• Journal of the Korean Mathematical Society
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• v.19 no.1
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• pp.55-60
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• 1982

• Honam Mathematical Journal
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• v.30 no.4
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• pp.677-683
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• 2008

On a compact n-dimensional manifold M, it has been conjectured that a critical point metric of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of volume 1, should be Einstein. The purpose of the present paper is to prove that a 3-dimensional manifold (M,g) is isometric to a standard sphere if ker $s^*_g{{\neq}}0$ and there is a lower Ricci curvature bound. We also study the structure of a compact oriented stable minimal surface in M.

• Bulletin of the Korean Mathematical Society
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• v.46 no.3
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• pp.401-408
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• 2009

The object of the present paper is to study certain curvature restriction on an LP-Sasakian manifold with a coefficient $\alpha$. Among others it is shown that if an LP-Sasakian manifold with a coefficient $\alpha$ is a manifold of constant curvature, then the manifold is the product manifold. Also it is proved that a 3-dimensional Ricci semisymmetric LP-Sasakian manifold with a constant coefficient $\alpha$ is a spaceform.

• Honam Mathematical Journal
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• v.44 no.3
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• pp.339-359
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• 2022

In this paper, we investigate k-Ricci curvature and k-scalar curvature on lightlike hypersurfaces of a real space form ${\tilde{M}}$(c) of constant sectional curvature c, endowed with semi-symmetric non-metric connection. Using this curvatures, we establish some inequalities for screen homothetic lightlike hypersurface of a real space form ${\tilde{M}}$(c) of constant sectional curvature c, endowed with semi-symmetric non-metric connection. Using these inequalities, we obtain some characterizations for such hypersurfaces. Considering the equality case, we obtain some results.

• Communications of the Korean Mathematical Society
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• v.24 no.2
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• pp.265-275
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• 2009

The object of the present paper is to study 3-dimensional normal almost contact metric manifolds satisfying certain curvature conditions. Among others it is proved that a parallel symmetric (0, 2) tensor field in a 3-dimensional non-cosympletic normal almost contact metric manifold is a constant multiple of the associated metric tensor and there does not exist a non-zero parallel 2-form. Also we obtain some equivalent conditions on a 3-dimensional normal almost contact metric manifold and we prove that if a 3-dimensional normal almost contact metric manifold which is not a ${\beta}$-Sasakian manifold satisfies cyclic parallel Ricci tensor, then the manifold is a manifold of constant curvature. Finally we prove the existence of such a manifold by a concrete example.

• Bulletin of the Korean Mathematical Society
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• v.48 no.6
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• pp.1245-1252
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• 2011

Consider the $L^2$-adjoint $s_g^{'*}$ of the linearization of the scalar curvature $s_g$. If ker $s_g^{'*}{\neq}0$ on an n-dimensional compact manifold, it is well known that the scalar curvature $s_g$ is a non-negative constant. In this paper, we study the structure of the level set ${\varphi}^{-1}$(0) and find the behavior of Ricci tensor when ker $s_g^{'*}{\neq}0$ with $s_g$ > 0. Also for a nontrivial solution (g, f) of $z=s_g^{'*}(f)$ on an n-dimensional compact manifold, we analyze the structure of the regular level set $f^{-1}$(-1). These results give a good understanding of the given manifolds.

• Korean Journal of Mathematics
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• v.8 no.1
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• pp.53-61
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• 2000

The object of this paper is to find the 4-dimensional compact Einstein manifold with negative Ricci curvature $r$.

• Journal of the Korean Mathematical Society
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• v.44 no.1
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• pp.211-235
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• 2007

In this paper, first we introduce the full expression of the curvature tensor of a real hypersurface M in complex two-plane Grass-mannians $G_2(\mathbb{C}^{m+2})$ from the equation of Gauss and derive a new formula for the Ricci tensor of M in $G_2(\mathbb{C}^{m+2})$. Next we prove that there do not exist any Hopf real hypersurfaces in complex two-plane Grassmannians $G_2(\mathbb{C}^{m+2})$ with parallel and commuting Ricci tensor. Finally we show that there do not exist any Einstein Hopf hypersurfaces in $G_2(\mathbb{C}^{m+2})$.

• Bulletin of the Korean Mathematical Society
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• v.52 no.5
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• pp.1535-1547
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• 2015

A vector field on a Riemannian manifold (M, g) is called concircular if it satisfies ${\nabla}X^v={\mu}X$ for any vector X tangent to M, where ${\nabla}$ is the Levi-Civita connection and ${\mu}$ is a non-trivial function on M. A smooth vector field ${\xi}$ on a Riemannian manifold (M, g) is said to define a Ricci soliton if it satisfies the following Ricci soliton equation: $$\frac{1}{2}L_{\xi}g+Ric={\lambda}g$$, where $L_{\xi}g$ is the Lie-derivative of the metric tensor g with respect to ${\xi}$, Ric is the Ricci tensor of (M, g) and ${\lambda}$ is a constant. A Ricci soliton (M, g, ${\xi}$, ${\lambda}$) on a Riemannian manifold (M, g) is said to have concircular potential field if its potential field is a concircular vector field. In the first part of this paper we determine Riemannian manifolds which admit a concircular vector field. In the second part we classify Ricci solitons with concircular potential field. In the last part we prove some important properties of Ricci solitons on submanifolds of a Riemannian manifold equipped with a concircular vector field.

• Bulletin of the Korean Mathematical Society
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• v.54 no.1
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• pp.177-186
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• 2017

In this paper, a type of Riemannian manifold (namely, pseudo semiconformally symmetric manifold) is introduced. Also the several geometric properties of such a manifold is investigated. Finally the existence of such a manifold is ensured by a proper example.