Homogeneous Einstein metrics on non-Kähler C-spaces
| Autor: | Ioannis Chrysikos, Yusuke Sakane |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Flag (linear algebra)
Simple Lie group 010102 general mathematics General Physics and Astronomy Rank (differential topology) Isometry (Riemannian geometry) 01 natural sciences Combinatorics symbols.namesake 0103 physical sciences symbols 010307 mathematical physics Geometry and Topology 0101 mathematics Einstein Invariant (mathematics) Indecomposable module Mathematical Physics Ricci curvature Mathematics |
| Zdroj: | Journal of Geometry and Physics. 160:103996 |
| ISSN: | 0393-0440 |
| DOI: | 10.1016/j.geomphys.2020.103996 |
| Popis: | 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 , 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 M l , 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 , n . For l = m we show the existence of two SU ( 2 m + n ) -invariant Einstein metrics on M m , m , n , and for l = m = n we obtain four SU ( 3 n ) -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 l , m , n , we provide the numerical form of all non-isometric invariant Einstein metrics. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|
Full Text from ScienceDirect