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In [[mathematics]], an '''isometry''' (or '''[[congruence (geometry)|congruence]]''', or '''congruent transformation''') is a [[distance]]-preserving transformation between [[metric spaces]], usually assumed to be [[Bijection|bijective]].<ref name=CoxeterIsometryDef>{{harvnb|Coxeter|1969|page=29}} <p>"We shall find it convenient to use the word ''transformation'' in the special sense of a one-to-one correspondence <math> P \to P'</math> among all points in the plane (or in space), that is, a rule for associating pairs of points, with the understanding that each pair has a first member {{mvar|P}} and a second member {{mvar|P'}} and that every point occurs as the first member of just one pair and also as the second member of just one pair...<p>In particular, an ''isometry'' (or "congruent transformation," or "congruence") is a transformation which preserves length..."</ref> 
[[File:Academ Reflections with parallel axis on wallpaper.svg|thumb|upright=1.4|A [[Function composition|composition]] of two [[Euclidean group#Direct and indirect isometries|opposite]] isometries is a direct isometry.  [[Reflection (mathematics)|A reflection]] in a line is an opposite isometry, like  {{math|''R''<sub> 1</sub>}}  or {{math|''R''<sub> 2</sub>}}  on the image. [[Translation (geometry)|Translation]] {{math|''T''}} is a direct isometry: [[Rigid body|a rigid motion]].<ref>{{harvnb|Coxeter|1969|p=46}} <p>'''3.51''' ''Any direct isometry is either a translation or a rotation. Any opposite isometry is either a reflection or a glide reflection.''</ref>]]

==Introduction==
Given a metric space (loosely, a set and a scheme for assigning distances between elements of the set), an isometry is a [[Transformation (geometry)|transformation]] which maps elements to the same or another metric space such that the distance between the image elements in the new metric space is equal to the distance between the elements in the original metric space. In a two-dimensional or three-dimensional [[Euclidean space]], two geometric figures are [[Congruence (geometry)|congruent]] if they are related by an isometry;<ref>{{harvnb|Coxeter|1969|page=39}}<p>'''3.11''' ''Any two congruent triangles are related by a unique isometry.''</ref> the isometry that relates them is either a rigid motion (translation or rotation), or a [[Function composition|composition]] of a rigid motion and a [[Reflection (mathematics)|reflection]].<!--commentary: i presume "they" here means the geometric figures. still commenting out because it doesn't seem to help.--> <!--They are equal, up to an action of a rigid motion, if related by a [[Euclidean group#Direct and indirect isometries|direct isometry]] (orientation preserving).-->

Isometries are often used in constructions where one space is [[Embedding|embedded]] in another space. For instance, the [[Complete space#Completion|completion]] of a metric space ''M'' involves an isometry from ''M'' into ''M''', a [[quotient set]] of the space of [[Cauchy sequence]]s on ''M''. The original space ''M'' is thus isometrically [[isomorphism|isomorphic]] to a subspace of a [[complete metric space]], and it is usually identified with this subspace. Other embedding constructions show that every metric space is isometrically isomorphic to a [[closed set|closed subset]] of some [[normed vector space]] and that every complete metric space is isometrically isomorphic to a closed subset of some [[Banach space]].

An isometric surjective linear operator on a [[Hilbert space]] is called a [[unitary operator]].

==Formal definitions==

Let ''X'' and ''Y'' be [[metric space]]s with metrics ''d''<sub>''X''</sub> and ''d''<sub>''Y''</sub>. A [[function (mathematics)|map]] ''f'' : ''X'' → ''Y'' is called an '''isometry''' or '''distance preserving''' if for any ''a'',''b'' ∈ ''X'' one has

:<math>d_Y\!\left(f(a),f(b)\right)=d_X(a,b).</math><ref>{{cite journal
 | last1 = Beckman | first1 = F. S.
 | last2 = Quarles | first2 = D. A., Jr.
 | journal = [[Proceedings of the American Mathematical Society]]
 | mr = 0058193
 | pages = 810–815
 | title = On isometries of Euclidean spaces
 | volume = 4
 | issue = 5
 | year = 1953
 | doi=10.2307/2032415
 | url=http://www.ams.org/journals/proc/1953-004-05/S0002-9939-1953-0058193-5/S0002-9939-1953-0058193-5.pdf
 | quote=<br>Let {{mvar|T}} be a transformation (possibly many-valued) of <math>E^n</math> (<math>2\leq n < \infty</math>) into itself.<br>Let <math>d(p,q)</math> be the distance between points {{mvar|p}} and {{mvar|q}} of <math>E^n</math>, and let {{mvar|Tp}}, {{mvar|Tq}} be any images of {{mvar|p}} and {{mvar|q}}, respectively.<br>If there is a length {{mvar|a}} > 0 such that <math>d(Tp,Tq)=a</math> whenever <math>d(p,q)=a</math>, then {{mvar|T}} is a Euclidean transformation of <math>E^n</math> onto itself.| jstor = 2032415
 }}</ref>

An isometry is automatically [[Injective function|injective]];<ref name=CoxeterIsometryDef/> otherwise two distinct points, ''a'' and ''b'', could be mapped to the same point, thereby contradicting the coincidence axiom of the metric ''d''. This proof is similar to the proof that an [[order embedding]] between [[partially ordered set]]s is injective. Clearly, every isometry between metric spaces is a topological embedding.

A '''global isometry''', '''isometric isomorphism''' or '''congruence mapping''' is a [[bijective]] isometry. Like any other bijection, a global isometry has a [[function inverse]].  The inverse of a global isometry is also a global isometry.

Two metric spaces ''X'' and ''Y'' are called '''isometric''' if there is a bijective isometry from ''X'' to ''Y''. The [[Set (mathematics)|set]] of bijective isometries from a metric space to itself forms a [[group (mathematics)|group]] with respect to [[function composition]], called the '''[[isometry group]]'''.

There is also the weaker notion of ''path isometry'' or ''arcwise isometry'':

A '''path isometry''' or '''arcwise isometry''' is a map which preserves the [[Arc length#Definition|lengths of curves]]; such a map is not necessarily an isometry in the distance preserving sense, and it need not necessarily be bijective, or even injective. This term is often abridged to simply ''isometry'', so one should take care to determine from context which type is intended.

==Examples==
* Any [[reflection (mathematics)|reflection]], [[translation (geometry)|translation]] and [[rotation]] is a global isometry on [[Euclidean space]]s. See also [[Euclidean group]] and {{slink|Euclidean space|Isometries}}.
*The map <math> x\mapsto |x|</math> in <math>{\mathbb R}</math> is a path isometry but not an isometry.  Note that unlike an isometry, it is not injective.
*The isometric [[linear map]]s from '''C'''<sup>''n''</sup> to itself are given by the [[unitary matrix|unitary matrices]].<ref>{{Cite journal | last1 = Roweis | first1 = S. T. | last2 = Saul | first2 = L. K. | title = Nonlinear Dimensionality Reduction by Locally Linear Embedding | doi = 10.1126/science.290.5500.2323 | journal = [[Science (journal)|Science]]| volume = 290 | issue = 5500 | pages = 2323–2326 | year = 2000 | pmid =  11125150| pmc = | citeseerx = 10.1.1.111.3313 }}</ref><ref>{{cite journal|last1=Saul |first1=Lawrence K. |last2=Roweis |first2=Sam T. | title= Think globally, fit locally: Unsupervised learning of nonlinear manifolds |journal=[[Journal of Machine Learning Research]]|volume=4|issue=June|pages=119–155|year=2003|quote=Quadratic optimisation of <math>\mathbf M =(I-W)^\top(I-W)</math> (page 135) such that <math>\mathbf M\equiv YY^\top</math> }}</ref><ref>{{Cite journal |last=Zhang |first=Zhenyue |last2=Zha|first2=Hongyuan  |title=Principal Manifolds and Nonlinear Dimension Reduction via Local Tangent Space Alignment |journal=SIAM Journal on Scientific Computing |volume=26 |issue=1 |year=2004 |pages=313–338 |doi=10.1137/s1064827502419154}}</ref><ref>{{cite journal|last1=Zhang|first1=Zhenyue|last2=Wang|first2=Jing|title=MLLE: Modified Locally Linear Embedding Using Multiple Weights|journal=[[Advances in Neural Information Processing Systems]]|year=2006|volume=19|url=https://papers.nips.cc/paper/3132-mlle-modified-locally-linear-embedding-using-multiple-weights|quote=It can retrieve the ideal embedding if MLLE is applied on data points sampled from an isometric manifold.}}</ref>

==Linear isometry==

Given two [[normed vector space]]s ''V'' and ''W'', a '''linear isometry''' is a [[linear map]] ''f'' : ''V'' → ''W'' that preserves the norms:
:<math>\|f(v)\| = \|v\|</math>
for all ''v'' in ''V''.<ref name="Thomsen 2017 p125">{{cite book |last=Thomsen |first=Jesper Funch |date=2017 |title=Lineær algebra |trans-title=Linear algebra |page=125 |language=Danish |location=Århus |publisher=Department of Mathematics, Aarhus University}}</ref> Linear isometries are distance-preserving maps in the above sense. They are global isometries if and only if they are [[surjective]].

By the [[Mazur–Ulam theorem]], any isometry of normed vector spaces over '''R''' is [[Affine transformation|affine]].

In an inner product space, the fact that any linear isometry is an orthogonal transformation can be shown by using [[Polarization identity|polarization identities]] to
prove ''<Ax, Ay> = <x, y>''<ref name="Thomsen 2017 p125"/> and then applying the [[Riesz representation theorem]].

==Manifolds==
An isometry of a [[manifold]] is any (smooth) mapping of that manifold into itself, or into another manifold that preserves the notion of distance between points. The definition of an isometry requires the notion of a [[metric (mathematics)|metric]] on the manifold; a manifold with a (positive-definite) metric is a [[Riemannian manifold]], one with an indefinite metric is a [[pseudo-Riemannian manifold]]. Thus, isometries are studied in [[Riemannian geometry]].

A '''local isometry''' from one ([[Pseudo-Riemannian manifold|pseudo]]-)[[Riemannian manifold]] to another is a map which [[pullback (differential geometry)|pulls back]] the [[metric tensor]] on the second manifold to the metric tensor on the first. When such a map is also a [[diffeomorphism]], such a map is called an '''isometry''' (or '''isometric isomorphism'''), and provides a notion of [[isomorphism]] ("sameness") in the [[category theory|category]] '''Rm''' of Riemannian manifolds.

===Definition===

Let <math>R = (M, g)</math> and <math>R' = (M', g')</math> be two (pseudo-)Riemannian manifolds, and let <math>f : R \to R'</math> be a diffeomorphism. Then <math>f</math> is called an '''isometry''' (or '''isometric isomorphism''') if

:<math>g = f^{*} g', \, </math>

where <math>f^{*} g'</math> denotes the [[pullback (differential geometry)|pullback]] of the rank (0, 2) metric tensor <math>g'</math> by <math>f</math>. Equivalently, in terms of the [[pushforward (differential)|pushforward]] <math>f_{*}</math>, we have that for any two vector fields <math>v, w</math> on <math>M</math> (i.e. sections of the [[tangent bundle]] <math>\mathrm{T} M</math>),

:<math>g(v, w) = g' \left( f_{*} v, f_{*} w \right). \, </math>

If <math>f</math> is a [[local diffeomorphism]] such that <math>g = f^{*} g'</math>, then <math>f</math> is called a '''local isometry'''.

==Generalizations==
* Given a positive real number ε, an '''ε-isometry''' or '''almost isometry''' (also called a '''[[Felix Hausdorff|Hausdorff]] approximation''') is a map <math>f:X\to Y</math> between metric spaces such that
*# for ''x'',''x''&prime; ∈ ''X'' one has |''d''<sub>''Y''</sub>(ƒ(''x''),ƒ(''x''&prime;))&minus;''d''<sub>''X''</sub>(''x'',''x''&prime;)| < ε, and
*# for any point ''y'' ∈ ''Y'' there exists a point ''x'' ∈ ''X'' with ''d''<sub>''Y''</sub>(''y'',ƒ(''x'')) < ε

:That is, an ε-isometry preserves distances to within ε and leaves no element of the codomain further than ε away from the image of an element of the domain.  Note that ε-isometries are not assumed to be [[continuous function|continuous]].

*The '''[[restricted isometry property]]''' characterizes nearly isometric matrices for sparse vectors.
*'''[[Quasi-isometry]]''' is yet another useful generalization.
* One may also define an element in an abstract unital C*-algebra to be an isometry: 
*:<math> a\in\mathfrak{A}</math> is an isometry if and only if <math> a^* \cdot a = 1 </math>.
:Note that as mentioned in the introduction this is not necessarily a unitary element because one does not in general have that left inverse is a right inverse.

*On a [[pseudo-Euclidean space]], the term ''isometry'' means a linear bijection preserving magnitude. See also [[Quadratic form#Quadratic spaces|Quadratic spaces]].

==See also==
{{Div col|colwidth=20em}}
*[[Motion (geometry)]]
*[[Beckman&ndash;Quarles theorem]]
*[[Semidefinite embedding]]
*[[Flat (geometry)]]
*[[Euclidean plane isometry]]
*[[Orthogonal group#3D isometries that leave the origin fixed|3D isometries that leave the origin fixed]]
*[[Space group]]
*[[Involution (mathematics)|Involution]]
*[[Symmetry in mathematics]]
*[[Homeomorphism group]]
*[[Partial isometry]]
*[[Dual norm#The second dual of a Banach space|The second dual of a Banach space as an isometric isomorphism]]
*[[Isometry group]]
*[[Myers–Steenrod theorem]]
{{Div col end}}

==References==
{{reflist}}

==Bibliography==
* {{cite book|last=Coxeter|first=H. S. M.|author-link1=Harold Scott MacDonald Coxeter|title=Introduction to Geometry, Second edition|year=1969|publisher=[[John Wiley & Sons|Wiley]]|isbn=9780471504580|ref=harv}}
* {{cite book | author=Lee, Jeffrey M. | title=Manifolds and Differential Geometry |location=Providence, RI |publisher=American Mathematical Society | year=2009 |isbn=978-0-8218-4815-9 |url=https://books.google.com/books?id=QqHdHy9WsEoC }}

[[Category:Functions and mappings]]
[[Category:Metric geometry]]
[[Category:Symmetry]]
[[Category:Equivalence (mathematics)]]
[[Category:Riemannian geometry]]