Isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective.[a] 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;[b] the isometry that relates them is either a rigid motion (translation or rotation), or a composition of a rigid motion and a reflection.a quotient set of the space of Cauchy sequences onOther 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.is called an isometry or distance-preserving map if for any, An isometry is automatically injective;[a] otherwise two distinct points, a and b, could be mapped to the same point, thereby contradicting the coincidence axiom of the metric d, i.e.,Clearly, every isometry between metric spaces is a topological embedding.Like any other bijection, a global isometry has a function inverse.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.[5][6] This term is often abridged to simply isometry, so one should take care to determine from context which type is intended.The following theorem is due to Mazur and Ulam.Theorem[7][8] — Let A : X → Y 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.If X and Y are complex vector spaces then A may fail to be linear as a map over[9] Linear isometries are distance-preserving maps in the above sense.This also implies that isometries preserve inner products, as Linear isometries are not always unitary operators, though, as those require additionally thatBy the Mazur–Ulam theorem, any isometry of normed vector spaces overThus, 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.denotes the pullback of the rank (0, 2) metric tensorThe Myers–Steenrod theorem states that every isometry between two connected Riemannian manifolds is smooth (differentiable).Symmetric spaces are important examples of Riemannian manifolds that have isometries defined at every point.3.11 Any two congruent triangles are related by a unique isometry.— Coxeter (1969) p. 39[3]3.51 Any direct isometry is either a translation or a rotation.
isometry (disambiguation)IsometricIsometric projectioncompositionA reflectionTranslationa rigid motiondistancemetric spacesbijectiveAncient Greekgeometric transformationmotiontransformationEuclidean spacecongruentreflectionembeddedquotient setCauchy sequencesisomorphiccomplete metric spaceclosed subsetnormed vector spaceBanach spaceHilbert spaceunitary operatorinjectiveorder embeddingpartially ordered setstopological embeddingfunction inversefunction compositionisometry grouprotationEuclidean spacesEuclidean groupnormed spacesStefan Banachnormed vector spaceslinear mapsurjectiveinner product spaceunitary operatorsdomaincodomaincoisometryMazur–Ulam theoremaffineconformal linear transformationdot productunitarymanifoldmetricRiemannian manifoldpseudo-Riemannian manifoldRiemannian geometrypseudopulls backmetric tensordiffeomorphismisomorphismcategorypullbackpushforwardtangent bundlelocal diffeomorphismcontinuous groupinfinitesimal generatorsKilling vector fieldsMyers–Steenrod theoremLie groupSymmetric spacesRiemannian manifoldsHausdorffcontinuousrestricted isometry propertyQuasi-isometrypseudo-Euclidean spaceBeckman–Quarles theoremConformal mapEuclidean plane isometryFlat (geometry)Homeomorphism groupInvolutionMotion (geometry)Partial isometryScaling (geometry)Semidefinite embeddingSpace groupSymmetry in mathematicsProceedings of the American Mathematical SocietyScienceBibcodeCiteSeerXJournal of Machine Learning ResearchAdvances in Neural Information Processing SystemsRudin, WalterMcGraw-Hill Science/Engineering/MathSchaefer, Helmut H.Trèves, FrançoisWilansky, AlbertCoxeter, H. S. M.Metric spaceCauchy sequenceCompletenessEquivalent metricsMetrizable spaceTriangle inequalityBaire category theoremBanach fixed-pointKuratowski embeddingLebesgue's number lemmaMetrization theoremsNagata–SmirnovUrysohn'sContractionMetric mapDilationEquicontinuityQuasi-Lipschitz continuityMetric derivativeMetric outer measureMetric projectionQuasisymmetric