About: Symmetric space

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In mathematics, a symmetric space is a Riemannian manifold (or more generally, 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.

Attributes	Values
rdf:type	yago:WikicatLieGroups yago:Abstraction100002137 yago:Attribute100024264 yago:Group100031264 yago:WikicatHomogeneousSpaces yago:Space100028651
rdfs:label	Symmetrischer Raum (de) Espace symétrique (fr) 대칭 공간 (ko) Symmetric space (en) Симметрическое пространство (ru) Симетричний простір (uk)
rdfs:comment	In der Mathematik sind symmetrische Räume eine Klasse von Riemannschen Mannigfaltigkeiten mit einem besonders hohen Grad an Symmetrien. Sie sind eine wichtige Klasse von Beispielen in Geometrie und Topologie und finden Anwendung unter anderem in Darstellungstheorie, harmonischer Analysis, Zahlentheorie, Modulformen und Physik. (de) ( 이 문서는 특별한 동차 공간에 관한 것입니다. 약한 분리 공리를 만족시키는 위상 공간에 대해서는 R0 공간 문서를 참고하십시오.) 리만 기하학과 리 군론에서 대칭 공간(對稱空間, 영어: symmetric space)은 일반점의 안정자군이 어떤 대합에 의하여 정의되는 동차 공간이다. (ko) Симетричний простір — ріманів многовид, група ізометрій якого містить центральні симетрії з центром в будь-якій точці. Початок вивченню симетричних просторів було покладено Елі Картаном. Зокрема їм була отримана їх класифікація в 1926 році. (uk) Симметрическое пространство — риманово многообразие, группа изометрий которого содержит центральные симметрии с центром в любой точке. (ru) In mathematics, a symmetric space is a Riemannian manifold (or more generally, 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. (en) En mathématiques, et plus spécifiquement en géométrie différentielle, un espace symétrique est une variété, espace courbe sur lequel on peut définir une généralisation convenable de la notion de symétrie centrale. La définition précise de la notion d'espace symétrique dépend du type de structure dont on munit la variété. Le plus couramment, on entend par espace symétrique une variété munie d'une métrique riemannienne pour laquelle l'application de symétrie le long des géodésiques constitue une isométrie. (fr)
dcterms:subject	Differential geometry Homogeneous spaces Lie groups Riemannian geometry
Wikipage page ID	1642877 (xsd:integer)
Wikipage revision ID	1061186625 (xsd:integer)
Link from a Wikipage to another Wikipage	Projective space Quaternion Representation theory De Sitter space Holonomy group Homogeneous space Riemann surface Unit quaternion Indecomposable module Lie group Lie theory Differential geometry Complete space Mathematics Gelfand pair Quaternionic vector space General relativity Grassmannian Bott periodicity theorem Minkowski space Lagrangian Grassmannian Anti-de Sitter space Antilinear Lie groups Homogeneous spaces Élie Cartan Freudenthal magic square Harmonic analysis Quaternion-Kähler symmetric space Tangent space Automorphism Irreducible representation Affine connection Lie groups Ernest Vinberg Euclidean space Parallel transport Lens space Pseudo-Riemannian manifold Riemannian geometry Riemannian manifold Involution (mathematics) Covariant derivative Hyperbolic space Simple Lie group Atle Selberg Riemannian geometry Jet bundle Katsumi Nomizu Killing form Hermitian symmetric space Holonomy Torsion tensor Differential geometry Direct sum of Lie algebras Marcel Berger Sphere Rosenfeld projective plane If and only if Metric tensor Octonion Orthogonal group Cartan–Ambrose–Hicks theorem Satake diagram Unitary representation Loop spaces Semisimple Lie algebra Exceptional Lie group Orthogonal symmetric Lie algebra Sectional curvature Inversion symmetry Universal cover Pseudo-Riemannian metric Simply connected Cartan involution Casimir operator

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