• 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.

• Communications of the Korean Mathematical Society
• /
• v.34 no.4
• /
• pp.1303-1313
• /
• 2019

Recently Hui et al. ([8,9]) studied contact CR-warped product submanifolds and also warped product pseudo-slant submanifolds of a $(LCS)_n$-manifold $\bar{M}$. The characterization for both these classes of warped product submanifolds have been studied here. It is also shown that there do not exists any proper warped product bi-slant submanifold of a $(LCS)_n$-manifold. Although the existence of a bi-slant submanifold of $(LCS)_n$-manifold is ensured by an example.

• Honam Mathematical Journal
• /
• v.27 no.2
• /
• pp.273-285
• /
• 2005

In this paper, when N is a compact Riemannian manifold, we discuss the method of using warped products to construct Lorentzian metrics on $M=[a,\;b){\times}_f\;N$ with specific scalar curvatures.

• 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.

• The Pure and Applied Mathematics
• /
• v.12 no.3 s.29
• /
• pp.211-221
• /
• 2005

The purpose of this paper is to study the geometry of null curves in a Lorentzian manifold (M, g). We show that it is possible to construct new type of Frenet equations of null curves in M, supported by two examples.

• 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 $(PSPS)_4$.

• The Pure and Applied Mathematics
• /
• v.5 no.2
• /
• pp.115-122
• /
• 1998

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 $M{\;}={\;}[a,{\;}{\infty}){\times}_f{\;}N$ with specific scalar curvatures.

• Journal of the Korean Mathematical Society
• /
• v.33 no.3
• /
• pp.507-512
• /
• 1996

A totally umbilic submanifold of a pseudo-Riemanian manifold is a submanifold whose first fundamental form and second fundamental form are proportiona. An ordinary hypersphere $S^n(r)$ of an affine (n + 1)-space of the Euclidean space $E^m$ is the best known example of totally umbilic submanifolds of $E^m$.

• Bulletin of the Korean Mathematical Society
• /
• v.50 no.4
• /
• pp.1367-1376
• /
• 2013

We study Einstein lightlike hypersurfaces M of a Lorentzian space form $\tilde{M}(c)$ admitting a semi-symmetric non-metric connection subject to the conditions; (1) M is screen conformal and (2) the structure vector field ${\zeta}$ of $\tilde{M}$ belongs to the screen distribution S(TM). The main result is a characterization theorem for such a lightlike hypersurface.