scholarly journals Screen conformal half-lightlike submanifolds

2004 ◽  
Vol 2004 (68) ◽  
pp. 3737-3753 ◽  
Author(s):  
K. L. Duggal ◽  
B. Sahin
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Ricci Tensor ◽  
Shape Operator ◽  
Close Relation ◽  
Screen Distribution ◽  
Lightlike Submanifolds ◽  
Half Lightlike Submanifold ◽  
Lightlike Submanifold

We study some properties of a half-lightlike submanifoldM, of a semi-Riemannian manifold, whose shape operator is conformal to the shape operator of its screen distribution. We show that any screen distributionS(TM)ofMis integrable and the geometry ofMhas a close relation with the nondegenerate geometry of a leaf ofS(TM). We prove some results on symmetric induced Ricci tensor and null sectional curvature of this class.

scholarly journals Co-screen conformal half-lightlike submanifolds

2013 ◽  
Vol 44 (4) ◽  
pp. 431-444
Author(s):  
Yaning Wang ◽  
Ximin Liu
Keyword(s):  
Riemannian Manifold ◽  
Curvature Tensor ◽  
Sufficient Conditions ◽  
Shape Operator ◽  
Geometric Condition ◽  
Screen Distribution ◽  
Unique Existence ◽  
Lightlike Submanifolds ◽  
Half Lightlike Submanifold ◽  
Lightlike Submanifold

In this paper, we introduce and study the geometry of half lightlike submanifold $M$ of a semi-Riemannian manifold $\overline{M}$ satisfying that the shape operator of screen transversal bundle is conformal to the shape operator of lightlike transversal bundle of $M$. Using this geometric condition we obtain some results to characterize the unique existence of screen distribution of $M$, also, we present some sufficient conditions for the induced Ricci curvature tensor of $M$ to be symmetric.

scholarly journals On the geometry of lightlike submanifolds of indefinite statistical manifolds

2020 ◽  
Vol 17 (07) ◽  
pp. 2050099
Author(s):  
Varun Jain ◽  
Amrinder Pal Singh ◽  
Rakesh Kumar
Keyword(s):  
Sectional Curvature ◽  
Classical Theory ◽  
Ricci Tensor ◽  
Statistical Manifolds ◽  
Lightlike Submanifolds ◽  
Statistical Submanifold ◽  
Lightlike Submanifold

We study lightlike submanifolds of indefinite statistical manifolds. Contrary to the classical theory of submanifolds of statistical manifolds, lightlike submanifolds of indefinite statistical manifolds need not to be statistical submanifold. Therefore, we obtain some conditions for a lightlike submanifold of indefinite statistical manifolds to be a lightlike statistical submanifold. We derive the expression of statistical sectional curvature and finally obtain some conditions for the induced statistical Ricci tensor on a lightlike submanifold of indefinite statistical manifolds to be symmetric.

Some characterizations on minimal lightlike submanifolds

2017 ◽  
Vol 14 (07) ◽  
pp. 1750103 ◽  
Author(s):  
Sangeet Kumar
Keyword(s):  
Sectional Curvature ◽  
Ricci Tensor ◽  
Minimal Condition ◽  
Characterization Theorems ◽  
Lightlike Submanifolds ◽  
Lightlike Submanifold

The present paper deals with the study of minimal lightlike submanifolds. We investigate a class of lightlike submanifolds namely, generic lightlike submanifolds under the minimal condition. We give one nontrivial example for minimal generic lightlike submanifolds and derive some characterization theorems for a generic lightlike submanifold to be a minimal lightlike submanifold. We also establish some conditions for the distributions for generic lightlike submanifolds to be minimal. We further derive the expressions for sectional curvature, null sectional curvature and induced Ricci tensor for a minimal lightlike submanifold. Finally, we prove that for a minimal lightlike submanifold, the null sectional curvature vanishes and the induced Ricci tensor is symmetric.

scholarly journals On Co-screen Conformality of 1-lightlike Submanifolds in a Lorentzian Manifold

2015 ◽  
Vol 7 (2) ◽  
pp. 76
Author(s):  
Erol Kilic ◽  
Sadik Keles ◽  
Mehmet Gulbahar
Keyword(s):  
Ricci Tensor ◽  
Lorentzian Manifold ◽  
Conformal Killing ◽  
Screen Distribution ◽  
Locally Conformal ◽  
Lightlike Submanifolds

In this paper, the co-screen conformal 1-lightlike submanifolds of a Lorentzian manifoldare introduced as a generalization of co-screen locally half-lightlike submanifolds in(Wang,  Wang {\&} Liu, 2013; Wang {\&} Liu, 2013) and two examples are given whichone is co-screen locally conformal andthe other is not. Some results are obtained on these submanifolds whichthe co-screen distribution is conformal Killing on the ambient manifold.The induced Ricci tensor of  co-screen conformal 1-lightlike submanifolds isinvestigated.

scholarly journals Some properties of lightlike submanifolds of semi-Riemannian manifolds

2010 ◽  
Vol 43 (3) ◽  
Author(s):  
Rakesh Kumar ◽  
Rachna Rani ◽  
R. K. Nagaich
Keyword(s):  
Riemannian Manifolds ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Totally Geodesic ◽  
Screen Distribution ◽  
Necessary And Sufficient ◽  
Lightlike Submanifolds ◽  
Lightlike Submanifold

AbstractWe initially obtain various relations and then establish necessary and sufficient condition for the integrability of screen distribution of a lightlike submanifold. We also establish necessary and sufficient condition for a lightlike submanifold to be totally geodesic.

scholarly journals Indefinite Almost Paracontact Metric Manifolds

2010 ◽  
Vol 2010 ◽  
pp. 1-19 ◽  
Author(s):  
Mukut Mani Tripathi ◽  
Erol Kılıç ◽  
Selcen Yüksel Perktaş ◽  
Sadık Keleş
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Curvature Tensor ◽  
Ricci Tensor ◽  
Sasakian Manifold ◽  
Constant Sectional Curvature ◽  
Sasakian Manifolds ◽  
Sasakian Structure

We introduce the concept of (ε)-almost paracontact manifolds, and in particular, of (ε)-para-Sasakian manifolds. Several examples are presented. Some typical identities for curvature tensor and Ricci tensor of (ε)-para Sasakian manifolds are obtained. We prove that if a semi-Riemannian manifold is one of flat, proper recurrent or proper Ricci-recurrent, then it cannot admit an (ε)-para Sasakian structure. We show that, for an (ε)-para Sasakian manifold, the conditions of being symmetric, semi-symmetric, or of constant sectional curvature are all identical. It is shown that a symmetric spacelike (resp., timelike) (ε)-para Sasakian manifoldMnis locally isometric to a pseudohyperbolic spaceHνn(1)(resp., pseudosphereSνn(1)). At last, it is proved that for an (ε)-para Sasakian manifold the conditions of being Ricci-semi-symmetric, Ricci-symmetric, and Einstein are all identical.

Radical screen transversal lightlike submanifolds of indefinite Kaehler manifolds admitting a quarter-symmetric non-metric connection

2018 ◽  
Vol 15 (02) ◽  
pp. 1850024
Author(s):  
Garima Gupta ◽  
Rakesh Kumar ◽  
Rakesh Kumar Nagaich
Keyword(s):  
Product Manifold ◽  
Necessary And Sufficient Condition ◽  
Kaehler Manifold ◽  
Screen Distribution ◽  
Kaehler Manifolds ◽  
Metric Connection ◽  
Necessary And Sufficient ◽  
Characterization Theorems ◽  
Lightlike Submanifolds ◽  
Lightlike Submanifold

We study radical screen transversal ([Formula: see text])-lightlike submanifolds of an indefinite Kaehler manifold admitting a quarter-symmetric non-metric connection and obtain a necessary and sufficient condition for the screen distribution of a radical [Formula: see text]-lightlike submanifold to be integrable. We also study totally umbilical radical [Formula: see text]-lightlike submanifolds and obtain some characterization theorems for a radical [Formula: see text]-lightlike submanifold to be a lightlike product manifold. Finally, we establish some results regarding the vanishes of null sectional curvature.

Non-immersibility of a space form as a totally umbilical hypersurface

1988 ◽  
Vol 109 (1-2) ◽  
pp. 17-21
Author(s):  
Masafumi Okumura ◽  
Hiroshi Takahashi
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Sectional Curvature ◽  
Mean Curvature ◽  
Ricci Tensor ◽  
Constant Mean Curvature ◽  
Space Form ◽  
Einstein Manifold ◽  
Totally Umbilical ◽  
Ambient Manifold

SynopsisSuppose that a space form is immersed into another Riemannian manifold as a totally umbilical hypersurface with constant mean curvature. Then, in the ambient manifold, the lengthof the curvature tensor, that of the Ricci tensor and the scalar curvature must satisfy an inequality. In this paper the authors proved the inequality. As applications of the inequality, some immersibility problems are investigated. For example, it is proved that if a space form is immersed in an Einstein manifold as a totally umbilical hypersurface, then the Einstein manifold has constant sectional curvature along the hypersurface. Moreover, it is proved that a space form cannot be immersed into some Kaehlerian manifolds as a totally umbilical hypersurface with constant mean curvature.

scholarly journals Characterization of Holomorphic Bisectional Curvature ofGCR-Lightlike Submanifolds

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Sangeet Kumar ◽  
Rakesh Kumar ◽  
R. K. Nagaich
Keyword(s):  
Sectional Curvature ◽  
Complex Space ◽  
Space Form ◽  
Complex Space Form ◽  
Holomorphic Sectional Curvature ◽  
Bisectional Curvature ◽  
Holomorphic Bisectional Curvature ◽  
Characterization Theorems ◽  
Lightlike Submanifolds ◽  
Lightlike Submanifold

We obtain the expressions for sectional curvature, holomorphic sectional curvature, and holomorphic bisectional curvature of aGCR-lightlike submanifold of an indefinite Kaehler manifold. We discuss the boundedness of holomorphic sectional curvature ofGCR-lightlike submanifolds of an indefinite complex space form. We establish a condition for aGCR-lightlike submanifold of an indefinite complex space form to be null holomorphically flat. We also obtain some characterization theorems for holomorphic sectional and holomorphic bisectional curvature.

scholarly journals Half lightlike submanifolds of a semi-Riemannian manifold of quasi-constant curvature

Filomat ◽  
2016 ◽  
Vol 30 (7) ◽  
pp. 1737-1745
Author(s):  
Dae Jin ◽  
Jae Lee
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Constant Curvature ◽  
Curvature Vector ◽  
Conformal Killing ◽  
Totally Geodesic ◽  
Totally Umbilical ◽  
Screen Distribution ◽  
Lightlike Submanifolds

We study the geometry of half lightlike submanifolds (M,g,S(TM), S(TM?)) of a semi-Riemannian manifold (M~,g~) of quasi-constant curvature subject to the following conditions; (1) the curvature vector field ? of M~ is tangent to M, (2) the screen distribution S(TM) of M is either totally geodesic or totally umbilical in M, and (3) the co-screen distribution S(TM?) of M is a conformal Killing distribution.

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