Homogeneous Kähler manifolds of non-degenerate Ricci curvature
According to the recently proven fundamental conjecture for homogeneous Kähler manifolds, every such manifold is analytically, but in general not as Kähler manifold, equivalent with the product of a flat homogeneous Kähler manifold, a simply connected compact homogeneous Kähler manifold and a homogeneous bounded domain. It is well known that the Ricci curvature for homogeneous Kähler manifolds is 0 for the flat type, positive for the compact type and negative for the domain type. The goal of this paper is to investigate homogeneous Kähler manifolds for which the Ricci curvature is nondegenerate. It is fairly easy to show (Theorem 1.1) that the Ricci curvature is nondegenerate if and only if the flat part does not occur in the product mentioned above. The main part of the paper is devoted to prove (mostly by Lie algebraic arguments): let (M,\(\phi\)) be a Kähler manifold and let G be a transitive group of holomorphic isometries. Then the Ricci curvature of (M,\(\phi\)) is nondegenerate if and only if the Kähler form \(\phi\) of M is exact. Hence in particular, a Kähler algebra (g,k,j,\(\phi\)) is nondegenerate if and only if (g,k,j) is a j-algebra.
scientific article; zbMATH DE number 4169616
scientific article