Symmetric Space - Classification of Riemannian Symmetric Spaces

Classification of Riemannian Symmetric Spaces

The algebraic description of Riemannian symmetric spaces enabled Élie Cartan to obtain a complete classification of them in 1926.

For a given Riemannian symmetric space M let (G,K,σ,g) be the algebraic data associated to it. To classify possibly isometry classes of M, first note that the universal cover of a Riemannian symmetric space is again Riemannian symmetric, and the covering map is described by dividing the connected isometry group G of the covering by a subgroup of its center. Therefore we may suppose without loss of generality that M is simply connected. (This implies K is connected by the long exact sequence of a fibration, because G is connected by assumption.)