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Symmetric space Symmetrischer Raum Симметрическое пространство Espace symétrique
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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. Симметрическое пространство — риманово многообразие, группа изометрий которого содержит центральные симметрии с центром в любой точке. 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. 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 made more precise, in either the language of Riemannian geometry or of Lie theory. The Riemannian definition is more geometric, and plays a deep role in the theory of holonomy. The Lie-theoretic definition is more algebraic. From the point of view of Lie theory, a symmetric space is the quotient G/H of Lie group G by a Lie subgroup H, where the Lie algebra of H is also required to be the
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Симметрическое пространство — риманово многообразие, группа изометрий которого содержит центральные симметрии с центром в любой точке. 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 made more precise, in either the language of Riemannian geometry or of Lie theory. The Riemannian definition is more geometric, and plays a deep role in the theory of holonomy. The Lie-theoretic definition is more algebraic. In Riemannian geometry, a complete, simply connected Riemannian manifold is a symmetric space if and only if its curvature tensor is invariant under parallel transport. More generally, a Riemannian manifold (M,g) is said to be symmetric if and only if, for each point p of M, there exists an isometry of M fixing p and acting on the tangent space of M at p by minus the identity. Any symmetric space is complete, and has a finite cover which is a simply connected symmetric space; thus these two characterizations in fact coincide up to finite covers. Both descriptions can also naturally be extended to the settingof pseudo-Riemannian manifolds. From the point of view of Lie theory, a symmetric space is the quotient G/H of Lie group G by a Lie subgroup H, where the Lie algebra of H is also required to be the -eigenspace of an involution of the Lie algebra of G. As stated, this characterization includes pseudo-Riemannian spaces as well as a Riemannian ones;extra algebraic conditions are needed to restrict to the Riemannian case. Riemannian symmetric spaces arise in a wide variety of situations in both mathematics and physics. They were first classified by Élie Cartan. Their central role in the theory of holonomy was discovered by Marcel Berger. They are important objects of study in representation theory and harmonic analysis as well as in differential geometry. 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. 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. Les espaces des géométries euclidiennes et non euclidiennes sont naturellement des espaces riemanniens symétriques. La plupart des espaces homogènes usuels de la géométrie différentielles sont soit des espaces symétriques (riemanniens ou non) soit ce que l'on appelle variétés de drapeaux généralisés (généralisation des espaces projectifs, des grassmanniennes, des quadriques projectives). Dans ce cas riemannien, ces espaces ont été définis et classifiés pour la première fois par Élie Cartan.Ils constituent un cadre naturel pour généraliser l'analyse harmonique classique sur les sphères.
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