Homogeneous Einstein metrics on non-Kahler C-spaces

CHRYSIKOS, Ioannis and Yusuke SAKANE. Homogeneous Einstein metrics on non-Kahler C-spaces. Journal of Geometry and Physics. Amsterdam: Elsevier Science BV, 2021, vol. 160, February, p. 1-31. ISSN 0393-0440. Available from: https://dx.doi.org/10.1016/j.geomphys.2020.103996.

Basic information

• Original name: Homogeneous Einstein metrics on non-Kahler C-spaces
• Authors: CHRYSIKOS, Ioannis and Yusuke SAKANE.
• Edition: Journal of Geometry and Physics, Amsterdam, Elsevier Science BV, 2021, 0393-0440.

Other information

• Original language: English
• Type of outcome: Article in a journal
• Field of Study: 10101 Pure mathematics
• Country of publisher: Netherlands
• Confidentiality degree: is not subject to a state or trade secret
• WWW: URL
• Impact factor: Impact factor: 1.380
• Doi: http://dx.doi.org/10.1016/j.geomphys.2020.103996
• UT WoS: 000623891500006
• Keywords in English: Homogeneous spaces; Invariant Einstein metrics; Non-Kahler C-spaces; Torus bundles
• Tags: RIV ne
• Tags: International impact, Reviewed

Abstract

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.