scholarly journals A Review on Unique Existence Theorems in Lightlike Geometry

Geometry ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-17
Author(s):  
K. L. Duggal
Keyword(s):  
Scalar Curvature ◽  
Ricci Tensor ◽  
Review Paper ◽  
Existence Theorems ◽  
Lorentzian Manifolds ◽  
Open Problems ◽  
Unique Existence ◽  
Null Curves ◽  
Levi Civita Connection ◽  
Lightlike Submanifolds

This is a review paper of up-to-date research done on the existence of unique null curves, screen distributions, Levi-Civita connection, symmetric Ricci tensor, and scalar curvature for a large variety of lightlike submanifolds of semi-Riemannian (in particular, Lorentzian) manifolds, supported by examples and an extensive bibliography. We also propose some open problems.

scholarly journals On a Metric Affine Manifold with Several Orthogonal Complementary Distributions

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 229
Author(s):  
Vladimir Rovenski ◽  
Sergey E. Stepanov
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Scalar Curvature ◽  
Existence Theorems ◽  
Linear Connection ◽  
Product Structure ◽  
Warped Products ◽  
Affine Manifold ◽  
Integral Formulas ◽  
Levi Civita Connection

A Riemannian manifold endowed with k>2 orthogonal complementary distributions (called here an almost multi-product structure) appears in such topics as multiply twisted or warped products and the webs or nets composed of orthogonal foliations. In this article, we define the mixed scalar curvature of an almost multi-product structure endowed with a linear connection, and represent this kind of curvature using fundamental tensors of distributions and the divergence of a geometrically interesting vector field. Using this formula, we prove decomposition and non-existence theorems and integral formulas that generalize results (for k=2) on almost product manifolds with the Levi-Civita connection. Some of our results are illustrated by examples with statistical and semi-symmetric connections.

scholarly journals A quarter-symmetric metric connection on almost contact B-metric manifolds

Filomat ◽  
2019 ◽  
Vol 33 (16) ◽  
pp. 5181-5190
Author(s):  
Şenay Bulut
Keyword(s):  
Scalar Curvature ◽  
Curvature Tensor ◽  
Ricci Tensor ◽  
Civita Connection ◽  
Metric Connection ◽  
Levi Civita Connection

The aim of this paper is to study the notion of a quarter-symmetric metric connection on an almost contact B-metric manifold (M,?,?,?,g). We obtain the relation between the Levi-Civita connection and the quarter-symmetric metric connection on (M,?,?,?,g).We investigate the curvature tensor, Ricci tensor and scalar curvature tensor with respect to the quarter-symmetric metric connection. In case the manifold (M,?,?,?,g) is a Sasaki-like almost contact B-metric manifold, we get some formulas. Finally, we give some examples of a quarter-symmetric metric connection.

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 On the Koszul formula in noncommutative geometry

2020 ◽  
Vol 32 (10) ◽  
pp. 2050032 ◽  
Author(s):  
Jyotishman Bhowmick ◽  
Debashish Goswami ◽  
Giovanni Landi
Keyword(s):  
Scalar Curvature ◽  
Noncommutative Geometry ◽  
Spectral Triple ◽  
Fuzzy Sphere ◽  
Civita Connection ◽  
Canonical Metric ◽  
Levi Civita Connection

We prove a Koszul formula for the Levi-Civita connection for any pseudo-Riemannian bilinear metric on a class of centered bimodule of noncommutative one-forms. As an application to the Koszul formula, we show that our Levi-Civita connection is a bimodule connection. We construct a spectral triple on a fuzzy sphere and compute the scalar curvature for the Levi-Civita connection associated to a canonical metric.

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.

Certain inequalities for the Casorati curvatures of submanifolds of generalized (k,μ)-space-forms

2018 ◽  
Vol 13 (02) ◽  
pp. 2050040
Author(s):  
Shyamal Kumar Hui ◽  
Pradip Mandal ◽  
Ali H. Alkhaldi ◽  
Tanumoy Pal
Keyword(s):  
Scalar Curvature ◽  
Space Form ◽  
Space Forms ◽  
Civita Connection ◽  
Metric Connection ◽  
Levi Civita Connection ◽  
Casorati Curvature ◽  
Optimal Inequalities

The paper deals with the study of Casorati curvature of submanifolds of generalized [Formula: see text]-space-form with respect to Levi-Civita connection as well as semisymmetric metric connection and derived two optimal inequalities between scalar curvature and Casorati curvature of such space forms. The equality cases are also considered.

scholarly journals Holomorphic Legendrian curves

2017 ◽  
Vol 153 (9) ◽  
pp. 1945-1986 ◽  
Author(s):  
Antonio Alarcón ◽  
Franc Forstnerič ◽  
Francisco J. López
Keyword(s):  
Riemann Surface ◽  
Contact Structure ◽  
Existence Theorems ◽  
Contact Manifold ◽  
Open Problems ◽  
General Existence ◽  
The Subject ◽  
Complex Contact Manifold ◽  
Holomorphic Contact ◽  
Legendrian Curves

In this paper we study holomorphic Legendrian curves in the standard holomorphic contact structure on$\mathbb{C}^{2n+1}$for any$n\in \mathbb{N}$. We provide several approximation and desingularization results which enable us to prove general existence theorems, settling some of the open problems in the subject. In particular, we show that every open Riemann surface$M$admits a proper holomorphic Legendrian embedding$M{\hookrightarrow}\mathbb{C}^{2n+1}$, and we prove that for every compact bordered Riemann surface$M={M\unicode[STIX]{x0030A}}\,\cup \,bM$there exists a topological embedding$M{\hookrightarrow}\mathbb{C}^{2n+1}$whose restriction to the interior is a complete holomorphic Legendrian embedding${M\unicode[STIX]{x0030A}}{\hookrightarrow}\mathbb{C}^{2n+1}$. As a consequence, we infer that every complex contact manifold$W$carries relatively compact holomorphic Legendrian curves, normalized by any given bordered Riemann surface, which are complete with respect to any Riemannian metric on$W$.

scholarly journals Time-Dependent Evolving Null Horizons of a Dynamical Spacetime

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
K. L. Duggal
Keyword(s):  
Black Hole ◽  
Energy Condition ◽  
Physical Models ◽  
Open Problems ◽  
Totally Umbilical ◽  
Unique Existence ◽  
Null Hypersurfaces ◽  
Isolated Horizons ◽  
Black Hole Spacetime ◽  
General Existence

Totally geodesic null hypersurfaces have been widely used in the study of isolated black holes. In this paper, we introduce a new quasilocal notion of a family of totally umbilical null hypersurfaces called evolving null horizons (ENH) of a dynamical spacetime, satisfied under an appropriate energy condition. We focus on a variety of examples of ENHs and in some cases establish their relation with event and isolated horizons. We also present two specific physical models of an ENH in a black hole spacetime. Beside the examples, for further study we propose two open problems on possible general existence of an ENH in a black hole spacetime and its canonical or unique existence. The results of this paper have ample scope of working on totally umbilical null hypersurfaces of Lorentzian and, in general, semiRiemannian manifolds.

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.

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