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.
关于黎曼均质结构的凯勒情形[3, 15、
18]的研究[1, 2, 6, 7、
13, 16] 等论文中进行了研究。Abbena 和 Garbiero [1] 给出了凯勒结构的分类。
给出了凯勒均相结构的分类,其中有四个基本类
[Kscr ]1,......,[Kscr ]4(见 [6, theorem
5-1] 的另一个证明,以及下文第 2 节的结果)。
本文旨在证明以下结果:定理 1-1.简单相连的不可还原同质凯勒流形具有
非消失的凯勒均相结构。
[Kscr ]2 [oplus ] [Kscr ]4if
且只有当它是复双曲空间,且配备了负常数全形截面的伯格曼度量时
负恒全形截面曲率的复双曲空间。