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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".

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 involves an isometry from into a quotient set of the space of Cauchy sequences on The original space 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.

Let and be metric spaces with metrics (e.g., distances) and A map is called an **isometry** or **distance preserving** if for any one has

^{ [4] }^{ [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 in is a
*path isometry*but not a (general) isometry. Note that unlike an isometry, this path isometry does not need to be injective.

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

Given two normed vector spaces and a **linear isometry** is a linear map that preserves the norms:

for all ^{ [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

for all which is equivalent to saying that This also implies that isometries preserve inner products, as

Linear isometries are not always unitary operators, though, as those require additionally that and

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

- Examples

- A linear map from to itself is an isometry (for the dot product) if and only if its matrix is unitary.
^{ [8] }^{ [9] }^{ [10] }^{ [11] }

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.

Let and be two (pseudo-)Riemannian manifolds, and let be a diffeomorphism. Then is called an **isometry** (or **isometric isomorphism**) if

where denotes the pullback of the rank (0, 2) metric tensor by Equivalently, in terms of the pushforward we have that for any two vector fields on (i.e. sections of the tangent bundle ),

If is a local diffeomorphism such that then is called a **local isometry**.

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.

- Given a positive real number ε, an
**ε-isometry**or**almost isometry**(also called a**Hausdorff approximation**) is a map between metric spaces such that- for one has and
- for any point there exists a point with

- 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:
- is an isometry if and only if

- 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 spaces.

- Beckman–Quarles theorem
- Conformal map – Mathematical function which preserves angles
- The second dual of a Banach space as an isometric isomorphism
- Euclidean plane isometry
- Flat (geometry)
- Homeomorphism group
- Involution
- Isometry group
- Motion (geometry)
- Myers–Steenrod theorem
- 3D isometries that leave the origin fixed
- Partial isometry
- Scaling (geometry)
- Semidefinite embedding
- Space group
- Symmetry in mathematics

- 1 2
"We shall find it convenient to use the word

*transformation*in the special sense of a one-to-one correspondence 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^{ [1] } - ↑
**3.11***Any two congruent triangles are related by a unique isometry.*— Coxeter (1969) p. 39^{ [3] } - ↑

Let T be a transformation (possibly many-valued) of () into itself.

Let be the distance between points p and q of , and let Tp, Tq be any images of p and q, respectively.

If there is a length a > 0 such that whenever , then T is a Euclidean transformation of onto itself.^{ [4] }

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 *g*_{p} on the tangent space *T*_{p}*M* 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 *K*(σ_{p}) depends on a two-dimensional linear subspace σ_{p} 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 σ_{p} as a tangent plane at *p*, obtained from geodesics which start at *p* in the directions of σ_{p}. 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, **H**^{2}, 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.

- ↑ Coxeter 1969, p. 29
- ↑ Coxeter 1969 , p. 46
**3.51***Any direct isometry is either a translation or a rotation. Any opposite isometry is either a reflection or a glide reflection.* - ↑ Coxeter 1969, p. 39
- 1 2 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: 10.2307/2032415 . JSTOR 2032415. MR 0058193. - 1 2 Narici & Beckenstein 2011, pp. 275–339.
- ↑ Wilansky 2013, pp. 21–26.
- ↑ Thomsen, Jesper Funch (2017).
*Lineær algebra*[*Linear Algebra*]. Department of Mathematics (in Danish). Århus: Aarhus University. p. 125. - ↑ Roweis, S.T.; Saul, L.K. (2000). "Nonlinear dimensionality reduction by locally linear embedding".
*Science*.**290**(5500): 2323–2326. CiteSeerX 10.1.1.111.3313 . doi:10.1126/science.290.5500.2323. PMID 11125150. - ↑ Saul, Lawrence K.; Roweis, Sam T. (June 2003). "Think globally, fit locally: Unsupervised learning of nonlinear manifolds".
*Journal of Machine Learning Research*.**4**(June): 119–155.Quadratic optimisation of (page 135) such that

- ↑ 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 10.1.1.211.9957 . doi:10.1137/s1064827502419154. - ↑ Zhang, Zhenyue; Wang, Jing (2006). "MLLE: Modified locally linear embedding using multiple weights". In Schölkopf, B.; Platt, J.; Hoffman, T. (eds.).
*Advances in Neural Information Processing Systems*. NIPS 2006. NeurIPS Proceedings. Vol. 19. pp. 1593–1600. ISBN 9781622760381.It can retrieve the ideal embedding if MLLE is applied on data points sampled from an isometric manifold.

- Rudin, Walter (1991).
*Functional Analysis*. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. - Narici, Lawrence; Beckenstein, Edward (2011).
*Topological Vector Spaces*. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. - Schaefer, Helmut H.; Wolff, Manfred P. (1999).
*Topological Vector Spaces*. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. - Trèves, François (2006) [1967].
*Topological Vector Spaces, Distributions and Kernels*. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. - Wilansky, Albert (2013).
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