Locally homogeneous Riemannian manifolds and Cartan triples

Abstract

Ann-dimensional Cartan triple is a triple (g, Γ,\(\bar \Omega \)) consisting of a Lie subalgebra g of so(n) (endowed with the Killing form), a linear map Γ:ℝn → g⊥ and a bilinear antisymmetric map Ω e Λ2(ℝn, g), which together satisfy (1.25)–(1.28) of Section 1. LetMn be the set ofmaximal n-dimensional Cartan triples, and letAn be thenatural action of the orthogonal group O(n) onMn (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 ofAn (Theorem 3.1(a)). Under this bijection, the classes of homogeneous Riemannian manifolds correspond to orbits ofclosed Cartan triples.