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

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

2009-11-30

arxiv.org/abs/0911.5655v3

Mathematics

Differential Geometry

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

Scientific paper

The study of quasi-K\"ahler 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-K\"ahler 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.

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