On Homogeneous Almost Kähler Einstein Manifolds of Negative Curvature

Abstract

A homogeneous almost Kähler manifold $M$ of negative curvature can be identified with a solvable Lie group $G$ with a left invariant metric $g$ and a left invariant almost complex structure $J$. We prove that if $g$ is an Einstein metric and $G$ is of Iwasawa type, then $J$ is integrable so that $M$ is Kähler, and hence is holomorphically isometric to a complex hyperbolic space of the same dimension.