• 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 $M_n(c)$.

• Bulletin of the Korean Mathematical Society
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• v.44 no.1
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• pp.157-172
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• 2007

In this paper, we prove that if the structure Jacobi operator $R_{\xi}-parallel\;and\;R_{\xi}$ commutes with the Ricci tensor S, then a real hypersurface with non-negative scalar curvature of a nonflat complex space form $M_{n}(C)$ is a Hopf hypersurface. Further, we characterize such Hopf hypersurface in $M_{n}(C)$.

• The Pure and Applied Mathematics
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• v.28 no.1
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• pp.55-60
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• 2021

Let M be a real hypersurface in a complex space form Mn(c), c ≠ 0. In this paper we prove that if the structure tensor field is Codazzi type, then M is a Hopf hypersurface. We characterize such Hopf hypersurfaces of Mn(c).

• The Pure and Applied Mathematics
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• v.28 no.3
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• pp.247-252
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• 2021

Let M be a real hypersurface in a complex space form M_n(c), c ≠ 0. In this paper, we prove that if (∇_Xϕ)Y + (∇_Yϕ)X = 0 holds on M, then M is a Hopf hypersurface, where ϕ is the tangential projection of the complex structure of M_n(c). We characterize such Hopf hypersurfaces of M_n(c).

• Journal of the Korean Mathematical Society
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• v.55 no.3
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• pp.735-747
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• 2018

Let M be a real hypersurface of a complex space form with constant curvature c. In this paper, we study the hypersurface M admitting Miao-Tam critical metric, i.e., the induced metric g on M satisfies the equation: $-({\Delta}_g{\lambda})g+{\nabla}^2_g{\lambda}-{\lambda}Ric=g$, where ${\lambda}$ is a smooth function on M. At first, for the case where M is Hopf, c = 0 and $c{\neq}0$ are considered respectively. For the non-Hopf case, we prove that the ruled real hypersurfaces of non-flat complex space forms do not admit Miao-Tam critical metrics. Finally, it is proved that a compact hypersurface of a complex Euclidean space admitting Miao-Tam critical metric with ${\lambda}$ > 0 or ${\lambda}$ < 0 is a sphere and a compact hypersurface of a non-flat complex space form does not exist such a critical metric.

• East Asian mathematical journal
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• v.37 no.1
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• pp.97-107
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• 2021

Let M be a real hypersurface with almost contact metric structure (��, ξ, ��, g) in a nonflat complex space form Mn(c). We denote S and Rξ by the Ricci tensor of M and by the structure Jacobi operator with respect to the vector field ξ respectively. In this paper, we prove that M is a Hopf hypersurface of type (A) in Mn(c) if it satisfies Rξ�� = ��Rξ and at the same time satisfies $({\nabla}_{{\phi}{\nabla}_{\xi}{\xi}}R_{\xi}){\xi}=0$ or Rξ��S = S��Rξ.

• Kyungpook Mathematical Journal
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• v.61 no.1
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• pp.181-190
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• 2021

Let M be a real hypersurface with almost contact metric structure (ϕ, ��, η, g) in a nonflat complex space form Mn(c). We denote S and R�� by the Ricci tensor of M and by the structure Jacobi operator with respect to the vector field �� respectively. In this paper, we prove that M is a Hopf hypersurface of type A in Mn(c) if it satisfies R��ϕ = ϕR�� and at the same time R��(Sϕ - ϕS) = 0.

• Bulletin of the Korean Mathematical Society
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• v.59 no.4
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• pp.897-904
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• 2022

In this paper, we obtain an inequality involving the squared norm of the covariant differentiation of the shape operator for a real hypersurface in nonflat complex space forms. It is proved that the equality holds for non-Hopf case if and only if the hypersurface is ruled and the equality holds for Hopf case if and only if the hypersurface is of type (A).

• Kyungpook Mathematical Journal
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• v.46 no.4
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• pp.463-476
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• 2006

It is known that there are no real hypersurfaces with parallel Ricci tensor in a nonflat complex space form ([6], [9]). In this paper we investigate real hypersurfaces in a complex projective space $P_n\mathbb{C}$ using some conditions of the Ricci tensor S which are weaker than ${\nabla}S=0$. We characterize Hopf hypersurfaces of $P_n\mathbb{C}$.

• Bulletin of the Korean Mathematical Society
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• v.44 no.1
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• pp.195-202
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• 2007

We prove that a real hypersurface M in a complex space form Mn(c), $c{\neq}0$, whose Ricci operator and structure tensor commute each other on the holomorphic distribution and the Ricci operator is ${\eta}-parallel$, is a Hopf hypersurface. We also give a characterization of this hypersurface.