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Encyclopedia > Riemannian symmetric space

In mathematics, including applications to general relativity, a (Riemannian) symmetric space in differential geometry is a certain kind of homogeneous space in the theory of Lie groups. Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ... Two-dimensional visualisation of space-time distortion. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In mathematics, in particular in the theory of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a manifold or topological space X on which G acts by symmetry in a transitive way; it is not assumed that the action of G is faithful. ... In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ...

A geometric characterization is that it is a Riemannian manifold such that for every point there exists an isometry fixing that point and inducing minus the identity on the tangent space at that point. A Lie group characterisation is as G/H where G is a Lie group and H a subgroup that is open in the fixed set of an automorphism of G of order 2. There is a classification of such spaces, by Elie Cartan. In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... Élie Joseph Cartan (9 April 1869 - 6 May 1951) was a French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications. ...

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