Homogeneous Einstein metrics on non-Kähler C-spaces

Abstract

We study homogeneous Einstein metrics on indecomposable non-Kählerian C-spaces, i.e. even-dimensional torus bundles $M=G/H$ with $\mathsf{rank}G>\mathsf{rank}H$ 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 \[ M_{\ell, m, n}:=\mathsf{SU}(\ell+m+n)/\mathsf{SU}(\ell)\times\mathsf{SU}(m)\times\mathsf{SU}(n)\,,\quad \ell, m, n\in\mathbb{Z}_{+} \] and examine several of its geometric properties. We show that invariant metrics on $M_{\ell ,m,n}$ 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 $\ell ,m,n$, by applying mapping degree theory we provide the existence of at least one $\mathsf{SU}(\ell +m+n)$-invariant Einstein metric on $M_{\ell ,m,n}$. For $\ell =m$ we show the existence of two $\mathsf{SU}(2m+n)$ invariant Einstein metrics on $M_{m,m,n}$, and for $\ell =m=n$ we obtain four $\mathsf{SU}(3n)$-invariant Einstein metrics on $M_{n,n,n}$. We also examine the isometry problem for these metrics, while for a plethora of cases induced by fixed $\ell ,m,n$, we provide the numerical form of all non-isometric invariant Einstein metrics.

42 pages

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