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

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.

arXiv comment15 pages. Final version to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci