scholarly journals On Einstein null hypersurfaces of $(LCS)_{n+2}$-space forms

2019 ◽  
Author(s):  
Samuel Ssekajja
Keyword(s):  
Ricci Tensor ◽  
Null Hypersurface ◽  
Space Forms ◽  
Null Hypersurfaces ◽  
Geometric Conditions ◽  
Null Curves ◽  
Products Of Spheres

We show that ascreen null hypersurfaces of an $(n+2)$-dimensional Lorentzian concircular structure $(LCS)_{n+2}$-manifold admits an induced Ricci tensor. We, therefore, prove, under some geometric conditions, that an Einstein ascreen null hypersurface is locally a product of null curves and products of spheres.

scholarly journals A characterization of minimal ascreen null hypersurfaces of $(LCS)$-space forms

2019 ◽  
Author(s):  
Samuel Ssekajja
Keyword(s):  
Constant Curvature ◽  
Ricci Tensor ◽  
Space Form ◽  
Null Hypersurface ◽  
Space Forms ◽  
Screen Distribution ◽  
Null Hypersurfaces ◽  
Null Curve

In the present paper, we study nontotally geodesic minimal ascreen null hypersurface, $M$, of a Lorentzian concircular structure $(LCS)$-space form of constant curvature $0$ or $1$. We prove that; if the Ricci tensor of $M$ is parallel with respect to any leaf of its screen distribution, then $M$ is isometric to a product of a null curve and spheres.

scholarly journals Geometry of isoparametric null hypersurfaces of Lorentzian manifolds

2019 ◽  
Author(s):  
Samuel Ssekajja
Keyword(s):  
Ambient Space ◽  
Null Hypersurface ◽  
Principal Curvatures ◽  
Lorentzian Manifolds ◽  
Space Forms ◽  
Lorentzian Space ◽  
Null Hypersurfaces ◽  
Shape Operators ◽  
Lorentzian Space Forms

We define two types of null hypersurfaces as; isoparametric and quasi isoparametric null hypersurfaces of Lorentzian space forms, based on the two shape operators associated with a null hypersurface. We prove that; on any screen conformal isoparametric null hypersurface, the screen geodesics lie on circles in the ambient space. Furthermore, we prove that the screen distributions of isoparametric (or quasi-parametric) null hypersurfaces with at most two principal curvatures are generally Riemannian products. Several examples are also given to illustrate the main concepts.

scholarly journals Null hypersurfaces in indefinite nearly Kaehlerian Finsler spaces

2019 ◽  
Author(s):  
Samuel Ssekajja
Keyword(s):  
Vector Fields ◽  
Finsler Space ◽  
Second Fundamental Form ◽  
Finsler Spaces ◽  
Space Forms ◽  
Null Hypersurfaces ◽  
Geometric Conditions ◽  
Cr Structure ◽  
Sectional Curvatures ◽  
New Inequalities

We study the geometry of null hypersurfaces, $M$, in indefinite nearly Kaehlerian Finsler space forms $\mathbb{F}^{2n}$. We prove new inequalities involving the point-wise vertical sectional curvatures of $\mathbb{F}^{2n}$, based on two special vector fields on an umbilic hypersurface. Such inequalities generalize some known results on null hypersurfaces of Kaehlerian space forms. Furthermore, under some geometric conditions, we show that the null hypersurface $(M, B)$, where $B$ is the local second fundamental form of $M$, is locally isometric to the null product $M_{D}\times M_{D'}$, where $M_{D}$ and $M_{D'}$ are the leaves of the distributions $D$ and $D'$ which constitutes the natural null-CR structure on $M$.

scholarly journals Some results on null hypersurfaces in $(LCS)$-manifolds

2019 ◽  
Author(s):  
Samuel Ssekajja
Keyword(s):  
Vector Field ◽  
Ricci Tensor ◽  
Characteristic Vector ◽  
Null Hypersurface ◽  
Conformally Flat ◽  
Totally Geodesic ◽  
Null Hypersurfaces ◽  
Characteristic Vector Field

We prove that a Lorentzian concircular structure $ (LCS)$-manifold does not admit any null hypersurface which is tangential or transversal to its characteristic vector field. Due to the above, we focus on its ascreen null hypersurfaces, and show that such hypersurfaces admit a symmetric Ricci tensor. Furthermore, we prove that there is no any totally geodesic ascreen null hypersurfaces of a conformally flat $(LCS)$-manifold.

scholarly journals Analysis of Obata’s Differential Equations on Pointwise Semislant Warped Product Submanifolds of Complex Space Forms via Ricci Curvature

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Amira A. Ishan
Keyword(s):  
Differential Equations ◽  
Ricci Curvature ◽  
Warped Product ◽  
Complex Space ◽  
Space Forms ◽  
Geometric Conditions ◽  
Isometric To A Sphere ◽  
Complex Space Forms

The present paper studies the applications of Obata’s differential equations on the Ricci curvature of the pointwise semislant warped product submanifolds. More precisely, by analyzing Obata’s differential equations on pointwise semislant warped product submanifolds, we demonstrate that, under certain conditions, the base of these submanifolds is isometric to a sphere. We also look at the effects of certain differential equations on pointwise semislant warped product submanifolds and show that the base is isometric to a special type of warped product under some geometric conditions.

Chen-like inequalities on null hypersurfaces with closed rigging of a Lorentzian manifold

2021 ◽  
pp. 2150125
Author(s):  
Amrinder Pal Singh ◽  
Cyriaque Atindogbe ◽  
Rakesh Kumar ◽  
Varun Jain
Keyword(s):  
Scalar Curvature ◽  
Space Form ◽  
Lorentzian Manifold ◽  
Null Hypersurface ◽  
Lorentzian Space ◽  
Null Hypersurfaces

We study null hypersurfaces of a Lorentzian manifold with a closed rigging for the hypersurface. We derive inequalities involving Ricci tensors, scalar curvature, squared mean curvatures for a null hypersurface with a closed rigging of a Lorentzian space form and for a screen homothetic null hypersurface of a Lorentzian manifold. We also establish a generalized Chen–Ricci inequality for a screen homothetic null hypersurface of a Lorentzian manifold with a closed rigging for the hypersurface.

RELATIVISTIC PARTICLES ALONG NULL CURVES IN 3D LORENTZIAN SPACE FORMS

2010 ◽  
Vol 20 (09) ◽  
pp. 2851-2859 ◽  
Author(s):  
ÁNGEL GIMÉNEZ
Keyword(s):  
Vector Fields ◽  
Lagrange Equation ◽  
Killing Vector ◽  
Space Forms ◽  
Lorentzian Space ◽  
Motion Equations ◽  
Critical Curves ◽  
Null Curves ◽  
Cartan Curvature ◽  
Relativistic Particles

We study relativistic particles modeled by actions whose Lagrangians are arbitrary functions on the curvature of null paths in (2 + 1)-dimensions backgrounds with constant curvature. We obtain first integrals of the Euler–Lagrange equation by using geometrical methods involving the search for Killing vector fields along critical curves of the action. In the case in which Lagrangian density depends quadratically on Cartan curvature, it is shown that the mechanical system is governed by a stationary Korteweg–De Vries system. Motion equations are completely integrated by quadratures in terms of elliptic and hyperelliptic functions.

NULL HELICES IN LORENTZIAN SPACE FORMS

2001 ◽  
Vol 16 (30) ◽  
pp. 4845-4863 ◽  
Author(s):  
ANGEL FERRÁNDEZ ◽  
ANGEL GIMÉNEZ ◽  
PASCUAL LUCAS
Keyword(s):  
Minkowski Space ◽  
Complete Classification ◽  
Space Forms ◽  
Lorentzian Space ◽  
Low Dimensions ◽  
Minkowski Space Time ◽  
Minimum Number ◽  
Null Curves ◽  
Lorentzian Space Forms

In this paper we introduce a reference along a null curve in an n-dimensional Lorentzian space with the minimum number of curvatures. That reference generalizes the reference of Bonnor for null curves in Minkowski space–time and it is called the Cartan frame of the curve. The associated curvature functions are called the Cartan curvatures of the curve. We characterize the null helices (that is, null curves with constant Cartan curvatures) in n-dimensional Lorentzian space forms and we obtain a complete classification of them in low dimensions.

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.

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