About: Symmetric space

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In differential geometry, representation theory and harmonic analysis, 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.

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rdf:type	yago:Space100028651 yago:WikicatHomogeneousSpaces yago:WikicatLieGroups yago:Abstraction100002137 yago:Attribute100024264 yago:Group100031264
rdfs:label	Symmetrischer Raum Symmetric space Espace symétrique 대칭 공간 Симметрическое пространство Симетричний простір
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. 리만 기하학과 리 군론에서, 대칭 공간(對稱空間, 영어: symmetric space)은 일반점의 안정자군이 어떤 대합에 의하여 정의되는 동차 공간이다. Симметрическое пространство — риманово многообразие, группа изометрий которого содержит центральные симметрии с центром в любой точке. 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. In differential geometry, representation theory and harmonic analysis, 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. En mathématiques, et plus spécifiquement en géométrie différentielle, un espace riemannien symétrique est une variété riemannienne qui, en chaque point, admet une isométrie involutive dont ce point est un point fixe isolé. Plus généralement, un espace symétrique est une variété différentielle munie, en chaque point, d'une involution, le tout vérifiant certaines conditions. Lorsqu'il n'y pas de risque de confusion, les espaces riemanniens symétriques sont appelés espaces symétriques. Les espaces symétriques connexes sont des espaces homogènes de groupes de Lie.
foaf:isPrimaryTopicOf	wikipedia-en:Symmetric_space
dct:subject	Homogeneous spaces Differential geometry Lie groups Riemannian geometry
Wikipage page ID	1642877(xsd:integer)
Wikipage revision ID	986166159(xsd:integer)
Link from a Wikipage to another Wikipage	Homogeneous spaces Cartan decomposition Cartan subgroup Cartan–Ambrose–Hicks theorem Casimir element Versor Satake diagram Lens space Differential geometry Grassmannian Group action Killing form Projective space Quaternion Harmonic analysis Holonomy Lagrangian Grassmannian Hyperbolic space Indecomposable module Simple Lie group Differential geometry Octonion Gelfand pair Point reflection Riemannian geometry Automorphism Freudenthal magic square Sectional curvature General relativity Katsumi Nomizu Élie Cartan Hopf–Rinow theorem Minkowski space Cayley plane Jet bundle Klein geometry Lie group Lie theory Pseudo-Riemannian manifold Sphere Orthogonal symmetric Lie algebra If and only if Involution (mathematics) Irreducible representation Semisimple Lie algebra Homogeneous space Homotopy group Anti-de Sitter space Atle Selberg Bott periodicity theorem Orthogonal group Riemann surface Affine connection Lie groups Unitary representation Quaternionic vector space Riemannian geometry Complete metric space Covering space Representation theory De Sitter space Covariant derivative Loop space Euclidean space Ernest Vinberg Antilinear map Mathematics Metric tensor Direct sum of modules Simply connected space Marcel Berger Parallel transport Hermitian symmetric space Quaternion-Kähler symmetric space Tangent space

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