Symmetric space: Difference between revisions

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In geometric terms, 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 <math>T_pM</math> as minus the identity (every symmetric space is complete, since any geodesic can be extended indefinitely via symmetries about the endpoints). Both descriptions can also naturally be extended to the setting of [[pseudo-Riemannian manifold]]s.
From the point of view of Lie theory, a symmetric space is the quotient ''G''/''H'' of a connected [[Lie group]] ''G'' by a Lie subgroup ''H'' which is (a connected component of) the invariant group of an [[involution (mathematics)|involution]] of ''G.'' This definition includes more than the Riemannian definition, and reduces to it when ''H'' is compact.
Riemannian symmetric spaces arise in a wide variety of situations in both mathematics and physics. 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.