TheInfoList

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 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 $P \to P\text{'}$ 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 a
transformation 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'' be
metric 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''''X'' and ''d''''Y''. A
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 :$d_Y\!\left\left(f\left(a\right),f\left(b\right)\right\right)=d_X\left(a,b\right).$ 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 ...
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 $x\mapsto , x,$ 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 two
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 ...
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 ...

$A : V \to W$ that preserves the norms: :$\, Av\, = \, v\,$ for all $v \in V$. 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 :$\langle v, v \rangle = \langle Av, Av \rangle$ for all $v \in V$, which is equivalent to saying that $A^\dagger A = \operatorname_V$. This also implies that isometries preserve inner products, as :$\langle A u, A v \rangle = \langle u, A^\dagger A v \rangle = \langle u, v \rangle.$ 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 $A A^\dagger = \operatorname_V$. 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 C''n'' to itself is an isometry (for the
dot 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 a
manifold 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 $R = \left(M, g\right)$ and $R\text{'} = \left(M\text{'}, g\text{'}\right)$ be two (pseudo-)Riemannian manifolds, and let $f : R \to R\text{'}$ be a diffeomorphism. Then $f$ is called an isometry (or isometric isomorphism) if :$g = f^ g\text{'}, \,$ where $f^ g\text{'}$ denotes the
pullback 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 ...

$\mathrm M$), :$g\left(v, w\right) = g\text{'} \left\left( f_ v, f_ w \right\right). \,$ 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, 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 ...
. 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 $f\colon X\to Y$ between metric spaces such that *# for $x, x\text{'} \in X$ one has $, d_Y\left(f\left(x\right),f\left(x\text{'}\right)\right) - d_X\left(x,x\text{'}\right), < \varepsilon$, and *# for any point $y \in Y$ there exists a point $x \in X$ with $d_Y\left(y, f\left(x\right)\right) < \varepsilon$ :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 be
continuous 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: *:$a\in\mathfrak$ is an isometry if and only if $a^* \cdot a = 1$. :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 ...

* 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 ...

* * * * *

# 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