Symmetric Space - Riemannian Symmetric Spaces Are Symmetric Spaces

Riemannian Symmetric Spaces Are Symmetric Spaces

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 (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 T_pM. 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 T_pM, hence compact. Moreover, if we denote by s_p: M → M the geodesic symmetry of M at p, the map

is an involutive Lie group automorphism such that the isotropy group K is contained between the fixed point group of σ and its identity component (hence an open subgroup).

To summarize, M is a symmetric space G/K with a compact isotropy group K. Conversely, symmetric spaces with compact isotropy group are Riemannian symmetric spaces, although not necessarily in a unique way. To obtain a Riemannian symmetric space structure we need to fix a K-invariant inner product on the tangent space to G/K at the identity coset eK: such an inner product always exists by averaging, since K is compact, and by acting with G, we obtain a G-invariant Riemannian metric g on G/K.

To show that G/K is Riemannian symmetric, consider any point p = hK (a coset of K, where h ∈ G) and define

where σ is the involution of G fixing K. Then one can check that s_p is an isometry with (clearly) s_p(p) = p and (by differentiating) ds_p equal to minus the identity on T_pM. Thus s_p is a geodesic symmetry and, since p was arbitrary, M is a Riemannian symmetric space.

If one starts with a Riemannian symmetric space M, and then performs these two constructions in sequence, then the Riemannian symmetric space yielded is isometric to the original one. This shows that the "algebraic data" (G,K,σ,g) completely describe the structure of M.