scholarly journals An example for the holomorphic sectional curvature of the Bergman metric

2010 ◽  
Vol 98 (2) ◽  
pp. 147-167 ◽  
Author(s):  
Żywomir Dinew
Keyword(s):  
Sectional Curvature ◽  
Holomorphic Sectional Curvature ◽  
Bergman Metric

scholarly journals The Bergman metric on a Thullen domain

1983 ◽  
Vol 89 ◽  
pp. 1-11 ◽  
Author(s):  
Kazuo Azukawa ◽  
Masaaki Suzuki
Keyword(s):  
Sectional Curvature ◽  
Holomorphic Sectional Curvature ◽  
Bergman Metric

In this paper we shall study the holomorphic sectional curvature of the Bergman metric on a domain

scholarly journals Characterizing the complex hyperbolic space by Kähler homogeneous structures

2000 ◽  
Vol 128 (1) ◽  
pp. 87-94 ◽  
Author(s):  
P. M. GADEA ◽  
A. MONTESINOS AMILIBIA ◽  
J. MUÑOZ MASQUÉ
Keyword(s):  
Sectional Curvature ◽  
Hyperbolic Space ◽  
Complex Hyperbolic Space ◽  
Homogeneous Structure ◽  
Holomorphic Sectional Curvature ◽  
Bergman Metric ◽  
Negative Constant ◽  
Homogeneous Structures ◽  
Simply Connected

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.

scholarly journals On a $K$-space of constant holomorphic sectional curvature

1973 ◽  
Vol 25 (3) ◽  
pp. 297-306 ◽  
Author(s):  
Yoshiyuki Watanabe ◽  
Kichiro Takamatsu
Keyword(s):  
Sectional Curvature ◽  
Holomorphic Sectional Curvature

Jacobi fields and geodesic spheres

1979 ◽  
Vol 82 (3-4) ◽  
pp. 233-240 ◽  
Author(s):  
L. Vanhecke ◽  
T. J. Willmore
Keyword(s):  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Vector Fields ◽  
Holomorphic Sectional Curvature ◽  
Principal Curvatures ◽  
Jacobi Fields ◽  
Geodesic Spheres ◽  
Jacobi Vector ◽  
Nearly Kähler ◽  
Nearly Kähler Manifolds

SynopsisThis is a contribution to the general problem of determining the extent to which the geometry of a riemannian manifold is determined by properties of its geodesic spheres. In particular we show that total umbilicity of geodesic spheres determines riemannian manifolds of constant sectional curvature; quasi-umbilicity of geodesic spheres determines Kähler and nearly-Kähler manifolds of constant holomorphic sectional curvature; and the condition that geodesic spheres have only two different principal curvatures, one having multiplicity 3, determines manifolds locally isometric to the quaternionic projective spaces. The use of Jacobi vector fields leads to a unified treatment of these different cases.

scholarly journals On totally umbilicalCR-submanifolds of a Kaehler manifold

1993 ◽  
Vol 16 (2) ◽  
pp. 405-408
Author(s):  
M. A. Bashir
Keyword(s):  
Sectional Curvature ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Lens Space ◽  
Holomorphic Sectional Curvature ◽  
Mean Curvature Vector ◽  
Kaehler Manifold ◽  
Totally Umbilical ◽  
The Mean

LetMbe a compact3-dimensional totally umbilicalCR-submanifold of a Kaehler manifold of positive holomorphic sectional curvature. We prove that if the length of the mean curvature vector ofMdoes not vanish, thenMis either diffeomorphic toS3orRP3or a lens spaceLp,q3.

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