Quasi-Kähler Chern-flat manifolds and complex 2-step nilpotent Lie algebras

2009/11/30 by Di Scala, Antonio J., Lauret, Jorge, Vezzoni, Luigi · 2 citations
#53B20 (Primary) #53C25 (Secondary) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.0911.5655

Abstract

The study of quasi-Kähler Chern-flat almost Hermitian manifolds is strictly related to the study of anti-bi-invariant almost complex Lie algebras. In the present paper we show that quasi-Kähler Chern-flat almost Hermitian structures on compact manifolds are in correspondence to complex parallelisable Hermitian structures satisfying the second Gray identity. From an algebraic point of view this correspondence reads as a natural correspondence between anti-bi-invariant almost complex structures on Lie algebras to bi-invariant complex structures. Some natural algebraic problems are approached and some exotic examples are carefully described.

Cited by

• Complex structures on two-step nilpotent Lie groups
• On balanced Hermitian threefolds with parallel Bismut torsion

Related

• On Hermitian manifolds with constant mixed curvature
• Hermitian structures on a class of almost nilpotent solvmanifolds
• Homogeneous almost Kähler manifolds and the Chern-Einstein equation
• Notes concerning Kähler and anti-Kähler structures on quasi-statistical manifolds
• Streets-Tian Conjecture on several special types of Hermitian manifolds