# 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. [1]

## 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; [3] 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.

## Isometry definition

Let X and Y be metric spaces with metrics dX and dY. A map f : XY is called an isometry or distance preserving if for any a,bX one has

${\displaystyle d_{Y}\!\left(f(a),f(b)\right)=d_{X}(a,b).}$ [4]

An isometry is automatically injective; [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

## Isometries between normed spaces

The following theorem is due to Mazur and Ulam.

Definition: [5] The midpoint of two elements x and y in a vector space is the vector 1/2(x + y).

Theorem [5] [6]   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 . If X and Y are complex vector spaces then A may fail to be linear as a map over .

### Linear isometry

Given two normed vector spaces ${\displaystyle V}$ and ${\displaystyle W}$, a linear isometry is a linear map ${\displaystyle A:V\to W}$ that preserves the norms:

${\displaystyle \|Av\|=\|v\|}$

for all ${\displaystyle v\in V}$. [7] 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

${\displaystyle \langle v,v\rangle =\langle Av,Av\rangle }$

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

${\displaystyle \langle Au,Av\rangle =\langle u,A^{\dagger }Av\rangle =\langle u,v\rangle .}$

Linear isometries are not always unitary operators, though, as those require additionally that ${\displaystyle V=W}$ and ${\displaystyle AA^{\dagger }=\operatorname {I} _{V}}$.

By the Mazur–Ulam theorem, any isometry of normed vector spaces over R is affine.

Examples

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

${\displaystyle g=f^{*}g',\,}$

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

${\displaystyle g(v,w)=g'\left(f_{*}v,f_{*}w\right).\,}$

If ${\displaystyle f}$ is a local diffeomorphism such that ${\displaystyle g=f^{*}g'}$, then ${\displaystyle f}$ 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 ${\displaystyle f:X\to Y}$ between metric spaces such that
1. for x,xX one has |dY(ƒ(x),ƒ(x))dX(x,x)| < ε, and
2. for any point yY there exists a point xX with dY(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.
• 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:
${\displaystyle a\in {\mathfrak {A}}}$ is an isometry if and only if ${\displaystyle a^{*}\cdot a=1}$.
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.

## 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 smooth.

Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century.

Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension, including the three-dimensional space and the Euclidean plane. It was introduced by the Ancient Greek mathematician Euclid of Alexandria, and the qualifier Euclidean is used to distinguish it from other spaces that were later discovered in physics and modern mathematics.

The Nash embedding theorems, named after John Forbes Nash, state that every Riemannian manifold can be isometrically embedded into some Euclidean space. Isometric means preserving the length of every path. For instance, bending but neither stretching nor tearing a page of paper gives an isometric embedding of the page into Euclidean space because curves drawn on the page retain the same arclength however the page is bent.

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) is a real, smooth manifold M equipped with a positive-definite inner product gp on the tangent space TpM at each point p. A common convention is to take g to be smooth, which means that for any smooth coordinate chart (U, x) on M, the n2 functions

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 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 with dimension greater than 2. 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, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a space.

In mathematics, a hyperbolic space is a homogeneous space that has a constant negative curvature, where in this case the curvature is the sectional curvature. It is hyperbolic geometry in more than 2 dimensions, and is distinguished from Euclidean spaces with zero curvature that define the Euclidean geometry, and elliptic geometry that have a constant positive curvature.

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 mathematics, a symmetric space is a pseudo-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, Mikhail Gromov's filling area conjecture asserts that the hemisphere has minimum area among the orientable surfaces that fill a closed curve of given length without introducing shortcuts between its points.

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 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 geometry, a motion is an isometry of a metric space. For instance, a plane equipped with the Euclidean distance metric is a metric space in which a mapping associating congruent figures is a motion. More generally, the term motion is a synonym for surjective isometry in metric geometry, including elliptic geometry and hyperbolic geometry. In the latter case, hyperbolic motions provide an approach to the subject for beginners.

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.

## References

1. Coxeter 1969 , p. 29

"We shall find it convenient to use the word transformation in the special sense of a one-to-one correspondence ${\displaystyle P\to P'}$ 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..."

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.

3. Coxeter 1969 , p. 39

3.11Any two congruent triangles are related by a unique isometry.

4. Beckman, F. S.; Quarles, D. A., Jr. (1953). "On isometries of Euclidean spaces" (PDF). Proceedings of the American Mathematical Society . 4 (5): 810–815. doi:. JSTOR   2032415. MR   0058193.
Let T be a transformation (possibly many-valued) of ${\displaystyle E^{n}}$ (${\displaystyle 2\leq n<\infty }$) into itself.
Let ${\displaystyle d(p,q)}$ be the distance between points p and q of ${\displaystyle E^{n}}$, and let Tp, Tq be any images of p and q, respectively.
If there is a length a > 0 such that ${\displaystyle d(Tp,Tq)=a}$ whenever ${\displaystyle d(p,q)=a}$, then T is a Euclidean transformation of ${\displaystyle E^{n}}$ onto itself.
5. Narici & Beckenstein 2011, pp. 275-339.
6. Wilansky 2013, pp. 21-26.
7. Thomsen, Jesper Funch (2017). Lineær algebra[Linear algebra] (in Danish). Århus: Department of Mathematics, Aarhus University. p. 125.
8. Roweis, S. T.; Saul, L. K. (2000). "Nonlinear Dimensionality Reduction by Locally Linear Embedding". Science . 290 (5500): 2323–2326. CiteSeerX  . doi:10.1126/science.290.5500.2323. PMID   11125150.
9. Saul, Lawrence K.; Roweis, Sam T. (2003). "Think globally, fit locally: Unsupervised learning of nonlinear manifolds". Journal of Machine Learning Research . 4 (June): 119–155. Quadratic optimisation of ${\displaystyle \mathbf {M} =(I-W)^{\top }(I-W)}$ (page 135) such that ${\displaystyle \mathbf {M} \equiv YY^{\top }}$
10. Zhang, Zhenyue; Zha, Hongyuan (2004). "Principal Manifolds and Nonlinear Dimension Reduction via Local Tangent Space Alignment". SIAM Journal on Scientific Computing. 26 (1): 313–338. CiteSeerX  . doi:10.1137/s1064827502419154.
11. Zhang, Zhenyue; Wang, Jing (2006). "MLLE: Modified Locally Linear Embedding Using Multiple Weights". Advances in Neural Information Processing Systems . 19. It can retrieve the ideal embedding if MLLE is applied on data points sampled from an isometric manifold.