Homogeneous Einstein metrics on non-Kähler C-spaces

Abstract

We study homogeneous Einstein metrics on indecomposable non-Kähler C-spaces, i.e. even-dimensional torus bundles M=G∕H with rankG>rankH over flag manifolds F=G∕K of a compact simple Lie group G. Based on the theory of painted Dynkin diagrams we present the classification of such spaces. Next we focus on the family Mℓ,m,n≔SU(ℓ+m+n)∕SU(ℓ)×SU(m)×SU(n),ℓ,m,n∈Z+and examine several of its geometric properties. We show that invariant metrics on Mℓ,m,n are not diagonal and beyond certain exceptions their parametrization depends on six real parameters. By using such an invariant Riemannian metric, we compute the diagonal and the non-diagonal part of the Ricci tensor and present explicitly the algebraic system of the homogeneous Einstein equation. For general positive integers ℓ,m,n, by applying mapping degree theory we provide the existence of at least one SU(ℓ+m+n)-invariant Einstein metric on Mℓ,m,n. For ℓ=m we show the existence of two SU(2m+n)-invariant Einstein metrics on Mm,m,n, and for ℓ=m=n we obtain four SU(3n)-invariant Einstein metrics on Mn,n,n. We also examine the isometry problem for these metrics, while for a plethora of cases induced by fixed ℓ,m,n, we provide the numerical form of all non-isometric invariant Einstein metrics.