Einstein-Kaehler Manifolds Immersed in a Complex Projective Space

1976 ◽  
Vol 28 (1) ◽  
pp. 1-8 ◽  
Author(s):  
Hisao Nakaga
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Complex Space ◽  
Space Form ◽  
Complex Space Form ◽  
Local Version ◽  
Kaehler Manifold ◽  
Kaehler Manifolds ◽  
Simply Connected ◽  
Complex Projective

A Kaehler manifold of constant holomorphic curvature is called a complex space form. By a Kaehler submanifold we mean a complex submanifold with the induced Kaehler metric. B. Smyth [5] has studied a complete Einstein- Kaehler hypersurface in a complete and simply connected complex space form and classified completely the hypersurface. The local version of this result has been shown to be true by S. S. Chern [1], and partially by T. Takahashi [6] independently.

scholarly journals On some inequalities for submanifolds of Bochner-Kaehler manifolds

Filomat ◽  
2017 ◽  
Vol 31 (18) ◽  
pp. 5511-5523
Author(s):  
Mehraj Lone ◽  
Mohammed Jamali ◽  
Mohammad Shahid
Keyword(s):  
Mean Curvature ◽  
Complex Space ◽  
Space Form ◽  
Real Space ◽  
Complex Space Form ◽  
Warping Function ◽  
Kaehler Manifold ◽  
Kaehler Manifolds

Chen established sharp inequalities between certain Riemannian invariants and the squared norm of mean curvature for submanifolds in real space form as well as in complex space form. In this paper we generalize Chen inequalities for submanifolds of Bochner-Kaehler manifolds. Moreover, we study CRwarped product submanifolds of Bochner-Kaehler manifold and establish an inequality for the Laplacian of the warping function, from which we conclude some obstructions to the existence of such immersions.

scholarly journals Special lightlike hypersurfaces of indefinite Kaehler manifolds

Filomat ◽  
2016 ◽  
Vol 30 (7) ◽  
pp. 1919-1930 ◽  
Author(s):  
Dae Jin
Keyword(s):  
Complex Space ◽  
Space Form ◽  
Complex Space Form ◽  
Kaehler Manifold ◽  
Kaehler Manifolds ◽  
Almost Complex ◽  
Lightlike Hypersurfaces

In this paper, we define three types of lightlike hypersurfaces of an indefinite Kaehler manifold, which are called Hopf, recurrent and Lie recurrent lightlike hypersurfaces. After that we provide several new results on such three type lightlike hypersurfaces of an indefinite Kaehler manifold or an indefinite almost complex space form.

scholarly journals Holomorphic Mappings into Projective Space with Lacunary Hyperplanes

1973 ◽  
Vol 50 ◽  
pp. 199-216 ◽  
Author(s):  
Peter Kiernan ◽  
Shoshichi Kobayashi
Keyword(s):  
Projective Space ◽  
General Position ◽  
Complex Projective Space ◽  
Complex Space ◽  
General Setting ◽  
Holomorphic Mappings ◽  
Complex Projective

In this note, we shall examine some results of Bloch [2] and Cartan [3] concerning complex projective space minus hyperplanes in general position. The purpose is to restate their results in a more general setting by using the intrinsic pseudo-distance defined on a complex space [16] and the concept of tautness introduced by Wu in [18]. In the process we shall generalize some results of Dufresnoy [4] and Fuj imoto [7].

scholarly journals Chen's problem on mixed foliate CR-submanifolds

1989 ◽  
Vol 40 (1) ◽  
pp. 157-160 ◽  
Author(s):  
Mohammed Ali Bashir
Keyword(s):  
Complex Space ◽  
Space Form ◽  
Complex Space Form ◽  
Complex Submanifold ◽  
Hyperbolic Complex ◽  
Real Submanifold ◽  
Totally Real Submanifold ◽  
Simply Connected ◽  
Totally Real ◽  
Cr Submanifolds

We prove that the simply connected compact mixed foliate CR-submanifold in a hyperbolic complex space form is either a complex submanifold or a totally real submanifold. This is the problem posed by Chen.

LEVI-UMBILICAL REAL HYPERSURFACES IN A COMPLEX SPACE FORM

2016 ◽  
Vol 229 ◽  
pp. 99-112 ◽  
Author(s):  
JONG TAEK CHO ◽  
MAKOTO KIMURA
Keyword(s):  
Complex Space ◽  
Space Form ◽  
Levi Form ◽  
Complex Space Form ◽  
Real Hypersurfaces ◽  
Complex Projective Plane ◽  
Nonzero Constant ◽  
Local Construction ◽  
Complex Projective

We give a classification of Levi-umbilical real hypersurfaces in a complex space form $\widetilde{M}_{n}(c)$, $n\geqslant 3$, whose Levi form is proportional to the induced metric by a nonzero constant. In a complex projective plane $\mathbb{C}\mathbb{P}^{2}$, we give a local construction of such hypersurfaces and moreover, we give new examples of Levi-flat real hypersurfaces in $\mathbb{C}\mathbb{P}^{2}$.

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.

Examples of simply-connected K-contact non-Sasakian manifolds of dimension 5

2015 ◽  
Vol 12 (03) ◽  
pp. 1550027 ◽  
Author(s):  
Jin Hong Kim
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Contact Manifold ◽  
Existence Problem ◽  
Sasakian Manifolds ◽  
Contact Manifolds ◽  
Sasakian Structure ◽  
Simply Connected ◽  
Complex Projective

The existence of compact simply-connected K-contact, but not Sasakian, manifolds has been unknown only for dimension 5. The aim of this paper is to show that the Kollár's simply-connected example which is a Seifert bundle over the complex projective space ℂℙ2 and does not carry any Sasakian structure is actually a K-contact manifold. As a consequence, we affirmatively answer the above existence problem in dimension 5, establishing that there are infinitely many compact simply-connected K-contact manifolds of dimension 5 which do not carry a Sasakian structure.

scholarly journals A Theorem on Analytic Mappings of Complex Manifolds

1966 ◽  
Vol 27 (2) ◽  
pp. 543-557 ◽  
Author(s):  
Minoru Kurita
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Complex Space ◽  
Main Idea ◽  
Complex Manifolds ◽  
Normal Contact ◽  
Contact Metric Structure ◽  
Complex Projective ◽  
Analytic Mappings

We prove in this paper a theorem on analytic mappings of the complex space Cn into the complex projective space Pn. The theorem is closely related to that of S. S. Chern in [1], and the main idea of the proof is the same with the latter, though the calculations are rather different. The background of our calculation is the normal contact metric structure which was found by S. Sasaki and Y. Hatakeyama [4].

scholarly journals OnCR-Lightlike Product of an Indefinite Kaehler Manifold

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Rakesh Kumar ◽  
Jasleen Kaur ◽  
R. K. Nagaich
Keyword(s):  
Euclidean Space ◽  
Complex Space ◽  
Space Form ◽  
Complex Space Form ◽  
Kaehler Manifold ◽  
Lightlike Submanifolds ◽  
Lightlike Submanifold

We have studied mixed foliateCR-lightlike submanifolds andCR-lightlike product of an indefinite Kaehler manifold and also obtained relationship between them. Mixed foliateCR-lightlike submanifold of indefinite complex space form has also been discussed and showed that the indefinite Kaehler manifold becomes the complex semi-Euclidean space.

THE SETS OF POINTS WITH BOUNDED ORBITS FOR GENERALIZED CHEBYSHEV MAPPINGS

2001 ◽  
Vol 11 (01) ◽  
pp. 91-107
Author(s):  
KEISUKE UCHIMURA
Keyword(s):  
Dynamical Systems ◽  
Projective Space ◽  
Complex Variable ◽  
Complex Projective Space ◽  
Cantor Set ◽  
Quadratic Polynomials ◽  
Holomorphic Map ◽  
Simply Connected ◽  
Complex Projective ◽  
Set Of Points

We study the dynamical systems given by generalized Chebyshev mappings [Formula: see text] and show that (1) the set of points with bounded orbits of Fc(z) is connected and its complement in C∪{∞} is simply connected if and only if -4 ≤ c ≤ 2; (2) if c > 2, then the set of points with bounded orbits of Fc(z) is Cantor set. These results are the analogue of the theory of filled Julia sets of quadratic polynomials in one complex variable. We show that the mapping Fc(z) has relation to an important holomorphic map on the complex projective space P2.

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