ABSTRACT

In the late 1920s, E´lie Cartan gave a local characterization of Riemannian symmetric spacesM = G/K by the differential condition

∇R= 0.

For any p ∈M, the curvature tensor Rp at p locally determines the symmetric space M. It also determines uniquely (up to isometry) a simply connected Riemannian symmetric space M˜ = G/K˜ whose curvature tensor is the same as the one of M. Indeed, from Rp together with the integrability condition (Rp)xy ·Rp = 0 we can construct an orthogonal symmetric Lie algebra g, which uniquely determines M˜. In terms of holonomy systems, the triple (TpM,Rp, K˜) is a symmetric holonomy system. As in Section 3.3.2, we put g = k⊕ TpM with k = span{(Rp)xy}x,y∈TpM (which is the Lie algebra of the holonomy group, as a consequence of the Ambrose-Singer Holonomy Theorem) and define a Lie bracket on g by

[B,C] = BC−CB for B,C ∈ k, [x,y] = (Rp)xy for x,y ∈ TpM, [A,z] = Az for A ∈ k and z ∈ TpM.