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.

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.