Almost K\"ahler 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 K\¨ahler structures on that group. The only K\¨ahler structure is the standard structure of the complex hyperbolic space and all
others are strictly almost K\¨ahler. We calculated the Chern connection and obtained the characterization by its torsion tensor. It is proved that the only SCF-soliton is the K\¨ahler one.