Locally homogeneous Riemannian manifolds and Cartan triples

Ann-dimensional Cartan triple is a triple (g, , ) consisting of a Lie subalgebra g of so(n) (endowed with the Killing form), a linear map : ⁿ g and a bilinear antisymmetric map ²( ⁿ , g), which together satisfy (1.25)–(1.28) of Section 1. LetM _n be the set ofmaximal n-dimensional Cartan triples, and letA _n be thenatural action of the orthogonal group O(n) onM _n (Section 3). One shows that there is a bijective mapping from the set of local isometry classes ofn-dimensional locally homogeneous Riemannian manifolds to the set of orbits ofA _n (Theorem 3.1(a)). Under this bijection, the classes of homogeneous Riemannian manifolds correspond to orbits ofclosed Cartan triples.