• Kyungpook Mathematical Journal
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• v.56 no.4
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• pp.1207-1235
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• 2016

Let M be a real hypersurface with constant mean curvature in a complex space form $M_n(c),c{\neq}0$. In this paper, we prove that if the structure Jacobi operator $R_{\xi}= R({\cdot},{\xi}){\xi}$ with respect to the structure vector field ${\xi}$ is ${\phi}{\nabla}_{\xi}{\xi}$-parallel and $R_{\xi}$ commute with the structure tensor field ${\phi}$, then M is a homogeneous real hypersurface of Type A.

• Communications of the Korean Mathematical Society
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• v.13 no.3
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• pp.545-560
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• 1998

Let ø and A be denoted by the structure tensor field of type (1,1) and by the shape operator of a real hypersurface in a complex space form $M_{n}$ (c), c $\neq$ 0 respectively. The main purpose of this paper is to prove that if a real hypersurface in $M_{n}$ (c) satisfies $R_{ξ}$ øA = $AøR_{ξ}$, then the structure vector field ξ is principal, where $R_{ξ}$ / is the Jacobi operator with respect to ξ.

• Bulletin of the Korean Mathematical Society
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• v.26 no.2
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• pp.159-164
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• 1989

Let M be a hypersurface of dimension n(.geq.2) in an (n+1)-dimensional real space form over bar M(c) with constant curvature c and H the second fundamental tensor of M. M is said to be birecurrent if here exists a covariant tensor field .alpha. of order 2 such that .del.$^{2}$H=H .alpha., where .del. is the connection of M. Also, M is said to be recurrent if there exists a 1-form .betha. such that .del.H=H .betha.. Matsuyama [2] recently proved that a recurrent hypersurface M in a real space form is locally symmetric and a complete irreducible birecurrent hypersurface M in a real space form is recurrent. The main purpose of this paper is to characterize the birecurrent or recurrent hypersurface M of a Riemannian manifold with constant curvature c and to prove that M is classified as a cylinder, $M^{n}$ (c) or ( $c_{1}$)* $M^{n-r}$ ( $c_{2}$) where 1/ $c_{1}$+1/ $c_{2}$=1/c.

• Bulletin of the Korean Mathematical Society
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• v.54 no.6
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• pp.2001-2011
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• 2017

In 1997, H. Li [12] proposed a conjecture: if $M^n(n{\geqslant}3)$ is a complete spacelike hypersurface in de Sitter space $S^{n+1}_1(1)$ with constant normalized scalar curvature R satisfying $\frac{n-2}{n}{\leqslant}R{\leqslant}1$, then is $M^n$ totally umbilical? Recently, F. E. C. Camargo et al. ([5]) partially proved the conjecture. In this paper, from a different viewpoint, we study closed convex spacelike hypersurface $M^n$ in locally symmetric Lorentz space $L^{n+1}_1$ and also prove that $M^n$ is totally umbilical if the square of length of second fundamental form of the closed convex spacelike hypersurface $M^n$ is constant, i.e., Theorem 1. On the other hand, we obtain that if the sectional curvature of the closed convex spacelike hypersurface $M^n$ in locally symmetric Lorentz space $L^{n+1}_1$ satisfies $K(M^n)$ > 0, then $M^n$ is totally umbilical, i.e., Theorem 2.

• Journal of the Korean Mathematical Society
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• v.52 no.5
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• pp.1097-1108
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• 2015

In this paper, we investigate p-biharmonic maps u : (M, g) $\rightarrow$ (N, h) from a Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. We obtain that if ${\int}_M|{\tau}(u)|^{{\alpha}+p}dv_g$ < ${\infty}$ and ${\int}_M|d(u)|^2dv_g$ < ${\infty}$, then u is harmonic, where ${\alpha}{\geq}0$ is a nonnegative constant and $p{\geq}2$. We also obtain that any weakly convex p-biharmonic hypersurfaces in space formN(c) with $c{\leq}0$ is minimal. These results give affirmative partial answer to Conjecture 2 (generalized Chen's conjecture for p-biharmonic submanifolds).

• Bulletin of the Korean Mathematical Society
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• v.50 no.4
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• pp.1201-1207
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• 2013

Let N be a complete Riemannian manifold with nonnegative Ricci curvature and let M be a complete noncompact oriented stable minimal hypersurface in N. We prove that if M has at least two ends and ${\int}_M{\mid}A{\mid}^2\;dv={\infty}$, then M admits a nonconstant harmonic function with finite Dirichlet integral, where A is the second fundamental form of M. We also show that the space of $L^2$ harmonic 1-forms on such a stable minimal hypersurface is not trivial. Our result is a generalization of one of main results in [12] because if N has nonnegative sectional curvature, then M admits no nonconstant harmonic functions with finite Dirichlet integral. And our result recovers a main theorem in [3] as a corollary.

• Bulletin of the Korean Mathematical Society
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• v.39 no.4
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• pp.671-680
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• 2002

In this article, we show that if a semi-Riemannian space form carries a conformal vector field V of which the tangential part $V^T$ on a connected hypersurface $M^N$ ecomes a conformal vector field and the normal part $V^N on $M^N$ does not vanish identically, then $M^N$ is totally umbilic. Furthermore, we give a complete description of conformal vector fields on semi-Riemannian space forms.

• Communications of the Korean Mathematical Society
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• v.25 no.2
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• pp.235-243
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• 2010

In a Riemannian manifold, the existence of a new connection is proved. In particular cases, this connection reduces to several symmetric, semi-symmetric and quarter symmetric connections, even some of them are not introduced so far. So, in this paper, we define a quarter symmetric semi-metric connection in an almost r-paracontact Riemannian manifold and consider invariant, non-invariant and anti-invariant hypersurfaces of an almost r-paracontact Riemannian manifold with that connection.

• Journal of the Korean Mathematical Society
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• v.46 no.3
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• pp.589-607
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• 2009

In this paper we study arithmetically Cohen-Macaulay (ACM for short) vector bundles $\varepsilon$ of rank k $\geq$ 3 on hypersurfaces $X_r\;{\subset}\;{\mathbb{P}}^4$ of degree r $\geq$ 1. We consider here mainly the case of degree r = 4, which is the first unknown case in literature. Under some natural conditions for the bundle $\varepsilon$ we derive a list of possible Chern classes ($c_1$, $c_2$, $c_3$) which may arise in the cases of rank k = 3 and k = 4, when r = 4 and we give several examples.

• Honam Mathematical Journal
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• v.35 no.1
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• pp.37-49
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• 2013

Consider a non-degenerate open convex cone C with vertex the origin in the $n$2-dimensional Euclidean space $E^n$. We study volume properties of strictly convex hypersurfaces in the cone C. As a result, for example, if the volume of the region of an elliptic cone C cut off by the tangent hyperplane P of M at $p$ is independent of the point $p{\in}M$, then it is shown that the hypersurface M is part of an elliptic hyperboloid.