Characteristic Properties of Almost Hermitian Structures on Homogeneous Reductive Spaces

Homogeneous reductive almost Hermitian spaces are considered. For such spaces satisfying a certain simple algebraic condition, criteria providing simple descriptions of Kähler, nearly Kähler, almost Kähler, quasi-Kähler, and $G_1$ structures are obtained. It is found that, under this condition, Kähler structures can occur only on locally symmetric spaces and nearly Kähler structures, on naturally reductive spaces. Almost Kähler, quasi-Kähler, and $G_1$ structures are described by simple conditions imposed on the Nomizu function $\alpha$ of the Riemannian connection of a homogeneous reductive almost Hermitian space.