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Characterizing the complex hyperbolic space by Kähler homogeneous structures

Published online by Cambridge University Press:  01 January 2000

P. M. GADEA ,
A. MONTESINOS AMILIBIA  and
J. MUÑOZ MASQUÉ
P. M. GADEA
Affiliation:
IMAFF, CSIC, Serrano 123, 28006 – Madrid, Spain
A. MONTESINOS AMILIBIA
Affiliation:
Department of Geometry and Topology, Faculty of Mathematics, 46100 – Burjasot, Valencia, Spain
J. MUÑOZ MASQUÉ
Affiliation:
IFA, CSIC, Serrano 144, 28006 – Madrid, Spain
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Abstract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 128 , Issue 1 , January 2000 , pp. 87 - 94
Copyright
The Cambridge Philosophical Society 2000

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