• Kyungpook Mathematical Journal
• /
• v.55 no.1
• /
• pp.149-156
• /
• 2015

We analyze the $(LCS)_n$-manifolds endowed with some symmetric properties, focusing on Ricci tensor and the 1-form ${\gamma}$. We study some properties of special Weakly Ricci-Symmetric $(LCS)_n$-manifolds and also shown that Weakly ${\phi}$-Ricci Symmetric $(LCS)_n$-manifold is an ${\eta}$-Einstein manifold.

• Communications of the Korean Mathematical Society
• /
• v.34 no.1
• /
• pp.303-319
• /
• 2019

The aim of the present paper is to study the Eisenhart problems of finding the properties of second order parallel tensors (symmetric and skew-symmetric) on a 3-dimensional LCS-manifold. We also investigate the properties of Ricci solitons, Ricci semisymmetric, locally ${\phi}$-symmetric, ${\eta}$-parallel Ricci tensor and a non-null concircular vector field on $(LCS)_3$-manifolds.

• Journal of the Korean Mathematical Society
• /
• v.32 no.1
• /
• pp.17-31
• /
• 1995

A Lorentzian para-Sasakian manifold M$(\varphi, \zeta, \eta, g)$ (abr. LPS-manifold) has been defined and studied in [9] and [10]. On the other hand, para-Sasakian (abr. PS)-manifolds are special semi-cosympletic manifolds (in the sense of [2]), that is, they are endowed with an almost cosympletic 2-form $\Omega$ such that $d^{2\eta}\Omega = \psi(d^\omega$ denotes the cohomological operator [6]), where the 3-form $\psi$ is the associated Lefebvre form of $\Omega$ ([8]). If $\eta$ is exact, $\psi$ is a $d^{2\eta}$-exact form, the manifold M is called an exact Ps-manifold. Clearly, any LPS-manifold is endowed with a semi-cosymplectic structure (abr. SC-structure).

• Communications of the Korean Mathematical Society
• /
• v.10 no.1
• /
• pp.163-173
• /
• 1995

Inspired in [2,9,10,17], pp. E. Ehrlich and S. B. Kim in [4] cultivated the Riccati equation related to the Raychaudhuri equation of General Relativity for the stable Jacobi tensor along the geodesics to extend the Hawking-Penrose conjugacy theorem to $$ f(t) = Ric(c(t)',c'(t)) + tr(\sigma(A)^2) $$ where $\sigma(A)$ is the shear tensor of A for the stable Jacobi tensor A with $A(t_0) = Id$ along the complete Riemannian or complete nonspacelike geodesics c.

• Journal of the Korean Mathematical Society
• /
• v.57 no.1
• /
• pp.195-213
• /
• 2020

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 isoparametric) null hypersurfaces with at most two principal curvatures are generally Riemannian products. Several examples are also given to illustrate the main concepts.

• Bulletin of the Korean Mathematical Society
• /
• v.56 no.1
• /
• pp.229-243
• /
• 2019

We use the so-called pseudoinversion of degenerate metrics technique on foliated compact null hypersurface, $M^{n+1}$, in Lorentzian manifold ${\overline{M}}^{n+2}$, to derive an integral formula involving the r-th order mean curvatures of its foliations, ${\mathcal{F}}^n$. We apply our formula to minimal foliations, showing that, under certain geometric conditions, they are isomorphic to n-dimensional spheres. We also use the formula to deduce expressions for total mean curvatures of such foliations.

• East Asian mathematical journal
• /
• v.30 no.1
• /
• pp.15-22
• /
• 2014

In this paper, we study lightlike hypersurfaces M of semi-Riemannian manifolds $\bar{M}$ of quasi-constant curvatures. Our main result is a characterization theorem for screen homothetic Einstein lightlike hypersurfaces of a Lorentzian manifold of quasi-constant curvature subject such that its curvature vector field ${\zeta}$ is tangent to M.

• Bulletin of the Korean Mathematical Society
• /
• v.41 no.4
• /
• pp.691-697
• /
• 2004

In this paper we find index form of the multiply warped product manifolds to investigate the physical properties from space-time.

• Bulletin of the Korean Mathematical Society
• /
• v.37 no.2
• /
• pp.317-336
• /
• 2000

In this paper, when N is a compact Riemannian manifold, we discuss the method of using warped products to construct timelike or null future (or past) complete Lorentzian metrics on $M=(-{\infty},{\;}\infty){\;}{\times}f^N$ with specific scalar curvatures.

• Journal of the Korean Mathematical Society
• /
• v.55 no.2
• /
• pp.391-413
• /
• 2018

In this paper we introduce a new tensor named semi-projective curvature tensor which generalizes the projective curvature tensor. First we deduce some basic geometric properties of semi-projective curvature tensor. Then we study pseudo semi-projective symmetric manifolds $(PSPS)_n$ which recover some known results of Chaki [5]. We provide several interesting results. Among others we prove that in a $(PSPS)_n$ if the associated vector field ${\rho}$ is a unit parallel vector field, then either the manifold reduces to a pseudosymmetric manifold or pseudo projective symmetric manifold. Moreover we deal with semi-projectively flat perfect fluid and dust fluid spacetimes respectively. As a consequence we obtain some important theorems. Next we consider the decomposability of $(PSPS)_n$. Finally, we construct a non-trivial Lorentzian metric of $(PSPS)_4$.