Homogeneous Einstein metrics on non-Kahler C-spaces

JOURNAL OF GEOMETRY AND PHYSICS

卷:160

Article

Chrysikos, Ioannis¹  Sakane, Yusuke² 

[1] Univ Hradec Kralove, Fac Sci, Rokitanskeho 62, Hradec Kralove 50003, Czech Republic

[2] Osaka Univ, Dept Pure & Appl Math, Grad Sch Informat Sci & Technol, Suita, Osaka 5650871, Japan

关键词: Homogeneous spaces;    Invariant Einstein metrics;    Non-Kahler C-spaces;    Torus bundles;   

DOI  :  10.1016/j.geomphys.2020.103996

来源: Elsevier

【 摘 要 】

We study homogeneous Einstein metrics on indecomposable non-Kahler C-spaces, i.e. even-dimensional torus bundles M = G/H with rank G > 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-l,M-m,M-n := SU(l + m + n)/SU(l) x SU(m) x SU(n) , l, m, n is an element of Z(+) and examine several of its geometric properties. We show that invariant metrics on M-l,M-m,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 M-l,M-m,M-n. For l = m we show the existence of two SU(2m + n)-invariant Einstein metrics on M-m,M-m,M-n, and for l = m = n we obtain four SU(3n)-invariant Einstein metrics on M-n,M-n,M-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. (C) 2020 Elsevier B.V. All rights reserved.

【 授权许可】

Free