# Isometry

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In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. [lower-alpha 1] The word isometry is derived from the Ancient Greek: ἴσος isos meaning "equal", and μέτρον metron meaning "measure".

## Contents A composition of two opposite isometries is a direct isometry. A reflection in a line is an opposite isometry, like R 1 or R 2 on the image. Translation T is a direct isometry: a rigid motion.

## Introduction

Given a metric space (loosely, a set and a scheme for assigning distances between elements of the set), an isometry is a 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 congruent if they are related by an isometry; [lower-alpha 2] the isometry that relates them is either a rigid motion (translation or rotation), or a composition of a rigid motion and a reflection.

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 of the space of Cauchy sequences on $\ M\ .$ The original space $\ M\$ is thus isometrically 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 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.

## Definition

Let $\ X\$ and $\ Y\$ be metric spaces with metrics (e.g., distances) $\ d_{X}\$ and $\ d_{Y}\ .$ A map $\ f:X\to Y\$ is called an isometry or distance preserving if for any $\ a,b\in X\$ one has

$d_{Y}\!\left(f(a),f(b)\right)=d_{X}(a,b).$ [lower-alpha 3]

An isometry is automatically injective; [lower-alpha 1] 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 sets 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 of bijective isometries from a metric space to itself forms a 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 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, translation and rotation is a global isometry on Euclidean spaces. See also Euclidean group and Euclidean space § Isometries.
• The map $\ x\mapsto |x|\$ in $\ \mathbb {R} \$ is a path isometry but not a (general) isometry. Note that unlike an isometry, this path isometry does not need to be injective.

## Isometries between normed spaces

The following theorem is due to Mazur and Ulam.

Definition:  The midpoint of two elements x and y in a vector space is the vector 1/2(x + y).

Theorem     Let A : XY be a surjective isometry between normed spaces that maps 0 to 0 (Stefan Banach called such maps rotations) where note that A is not assumed to be a linear isometry. Then A maps midpoints to midpoints and is linear as a map over the real numbers $\mathbb {R}$ . If X and Y are complex vector spaces then A may fail to be linear as a map over $\mathbb {C}$ .

### Linear isometry

Given two normed vector spaces $V$ and $W,$ a linear isometry is a linear map $A:V\to W$ 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 an inner product space, the above definition reduces to

$\langle v,v\rangle =\langle Av,Av\rangle$ for all $v\in V\ ,$ which is equivalent to saying that $\ A^{\dagger }A=\operatorname {I} _{V}\ .$ This also implies that isometries preserve inner products, as

$\langle Au,Av\rangle =\langle u,A^{\dagger }Av\rangle =\langle u,v\rangle \ .$ Linear isometries are not always unitary operators, though, as those require additionally that $V=W$ and $AA^{\dagger }=\operatorname {I} _{V}\ .$ By the Mazur–Ulam theorem, any isometry of normed vector spaces over $\mathbb {R}$ is affine.

Examples
• A linear map from $\mathbb {C} ^{n}$ to itself is an isometry (for the dot product) if and only if its matrix is unitary.    

## Manifold

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 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 to another is a map which 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 Rm of Riemannian manifolds.

### Definition

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

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

$\ g(v,w)=g'\left(f_{*}v,f_{*}w\right)\ .$ If $\ f\$ is a local diffeomorphism such that $\ g=f^{*}g'\ ,$ then $f$ is called a local isometry.

### Properties

A collection of isometries typically form a group, the isometry group. When the group is a continuous group, the infinitesimal generators of the group are the Killing vector fields.

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.

Riemannian manifolds that have isometries defined at every point are called symmetric spaces.

## 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
1. for $x,x'\in X$ one has $\ |d_{Y}(f(x),f(x'))-d_{X}(x,x')|<\varepsilon \ ,$ and
2. for any point $y\in Y$ there exists a point $\ x\in X$ with $d_{Y}(y,f(x))<\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.
• 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:
$\ a\in {\mathfrak {A}}\$ 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.

## Footnotes

1. "We shall find it convenient to use the word transformation in the special sense of a one-to-one correspondence $\ P\to P'\$ 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 P and a second member P' 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 ..." — Coxeter (1969) p. 29 

2. 3.11Any two congruent triangles are related by a unique isometry.— Coxeter (1969) p. 39 

3. Let T be a transformation (possibly many-valued) of $E^{n}$ ($2\leq n<\infty$ ) into itself.
Let $d(p,q)$ be the distance between points p and q of $E^{n}$ , and let Tp, Tq be any images of p and q, respectively.
If there is a length a > 0 such that $d(Tp,Tq)=a$ whenever $d(p,q)=a$ , then T is a Euclidean transformation of $E^{n}$ onto itself. 

## Related Research Articles In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension, including the three-dimensional space and the Euclidean plane. The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry.

In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.

In differential geometry, a Riemannian manifold or Riemannian space(M, g), so called after the German mathematician Bernhard Riemann, is a real, smooth manifold M equipped with a positive-definite inner product gp on the tangent space TpM at each point p.

In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space (TVS) for which the canonical evaluation map from into its bidual is an isomorphism of TVSs. Since a normable TVS is reflexive if and only if it is semi-reflexive, every normed space is reflexive if and only if the canonical evaluation map from into its bidual is surjective; in this case the normed space is necessarily also a Banach space. In 1951, R. C. James discovered a Banach space, now known as James' space, that is not reflexive but is nevertheless isometrically isomorphic to its bidual. In mathematical physics, Minkowski space is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded. Although initially developed by mathematician Hermann Minkowski for Maxwell's equations of electromagnetism, the mathematical structure of Minkowski spacetime was shown to be implied by the postulates of special relativity.

In Riemannian or pseudo Riemannian geometry, the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold that preserves the (pseudo-)Riemannian metric and is torsion-free.

In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature Kp) depends on a two-dimensional linear subspace σp of the tangent space at a point p of the manifold. It can be defined geometrically as the Gaussian curvature of the surface which has the plane σp as a tangent plane at p, obtained from geodesics which start at p in the directions of σp. The sectional curvature is a real-valued function on the 2-Grassmannian bundle over the manifold. In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. There are many ways to construct it as an open subset of with an explicitly written Riemannian metric; such constructions are referred to as models. Hyperbolic 2-space, H2, which was the first instance studied, is also called the hyperbolic plane.

In mathematics, Gromov–Hausdorff convergence, named after Mikhail Gromov and Felix Hausdorff, is a notion for convergence of metric spaces which is a generalization of Hausdorff convergence.

This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology.

This is a glossary of terms specific to differential geometry and differential topology. The following three glossaries are closely related: In differential geometry, an affine connection is a geometric object on a smooth manifold which connects nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space. Connections are among the simplest methods of defining differentiation of the sections of vector bundles. In mathematics, a symmetric space is a Riemannian manifold whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, leading to consequences in the theory of holonomy; or algebraically through Lie theory, which allowed Cartan to give a complete classification. Symmetric spaces commonly occur in differential geometry, representation theory and harmonic analysis.

In differential geometry, a branch of mathematics, a Riemannian submersion is a submersion from one Riemannian manifold to another that respects the metrics, meaning that it is an orthogonal projection on tangent spaces.

In mathematics, a Riemannian manifold is said to be flat if its Riemann curvature tensor is everywhere zero. Intuitively, a flat manifold is one that "locally looks like" Euclidean space in terms of distances and angles, e.g. the interior angles of a triangle add up to 180°.

In Riemannian geometry, the filling radius of a Riemannian manifold X is a metric invariant of X. It was originally introduced in 1983 by Mikhail Gromov, who used it to prove his systolic inequality for essential manifolds, vastly generalizing Loewner's torus inequality and Pu's inequality for the real projective plane, and creating systolic geometry in its modern form. In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsically, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.

In mathematics, the Cartan–Ambrose–Hicks theorem is a theorem of Riemannian geometry, according to which the Riemannian metric is locally determined by the Riemann curvature tensor, or in other words, behavior of the curvature tensor under parallel translation determines the metric.

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2. Coxeter 1969 , p. 46

3.51Any direct isometry is either a translation or a rotation. Any opposite isometry is either a reflection or a glide reflection.

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