• Bulletin of the Korean Mathematical Society
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• v.59 no.3
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• pp.789-799
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• 2022

We establish a sharp integral inequality related to compact (without boundary) linear Weingarten hypersurfaces (immersed) in a locally symmetric Einstein manifold and we apply it to characterize totally umbilical hypersurfaces and isoparametric hypersurfaces with two distinct principal curvatures, one which is simple, in such an ambient space. Our approach is based on the ideas and techniques introduced by Alías and Meléndez in [4] for the case of hypersurfaces with constant scalar curvature in an Euclidean round sphere.

• Kyungpook Mathematical Journal
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• v.59 no.4
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• pp.783-795
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• 2019

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 are no totally geodesic ascreen null hypersurfaces of a conformally flat (LCS)-manifold.

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

• Journal of the Chungcheong Mathematical Society
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• v.21 no.1
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• pp.29-48
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• 2008

We deal with the classification problem of real hypersurfaces in a complex hyperbolic space. In order to classify real hypersurfaces in a complex hyperbolic space we characterize a real hypersurface M in $H_n(\mathbb{C})$ whose structure vector field is not principal. We also construct extrinsically homogeneous real hypersurfaces with four distinct curvatures and their structure vector fields are not principal.

• Communications of the Korean Mathematical Society
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• v.33 no.1
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• pp.305-315
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• 2018

This note provides a study of lightlike hypersurfaces of a generalized Robertson-Walker(GRW) space-time with a certain screen distribution, which are integrable and have good properties. Focus is to investigate geometric features from the relation of the second fundamental forms between lightlike hypersurfaces and leaves of the integrable screen distribution. Also, we shall apply those results on lightlike hypersurfaces of a GRW space-time to lightlike hypersurfaces of a Robertson-Walker(RW) space-time.

• Communications of the Korean Mathematical Society
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• v.34 no.3
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• pp.863-879
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• 2019

In this paper we study the Myers' theorem for orientable Riemannian hypersurfaces under some restrictions on the mean curvature, the second-order mean curvature, or the divergence of the shape operator and give some estimates for the diameter of such hypersurfaces.

• Bulletin of the Korean Mathematical Society
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• v.49 no.4
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• pp.685-692
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• 2012

Consider a hypersurface $M^n$ in $\mathbb{R}^{n+1}$ with $n$ distinct principal curvatures, parametrized by lines of curvature with vanishing Laplace invariants. (1) If the lines of curvature are planar, then there are no such hypersurfaces for $n{\geq}4$, and for $n=3$, they are, up to M$\ddot{o}$bius transformations Dupin hypersurfaces with constant M$\ddot{o}$bius curvature. (2) If the principal curvatures are given by a sum of functions of separated variables, there are no such hypersurfaces for $n{\geq}4$, and for $n=3$, they are, up to M$\ddot{o}$bius transformations, Dupin hypersurfaces with constant M$\ddot{o}$bius curvature.

• Journal for History of Mathematics
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• v.11 no.1
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• pp.42-46
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• 1998

Using Fermi coordinates and the principle curvature on the tubula hypersurfaces, we characterize space of constant sectional curvature by analysing the shape operator on the tubular hypersurfaces.

• Honam Mathematical Journal
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• v.32 no.4
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• pp.747-761
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• 2010

It is known that there are no real hypersurfaces with parallel structure Jacobi operator $R_{\xi}$ (cf.[16], [17]). In this paper we investigate real hypersurfaces in a nonflat complex space form using some conditions of the structure Jacobi operator $R_{\xi}$ which are weaker than ${\nabla}R_{\xi}$ = 0. Under further condition $S\phi={\phi}S$ for the Ricci tensor S we characterize Hopf hypersurfaces in a complex space form.

• Journal of the Korean Mathematical Society
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• v.45 no.4
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• pp.1113-1133
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• 2008

In this paper we give some non-existence theorems for real hypersurfaces in complex two-plane Grassmannians $G_2({\mathbb{C}}^{m+2})$ with Lie ${\xi}$-parallel normal Jacobi operator $\bar{R}_N$ and another geometric conditions.