• Bulletin of the Korean Mathematical Society
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• v.49 no.2
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• pp.271-284
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• 2012

Let M be a linear Weingarten spacelike hypersurface in a locally symmetric Lorentz space with R = aH + b, where R and H are the normalized scalar curvature and the mean curvature, respectively. In this paper, we give some conditions for the complete hypersurface M to be totally umbilical.

• Journal of the Korean Mathematical Society
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• v.54 no.4
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• pp.1163-1173
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• 2017

In this paper, we introduce the condition (I) (cf. Section 2) and prove that there is no nontrivial tangential holomorphic vector field of a certain hypersurface of infinite type in ${\mathbb{C}}^2$.

• Journal of the Korean Mathematical Society
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• v.36 no.4
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• pp.725-736
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• 1999

In the present paper we will give a characterization of homogeneous real hypersurfaces of type A1, A2 and B of a complex projective space.

• Kyungpook Mathematical Journal
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• v.51 no.2
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• pp.125-138
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• 2011

We give some characterizations of non-existence of lightlike hypersurfaces of an indefinite space form. Some geometric objects for the induced Ricci tensor to be symmetric are studied.

• Honam Mathematical Journal
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• v.40 no.3
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• pp.521-528
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• 2018

The object of the present paper is to study a decomposable $(APS)_n$ and hypersurfaces of $(APS)_n$. Also, the existence of a decomposable $(APS)_n$ is ensured by a proper example.

• Bulletin of the Korean Mathematical Society
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• v.58 no.4
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• pp.877-884
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• 2021

We prove that if a compact convex hypersurface of ℝⁿ⁺¹ has almost maximal extinction time when it is evolved by the mean curvature flow, then it must be nearly round in the C⁰-norm.

• East Asian mathematical journal
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• v.33 no.5
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• pp.571-585
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• 2017

Let M be a connected n-dimensional submanifold of a Euclidean space $E^{n+k}$ equipped with the induced metric and ${\Delta}$ its Laplacian. If the position vector x of M is decomposed as a sum of three vectors $x=x_1+x_2+x_0$ where two vectors $x_1$ and $x_2$ are non-constant eigen vectors of the Laplacian, i.e., ${\Delta}x_i={\lambda}_ix_i$, i = 1, 2 (${\lambda}_i{\in}R$) and $x_0$ is a constant vector, then, M is called a 2-type submanifold. In this paper we showed that a 2-type surface M in $E^3$ satisfies ${\langle}{\Delta}x,x-x_0{\rangle}=c$ for a constant c, where ${\langle},{\rangle}$ is the usual inner product in $E^3$, then M is an open part of a circular cylinder. Also we showed that if a quadric hypersurface M in a Euclidean space satisfies ${\langle}{\Delta}x,x{\rangle}=c$ for a constant c, then it is one of a minimal quadric hypersurface, a genaralized cone, a hypersphere, and a spherical cylinder.

• Bulletin of the Korean Mathematical Society
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• v.50 no.1
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• pp.97-103
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• 2013

As a suitable application of the well known generalized maximum principle of Omori-Yau, we obtain rigidity results concerning to a complete hypersurface immersed with bounded mean curvature in the $(n+1)$-dimensional hyperbolic space $\mathbb{H}^{n+1}$. In our approach, we explore the existence of a natural duality between $\mathbb{H}^{n+1}$ and the half $\mathcal{H}^{n+1}$ of the de Sitter space $\mathbb{S}_1^{n+1}$, which models the so-called steady state space.

• Honam Mathematical Journal
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• v.28 no.4
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• pp.573-589
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• 2006

Let M be a real hypersurface with almost contact metric structure $({\phi},{\xi},{\eta},g)$ in a non flat complex space form $M_n(c)$. In this paper, we prove that if the structure Jacobi operator $R_{\xi}$ is ${\xi}$-parallel and the Ricci tensor S commutes with the structure operator $\phi$, then a real hypersurface in $M_n(c)$ is a Hopf hypersurface. Further, we characterize such Hopf hypersurface in $M_n(c)$.

• Honam Mathematical Journal
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• v.32 no.3
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• pp.363-374
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• 2010

We define a semi-symmetric metric connection in an almost r-paracontact Riemannian manifold and we consider submanifolds of an almost r-paracontact Riemannian manifold endowed with a semi-symmetric metric connection and obtain Gauss and Codazzi equations, Weingarten equation and curvature tensor for submanifolds of an almost r-paracontact Riemannian manifold endowed with a semi-symmetric metric connection.