Topic summary
Symmetric spaces
Extracted from the Wikipedia article Symmetric space.
Riemannian symmetric spaces satisfy the Lie-theoretic characterization
If M is a Riemannian symmetric space, the identity component G of the isometry group of M is a Lie group acting transitively on M (that is, M is Riemannian homogeneous). Therefore, if we fix some point p of M, M is diffeomorphic to the quotient G/K, where K denotes the isotropy group of the action of G on M at p. By differentiating the action at p we obtain an isometric action of K on TpM. This action is faithful (e.g., by a theorem of Kostant, any isometry in the identity component is determined by its 1-jet at any point) and so K is a subgroup of the orthogonal group of TpM, hence compact. Moreover, if we denote by sp: M → M the geodesic symmetry of M at p, the map