Symmetric Space - Symmetric Spaces in General

Symmetric Spaces in General

An important class of symmetric spaces generalizing the Riemannian symmetric spaces are pseudo-Riemannian symmetric spaces, in which the Riemannian metric is replaced by a pseudo-Riemannian metric (nondegenerate instead of positive definite on each tangent space). In particular, Lorentzian symmetric spaces, i.e., n dimensional pseudo-Riemannian symmetric spaces of signature (n − 1,1), are important in general relativity, the most notable examples being Minkowski space, De Sitter space and anti de Sitter space (with zero, positive and negative curvature respectively). De Sitter space of dimension n may be identified with the 1-sheeted hyperboloid in a Minkowski space of dimension n + 1.

Symmetric and locally symmetric spaces in general can be regarded as affine symmetric spaces. If M = G/H is a symmetric space, then Nomizu showed that there is a G-invariant torsion-free affine connection on M whose curvature is parallel. Conversely a manifold with such a connection is locally symmetric (i.e., its universal cover is a symmetric space). Such manifolds can also be described as those affine manifolds whose geodesic symmetries are all globally defined affine diffeomorphisms, generalizing the Riemannian and pseudo-Riemannian case.