• Title, Summary, Keyword: Hopf real hypersurface

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ξ-PARALLEL STRUCTURE JACOBI OPERATORS OF REAL HYPERSURFACES IN A NONFLAT COMPLEX SPACE FORM

  • KIM, NAM-GIL;KI, U-HANG
    • 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 $M_n(c)$.

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SOME CHARACTERIZATIONS OF REAL HYPERSURFACES OF TYPE (A) IN A NONFLAT COMPLEX SPACE FORM

  • Ki, U-Hang;Liu, Hui-Li
    • 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 $M_{n}(C)$.

REAL HYPERSURFACES WITH MIAO-TAM CRITICAL METRICS OF COMPLEX SPACE FORMS

  • Chen, Xiaomin
    • 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.

Structure Eigenvectors of the Ricci Tensor in a Real Hypersurface of a Complex Projective Space

  • Li, Chunji;Ki, U-Hang
    • 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 $P_n\mathbb{C}$.

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CHARACTERIZATIONS OF REAL HYPERSURFACES OF COMPLEX SPACE FORMS IN TERMS OF RICCI OPERATORS

  • Sohn, Woon-Ha
    • 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.

CHARACTERIZATIONS OF REAL HYPERSURFACES OF TYPE A IN A COMPLEX SPACE FORM

  • Ki, U-Hang;Kim, In-Bae;Lim, Dong-Ho
    • Bulletin of the Korean Mathematical Society
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    • v.47 no.1
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    • pp.1-15
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    • 2010
  • Let M be a real hypersurface with almost contact metric structure $(\phi,g,\xi,\eta)$ in a complex space form $M_n(c)$, $c\neq0$. In this paper we prove that if $R_{\xi}L_{\xi}g=0$ holds on M, then M is a Hopf hypersurface in $M_n(c)$, where $R_{\xi}$ and $L_{\xi}$ denote the structure Jacobi operator and the operator of the Lie derivative with respect to the structure vector field $\xi$ respectively. We characterize such Hopf hypersurfaces of $M_n(c)$.

Hopf Hypersurfaces in Complex Two-plane Grassmannians with Generalized Tanaka-Webster Reeb-parallel Structure Jacobi Operator

  • Kim, Byung Hak;Lee, Hyunjin;Pak, Eunmi
    • Kyungpook Mathematical Journal
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    • v.59 no.3
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    • pp.525-535
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    • 2019
  • In relation to the generalized Tanaka-Webster connection, we consider a new notion of parallel structure Jacobi operator for real hypersurfaces in complex two-plane Grassmannians and prove the non-existence of real hypersurfaces in $G_2({\mathbb{C}}^{m+2})$ with generalized Tanaka-Webster parallel structure Jacobi operator.

Real Hypersurfaces with k-th Generalized Tanaka-Webster Connection in Complex Grassmannians of Rank Two

  • Jeong, Imsoon;Lee, Hyunjin
    • Kyungpook Mathematical Journal
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    • v.57 no.3
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    • pp.525-535
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    • 2017
  • In this paper, we consider two kinds of derivatives for the shape operator of a real hypersurface in a $K{\ddot{a}}hler$ manifold which are named the Lie derivative and the covariant derivative with respect to the k-th generalized Tanaka-Webster connection ${\hat{\nabla}}^{(k)}$. The purpose of this paper is to study Hopf hypersurfaces in complex Grassmannians of rank two, whose Lie derivative of the shape operator coincides with the covariant derivative of it with respect to ${\hat{\nabla}}^{(k)}$ either in direction of any vector field or in direction of Reeb vector field.

Real Hypersurfaces in Complex Hyperbolic Space with Commuting Ricci Tensor

  • Ki, U-Hang;Suh, Young-Jin
    • Kyungpook Mathematical Journal
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    • v.48 no.3
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    • pp.433-442
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    • 2008
  • In this paper we consider a real hypersurface M in complex hyperbolic space $H_n\mathbb{C}$ satisfying $S{\phi}A\;=\;{\phi}AS$, where $\phi$, A and S denote the structure tensor, the shape operator and the Ricci tensor of M respectively. Moreover, we give a characterization of real hypersurfaces of type A in $H_n\mathbb{C}$ by such a commuting Ricci tensor.