Space B C)
  (f : isometry-Metric-Space A B)
  where

  comp-isometry-Metric-Space :
    isometry-Metric-Space A C
  comp-isometry-Metric-Space =
    ( map-isometry-Metric-Space B C g ∘
      map-isometry-Metric-Space A B f) ,
    ( preserves-isometry-comp-function-Premetric-Space
      ( premetric-Metric-Space A)
      ( premetric-Metric-Space B)
      ( premetric-Metric-Space C)
      ( map-isometry-Metric-Space B C g)
      ( map-isometry-Metric-Space A B f)
      ( is-isometry-map-isometry-Metric-Space B C g)
      ( is-isometry-map-isometry-Metric-Space A B f))

Unit laws for composition of isometries between metric spaces

module _
  {l1a l2a l1b l2b : Level}
  (A : Metric-Space l1a l2a)
  (B : Metric-Space l1b l2b)
  (f : isometry-Metric-Space A B)
  where

  left-unit-law-comp-isometry-Metric-Space :
    ( comp-isometry-Metric-Space A B B
      (isometry-id-Metric-Space B)
      ( f)) ＝
    ( f)
  left-unit-law-comp-isometry-Metric-Space =
    eq-htpy-map-isometry-Metric-Space
      ( A)
      ( B)
      ( comp-isometry-Metric-Space
        ( A)
        ( B)
        ( B)
        (isometry-id-Metric-Space B)
        ( f))
      ( f)
      ( λ x → refl)

  right-unit-law-comp-isometry-Metric-Space :
    ( comp-isometry-Metric-Space A A B
      ( f)
      ( isometry-id-Metric-Space A)) ＝
    ( f)
  right-unit-law-comp-isometry-Metric-Space =
    eq-htpy-map-isometry-Metric-Space
      ( A)
      ( B)
      ( f)
      ( comp-isometry-Metric-Space
        ( A)
        ( A)
        ( B)
        ( f)
        ( isometry-id-Metric-Space A))
      ( λ x → refl)

Associativity of composition of isometries between metric spaces

module _
  {l1a l2a l1b l2b l1c l2c l1d l2d : Level}
  (A : Metric-Space l1a l2a)
  (B : Metric-Space l1b l2b)
  (C : Metric-Space l1c l2c)
  (D : Metric-Space l1d l2d)
  (h : isometry-Metric-Space C D)
  (g : isometry-Metric-Space B C)
  (f : isometry-Metric-Space A B)
  where

  associative-comp-isometry-Metric-Space :
    ( comp-isometry-Metric-Space A B D
      ( comp-isometry-Metric-Space B C D h g)
      ( f)) ＝
    ( comp-isometry-Metric-Space A C D
      ( h)
      ( comp-isometry-Metric-Space A B C g f))
  associative-comp-isometry-Metric-Space =
    eq-htpy-map-isometry-Metric-Space
      ( A)
      ( D)
      ( comp-isometry-Metric-Space A B D
        ( comp-isometry-Metric-Space B C D h g)
        ( f))
      ( comp-isometry-Metric-Space A C D
        ( h)
        ( comp-isometry-Metric-Space A B C g f))
      ( λ x → refl)

The inverse of an isometric equivalence is an isometry

module _
  {l1 l2 l1' l2' : Level}
  (A : Metric-Space l1 l2) (B : Metric-Space l1' l2')
  (f : isometry-Metric-Space A B)
  (E : is-equiv (map-isometry-Metric-Space A B f))
  where

  isometry-inv-is-equiv-isometry-Metric-Space :
    isometry-Metric-Space B A
  isometry-inv-is-equiv-isometry-Metric-Space =
    ( map-inv-is-equiv E) ,
    ( is-isometry-map-inv-is-equiv-is-isometry-Premetric-Space
      ( premetric-Metric-Space A)
      ( premetric-Metric-Space B)
      ( map-isometry-Metric-Space A B f)
      ( E)
      ( is-isometry-map-isometry-Metric-Space A B f))

  is-section-isometry-inv-is-equiv-isometry-Metric-Space :
    ( comp-isometry-Metric-Space
      B
      A
      B
      f
      isometry-inv-is-equiv-isometry-Metric-Space) ＝
    ( isometry-id-Metric-Space B)
  is-section-isometry-inv-is-equiv-isometry-Metric-Space =
    eq-htpy-map-isometry-Metric-Space
      ( B)
      ( B)
      ( comp-isometry-Metric-Space
        B
        A
        B
        f
        isometry-inv-is-equiv-isometry-Metric-Space)
      ( isometry-id-Metric-Space B)
      ( is-section-map-inv-is-equiv E)

  is-retraction-isometry-inv-is-equiv-isometry-Metric-Space :
    ( comp-isometry-Metric-Space
      A
      B
      A
      isometry-inv-is-equiv-isometry-Metric-Space
      f) ＝
    ( isometry-id-Metric-Space A)
  is-retraction-isometry-inv-is-equiv-isometry-Metric-Space =
    eq-htpy-map-isometry-Metric-Space
      ( A)
      ( A)
      ( comp-isometry-Metric-Space
        A
        B
        A
        isometry-inv-is-equiv-isometry-Metric-Space
        f)
      ( isometry-id-Metric-Space A)
      ( is-retraction-map-inv-is-equiv E)

Any isometry between metric spaces is an embedding

module _
  {l1 l2 l1' l2' : Level}
  (A : Metric-Space l1 l2) (B : Metric-Space l1' l2')
  where

  is-emb-map-isometry-Metric-Space :
    (f : isometry-Metric-Space A B) → is-emb (map-isometry-Metric-Space A B f)
  is-emb-map-isometry-Metric-Space =
    is-emb-map-isometry-is-extensional-Premetric-Space
      ( premetric-Metric-Space A)
      ( premetric-Metric-Space B)
      ( is-extensional-structure-Metric-Space A)
      ( is-extensional-structure-Metric-Space B)

Any isometry between metric spaces is uniformly continuous

module _
  {l1 l2 l3 l4 : Level}
  (A : Metric-Space l1 l2) (B : Metric-Space l3 l4)
  where

  is-uniformly-continuous-map-isometry-Metric-Space :
    (f : isometry-Metric-Space A B) →
    is-uniformly-continuous-map-Metric-Space A B
      (map-isometry-Metric-Space A B f)
  is-uniformly-continuous-map-isometry-Metric-Space =
    is-uniformly-continuous-map-isometry-Premetric-Space
      ( premetric-Metric-Space A)
      ( premetric-Metric-Space B)

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