The Kähler case of Riemannian homogeneous structures [3, 15,
18] has been studied in [1, 2, 6, 7,
13, 16], among other papers. Abbena and Garbiero [1]
gave a classification of Kähler homogeneous structures, which has four primitive classes
[Kscr ]1, …, [Kscr ]4 (see [6, theorem
5·1] for another proof and Section 2 below for the result).
The purpose of the present paper is to prove the following result:THEOREM 1·1. A simply connected irreducible homogeneous Kähler manifold admits
a nonvanishing Kähler homogeneous structure in Abbena–Garbiero's class
[Kscr ]2 [oplus ] [Kscr ]4if
and only if it is the complex hyperbolic space equipped with the Bergman metric of
negative constant holomorphic sectional curvature.
The Bergman metric on a Thullen domain