Almost Kähler structures on complex hyperbolic space

Abstract

Complex hyperbolic space has the structure of a solvable Lie group. Based on the recent classification of Riemannian left-invariant metrics we describe all left-invariant almost Kähler structures on that group. The only Kähler structure is the standard structure of the complex hyperbolic space and all others are strictly almost Kähler. We calculated the Chern connection and obtained the characterization by its torsion tensor. It is proved that the only SCF-soliton is the Kähler one.