scholarly journals Complete 2-transnormal hypersurfaces in a Kaehler manifold of negative constant holomorphic sectional curvature

1987 ◽  
Vol 11 (2) ◽  
pp. 361-366
Author(s):  
Fumiko Ohtsuka
Keyword(s):  
Sectional Curvature ◽  
Holomorphic Sectional Curvature ◽  
Kaehler Manifold ◽  
Negative Constant

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.

On Kaehler Immersions

1972 ◽  
Vol 24 (6) ◽  
pp. 1178-1182 ◽  
Author(s):  
Koichi Ogiue
Keyword(s):  
Projective Space ◽  
Sectional Curvature ◽  
Complex Projective Space ◽  
Holomorphic Sectional Curvature ◽  
Kaehler Manifold ◽  
Totally Geodesic ◽  
Complex Submanifold ◽  
Complex Projective

Let be an (n + p)-dimensional Kaehler manifold of constant holomorphic sectional curvature . B. O'Neill [3] proved the following result.PROPOSITION A. Let M be an n-dimensional complex submanifold immersed in . If and if the holomorphic sectional curvature of M with respect to the induced Kaehler metric is constant, then M is totally geodesic.He also gave the following example: There is a Kaehler imbedding of an w-dimensional complex projective space of constant holomorphic sectional curvature ½ into an -dimensional complex projective space of constant holomorphic sectional curvature 1. This shows that Proposition A is the best possible.

scholarly journals Complete Kähler–Einstein Metric on Stein Manifolds With Negative Curvature

2020 ◽  
Author(s):  
Man-Chun Lee
Keyword(s):  
Sectional Curvature ◽  
Ricci Flow ◽  
Negative Curvature ◽  
Einstein Metric ◽  
Holomorphic Sectional Curvature ◽  
Kähler Metrics ◽  
Stein Manifolds ◽  
Kahler Metrics ◽  
Negative Constant

Abstract We show the existence of complete negative Kähler–Einstein metric on Stein manifolds with holomorphic sectional curvature bounded from above by a negative constant. We prove that any Kähler metrics on such manifolds can be deformed to the complete negative Kähler–Einstein metric using the normalized Kähler–Ricci flow.

On Some Complex Submanifolds in Kaehler Manifolds

1974 ◽  
Vol 26 (6) ◽  
pp. 1442-1449 ◽  
Author(s):  
Masahiro Kon
Keyword(s):  
Sectional Curvature ◽  
Ricci Tensor ◽  
Holomorphic Sectional Curvature ◽  
Kaehler Manifold ◽  
Ricci Operator ◽  
Kaehler Manifolds ◽  
Complex Submanifold ◽  
Complex Submanifolds ◽  
Parallel Ricci Tensor ◽  
Local Result

The purpose of this paper is to give some conditions for complex submanifolds in a Kaehler manifold of constant holomorphic sectional curvature to be Einstein.For a complex hypersurface which is Einstein, Smyth [8] has obtained its classification and Chern [2] has proved the corresponding local result. Moreover, Takahashi [9] and Nomizu-Smyth [3] generalized this to a complex hypersurface with parallel Ricci tensor. We shall consider a condition weaker than the requirement that the Ricci tensor be parallel, that is we shall consider a complex submanifold with commuting curvature and Ricci operator, which condition was treated by Bishop-Goldberg [1].

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.

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