In

_{''X''} and ''d''_{''Y''}.
A

^{''n''} to itself is an isometry (for the

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, an isometry (or congruence
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In modu ...

, or congruent transformation) is a distance
Distance is a numerical measurement
'
Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or eve ...

-preserving transformation between metric spaces
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
Mathematics
* Metric (mathematics), an abstraction of the notion of ''distance'' in a metric space
* Metric tensor, in differential geometr ...

, usually assumed to be bijective
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. "We shall find it convenient to use the word ''transformation'' in the special sense of a one-to-one correspondence $P\; \backslash to\; P\text{'}$ 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 and a second member and that every point occurs as the first member of just one pair and also as the second member of just one pair...

In particular, an ''isometry'' (or "congruent transformation," or "congruence") is a transformation which preserves length..."

Introduction

Given a metric space (loosely, a set and a scheme for assigning distances between elements of the set), an isometry is atransformation
Transformation may refer to:
Science and mathematics
In biology and medicine
* Metamorphosis, the biological process of changing physical form after birth or hatching
* Malignant transformation, the process of cells becoming cancerous
* Transf ...

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
Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...

, two geometric figures are congruent
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In modu ...

if they are related by an isometry; the isometry that relates them is either a rigid motion (translation or rotation), or a composition
Composition or Compositions may refer to:
Arts
* Composition (dance), practice and teaching of choreography
* Composition (music), an original piece of music and its creation
*Composition (visual arts)
The term composition means "putting togethe ...

of a rigid motion and a reflectionReflection or reflexion may refer to:
Philosophy
* Self-reflection
Science
* Reflection (physics), a common wave phenomenon
** Specular reflection, reflection from a smooth surface
*** Mirror image, a reflection in a mirror or in water
** Signal r ...

.
Isometries are often used in constructions where one space is embedded in another space. For instance, the completion of a metric space ''M'' involves an isometry from ''M'' into ''M, a quotient set
Set, The Set, or SET may refer to:
Science, technology, and mathematics Mathematics
* Set (mathematics), a collection of distinct elements or members
* Category of sets, the category whose objects and morphisms are sets and total functions, respe ...

of the space of Cauchy sequence
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s on ''M''.
The original space ''M'' is thus isometrically isomorphic
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

to a subspace of a complete metric space
In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in or, alternatively, if every Cauchy sequence in converges in .
Intuitively, a space is complet ...

, and it is usually identified with this subspace.
Other embedding constructions show that every metric space is isometrically isomorphic to a closed subset
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...

of some normed vector space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

and that every complete metric space is isometrically isomorphic to a closed subset of some Banach space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

.
An isometric surjective linear operator on a Hilbert space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

is called a unitary operator
In functional analysis
Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional ...

.
Isometry definition

Let ''X'' and ''Y'' bemetric space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

s with metrics (e.g., distances) ''d''map
A map is a symbol
A symbol is a mark, sign, or that indicates, signifies, or is understood as representing an , , or . Symbols allow people to go beyond what is n or seen by creating linkages between otherwise very different s and s. A ...

''f'' : ''X'' → ''Y'' is called an isometry or distance preserving if for any ''a'',''b'' ∈ ''X'' one has
:$d\_Y\backslash !\backslash left(f(a),f(b)\backslash right)=d\_X(a,b).$
An isometry is automatically injective
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

; 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 embeddingIn order theory
Order theory is a branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, cha ...

between partially ordered set
upright=1.15, Fig.1 The Hasse diagram of the Power set, set of all subsets of a three-element set \, ordered by set inclusion, inclusion. Sets connected by an upward path, like \emptyset and \, are comparable, while e.g. \ and \ are not.
In mathem ...

s is injective. Clearly, every isometry between metric spaces is a topological embedding.
A global isometry, isometric isomorphism or congruence mapping is a bijective
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

isometry. Like any other bijection, a global isometry has a function inverse
In mathematics, an inverse function (or anti-function) is a function (mathematics), function that "reverses" another function: if the function applied to an input gives a result of , then applying its inverse function to gives the result , i ...

.
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 of bijective isometries from a metric space to itself forms a group
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

with respect to function composition
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

, called the isometry group In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...

.
There is also the weaker notion of ''path isometry'' or ''arcwise isometry'':
A path isometry or arcwise isometry is a map which preserves the ; 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 reflectionReflection or reflexion may refer to:
Philosophy
* Self-reflection
Science
* Reflection (physics), a common wave phenomenon
** Specular reflection, reflection from a smooth surface
*** Mirror image, a reflection in a mirror or in water
** Signal r ...

, translation
Translation is the communication of the meaning
Meaning most commonly refers to:
* Meaning (linguistics), meaning which is communicated through the use of language
* Meaning (philosophy), definition, elements, and types of meaning discusse ...

and rotation
A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotation occurs is called the ''rotation plane'', and the imaginary line extending from the center an ...

is a global isometry on Euclidean space
Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...

s. See also Euclidean group
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

and .
* The map $x\backslash mapsto\; ,\; x,$ in $$ is a path isometry but not an isometry. Note that unlike an isometry, it is not injective.
Isometries between normed spaces

The following theorem is due to Mazur and Ulam. :Definition: The midpoint of two elements and in a vector space is the vector .Linear isometry

Given twonormed vector space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

s $V$ and $W$, a linear isometry is a linear map
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

$A\; :\; V\; \backslash to\; W$ that preserves the norms:
:$\backslash ,\; Av\backslash ,\; =\; \backslash ,\; v\backslash ,$
for all $v\; \backslash in\; V$.
Linear isometries are distance-preserving maps in the above sense.
They are global isometries if and only if they are surjective
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

.
In an inner product space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

, the above definition reduces to
:$\backslash langle\; v,\; v\; \backslash rangle\; =\; \backslash langle\; Av,\; Av\; \backslash rangle$
for all $v\; \backslash in\; V$, which is equivalent to saying that $A^\backslash dagger\; A\; =\; \backslash operatorname\_V$. This also implies that isometries preserve inner products, as
:$\backslash langle\; A\; u,\; A\; v\; \backslash rangle\; =\; \backslash langle\; u,\; A^\backslash dagger\; A\; v\; \backslash rangle\; =\; \backslash langle\; u,\; v\; \backslash rangle.$
Linear isometries are not always unitary operator
In functional analysis
Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional ...

s, though, as those require additionally that $V\; =\; W$ and $A\; A^\backslash dagger\; =\; \backslash operatorname\_V$.
By the Mazur–Ulam theorem, any isometry of normed vector spaces over R is affine
Affine (pronounced /əˈfaɪn/) relates to connections or affinities. It may refer to:
*Affine, a Affinity_(law)#Terminology, relative by marriage in law and anthropology
*Affine cipher, a special case of the more general substitution cipher
*Aff ...

.
;Examples
* A linear map
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

from Cdot product
In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...

) if and only if its matrix is unitary
Unitary may refer to:
* Unitary construction, in automotive design a common term for unibody (unitary body/chassis) construction
* Lethal Unitary Chemical Agents and Munitions (Unitary), as chemical weapons opposite of Binary
* Unitarianism, in Chr ...

.
Manifold

An isometry of amanifold
The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface.
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

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
METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model
Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ...

on the manifold; a manifold with a (positive-definite) metric is a Riemannian manifold
In differential geometry
Differential geometry is a mathematical
Mathematics (from Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a c ...

, one with an indefinite metric is a pseudo-Riemannian manifold
In differential geometry
Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differenti ...

. Thus, isometries are studied in Riemannian geometry#REDIRECT Riemannian geometry
Riemannian geometry is the branch of differential geometry
Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and m ...

.
A local isometry from one (pseudo
The prefix
A prefix is an affix
In linguistics
Linguistics is the scientific study of language
A language is a structured system of communication used by humans, including speech (spoken language), gestures (Signed language, ...

-)Riemannian manifold
In differential geometry
Differential geometry is a mathematical
Mathematics (from Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a c ...

to another is a map which pulls back the metric tensor
In the mathematical
Mathematics (from Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population ...

on the second manifold to the metric tensor on the first. When such a map is also a diffeomorphism
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

, such a map is called an isometry (or isometric isomorphism), and provides a notion of isomorphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

("sameness") in the category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization
Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...

Rm of Riemannian manifolds.
Definition

Let $R\; =\; (M,\; g)$ and $R\text{'}\; =\; (M\text{'},\; g\text{'})$ be two (pseudo-)Riemannian manifolds, and let $f\; :\; R\; \backslash to\; R\text{'}$ be a diffeomorphism. Then $f$ is called an isometry (or isometric isomorphism) if :$g\; =\; f^\; g\text{'},\; \backslash ,$ where $f^\; g\text{'}$ denotes thepullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a Pushforward (disambiguation), pushforward.
Precomposition
Precomposition with a Function (mathematics), function probabl ...

of the rank (0, 2) metric tensor $g\text{'}$ by $f$.
Equivalently, in terms of the pushforward
The notion of pushforward in mathematics is "dual" to the notion of pullback, and can mean a number of different but closely related things.
* Pushforward (differential), the differential of a smooth map between manifolds, and the "pushforward" ope ...

$f\_$, we have that for any two vector fields $v,\; w$ on $M$ (i.e. sections of the tangent bundle
Image:Tangent bundle.svg, Informally, the tangent bundle of a manifold (which in this case is a circle) is obtained by considering all the tangent spaces (top), and joining them together in a smooth and non-overlapping manner (bottom).
In differen ...

$\backslash mathrm\; M$),
:$g(v,\; w)\; =\; g\text{'}\; \backslash left(\; f\_\; v,\; f\_\; w\; \backslash right).\; \backslash ,$
If $f$ is a local diffeomorphism such that $g\; =\; f^\; g\text{'}$, then $f$ is called a local isometry.
Properties

A collection of isometries typically form a group, theisometry group In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...

. When the group is a continuous group, the infinitesimal generators of the group are the Killing vector fieldIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

s.
The Myers–Steenrod theorem states that every isometry between two connected Riemannian manifolds is smooth (differentiable). A second form of this theorem states that the isometry group of a Riemannian manifold is a Lie group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

.
Riemannian manifold
In differential geometry
Differential geometry is a mathematical
Mathematics (from Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a c ...

s that have isometries defined at every point are called symmetric space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s.
Generalizations

* Given a positive real number ε, an ε-isometry or almost isometry (also called a Hausdorff approximation) is a map $f\backslash colon\; X\backslash to\; Y$ between metric spaces such that *# for $x,\; x\text{'}\; \backslash in\; X$ one has $,\; d\_Y(f(x),f(x\text{'}))\; -\; d\_X(x,x\text{'}),\; <\; \backslash varepsilon$, and *# for any point $y\; \backslash in\; Y$ there exists a point $x\; \backslash in\; X$ with $d\_Y(y,\; f(x))\; <\; \backslash varepsilon$ :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 becontinuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ga ...

.
* The restricted isometry propertyIn linear algebra, the restricted isometry property (RIP) characterizes matrices
Matrix or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics)
In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, ...

characterizes nearly isometric matrices for sparse vectors.
* Quasi-isometryIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

is yet another useful generalization.
* One may also define an element in an abstract unital C*-algebra to be an isometry:
*:$a\backslash in\backslash mathfrak$ is an isometry if and only if $a^*\; \backslash cdot\; a\; =\; 1$.
: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 spaceIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

, the term ''isometry'' means a linear bijection preserving magnitude. See also Quadratic spaces.
See also

* Beckman–Quarles theorem * * The second dual of a Banach space as an isometric isomorphism *Euclidean plane isometry
Euclidean (or, less commonly, Euclidian) is an adjective derived from the name of Euclid, an ancient Greek mathematician. It is the name of:
Geometry
*Euclidean space, the two-dimensional plane and three-dimensional space of Euclidean geometry a ...

* Flat (geometry)
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...

* Homeomorphism groupIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

* Involution
Involution may refer to:
* Involute, a construction in the differential geometry of curves
* ''Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour input ...

* Isometry group In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...

* Motion (geometry)
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...

* Myers–Steenrod theorem
* 3D isometries that leave the origin fixed
* Partial isometryIn functional analysis
Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional an ...

* Scaling (geometry)
In Euclidean geometry
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's metho ...

* Semidefinite embedding
* Space group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

* Symmetry in mathematics
Symmetry occurs not only in geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned w ...

References

Bibliography

* * * * *Bibliography

* * {{cite book , last=Lee , first= 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 Functions and mappings Metric geometry Symmetry Equivalence (mathematics) Riemannian geometry