• Bulletin of the Korean Mathematical Society
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• v.53 no.4
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• pp.1249-1257
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• 2016

We prove that the Ricci operator S of an almost cosymplectic three-manifold M is invariant along the Reeb flow, that is, M satisfies ${\pounds}_{\xi}S=0$ if and only if M is either cosymplectic or locally isometric to the group E(1, 1) of rigid motions of Minkowski 2-space with a left invariant almost cosymplectic structure.

• Bulletin of the Korean Mathematical Society
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• v.48 no.2
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• pp.427-444
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• 2011

In this paper we give some non-existence theorems for Hopf hypersurfaces in the complex two-plane Grassmannian $G_2(\mathbb{C}^{m+2})$ with Lie parallel normal Jacobi operator $\bar{R}_N$ and totally geodesic D and $D^{\bot}$ components of the Reeb flow.

• Honam Mathematical Journal
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• v.36 no.4
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• pp.805-812
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• 2014

For unit tangent sphere bundles $T_1M$ with the standard contact metric structure (${\eta},\bar{g},{\phi},{\xi}$), we have two fundamental operators that is, $h=\frac{1}{2}{\pounds}_{\xi}{\phi}$ and ${\ell}=\bar{R}({\cdot},{\xi}){\xi}$, where ${\pounds}_{\xi}$ denotes Lie differentiation for the Reeb vector field ${\xi}$ and $\bar{R}$ denotes the Riemmannian curvature tensor of $T_1M$. In this paper, we study the Reeb ow invariancy of the corresponding (0, 2)-tensor fields H and L of h and ${\ell}$, respectively.

• Bulletin of the Korean Mathematical Society
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• v.46 no.3
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• pp.447-461
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• 2009

In this paper we give a non-existence theorem for Hopf real hypersurfaces in complex two-plane Grassmannians $G_2(\mathbb{C}^{m+2})$ satisfying the condition that the structure Jacobi operator $R_{\xi}$ commutes with the 3-structure tensors ${\phi}_i$, i = 1, 2, 3.

• Bulletin of the Korean Mathematical Society
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• v.45 no.3
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• pp.495-507
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• 2008

In this paper we give a complete classification of real hyper-surfaces in complex two-plane Grassmannians $G_2({\mathbb{C}}^{m+2})$ with commuting structure Jacobi operator $R_{\xi}$ and another geometric condition.

• Bulletin of the Korean Mathematical Society
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• v.53 no.6
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• pp.1715-1723
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• 2016

We study the characteristic Jacobi operator ${\ell}={\bar{R}({\cdot},{\xi}){\xi}$ (along the Reeb flow ${\xi}$) on the unit tangent sphere bundle $T_1M$ over a Riemannian manifold ($M^n$, g). We prove that if ${\ell}$ is pseudo-parallel, i.e., ${\bar{R}{\cdot}{\ell}=L{\mathcal{Q}}({\bar{g}},{\ell})$, by a non-positive function L, then M is locally flat. Moreover, when L is a constant and $n{\neq}16$, M is of constant curvature 0 or 1.

• Journal of the Korean Mathematical Society
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• v.55 no.1
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• pp.161-174
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• 2018

The aim of this article is to study the k-almost Ricci soliton and k-almost gradient Ricci soliton on contact metric manifold. First, we prove that if a compact K-contact metric is a k-almost gradient Ricci soliton, then it is isometric to a unit sphere $S^{2n+1}$. Next, we extend this result on a compact k-almost Ricci soliton when the flow vector field X is contact. Finally, we study some special types of k-almost Ricci solitons where the potential vector field X is point wise collinear with the Reeb vector field ${\xi}$ of the contact metric structure.

• Korean Journal of Mathematics
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• v.27 no.3
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• pp.691-705
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• 2019

In the present paper is to classify Beta-almost (${\beta}$-almost) Ricci solitons and ${\beta}$-almost gradient Ricci solitons on almost $CoK{\ddot{a}}hler$ manifolds with ${\xi}$ belongs to ($k,{\mu}$)-nullity distribution. In this paper, we prove that such manifolds with V is contact vector field and $Q{\phi}={\phi}Q$ is ${\eta}$-Einstein and it is steady when the potential vector field is pointwise collinear to the reeb vectoer field. Moreover, we prove that a ($k,{\mu}$)-almost $CoK{\ddot{a}}hler$ manifolds admitting ${\beta}$-almost gradient Ricci solitons is isometric to a sphere.

• Journal of the Korean Mathematical Society
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• v.59 no.2
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• pp.255-278
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• 2022

In this paper, first we introduce a new notion of generalized Killing structure Jacobi operator for a real hypersurface M in complex hyperbolic two-plane Grassmannians SU_2,m/S (U₂·U_m). Next we prove that there does not exist a Hopf real hypersurface in complex hyperbolic two-plane Grassmannians SU_2,m/S (U₂·U_m) with generalized Killing structure Jacobi operator.