• Title, Summary, Keyword: H-Jacobi tensor

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REAL HYPERSURFACES OF THE JACOBI OPERATOR WITH RESPECT TO THE STRUCTURE VECTOR FIELD IN A COMPLEX SPACE FORM

  • AHN, SEONG-SOO
    • Bulletin of the Korean Mathematical Society
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    • v.42 no.2
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    • pp.279-294
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    • 2005
  • We study a real hypersurface M satisfying $L_{\xi}S=0\;and\;R_{\xi}S=SR_{\xi}$ in a complex hyperbolic space $H_n\mathbb{C}$, where S is the Ricci tensor of type (1,1) on M, $L_{\xi}\;and\;R_{\xi}$ denotes the operator of the Lie derivative and the Jacobi operator with respect to the structure vector field e respectively.

CHARACTERIZATIONS OF REAL HYPERSURFACES OF TYPE A IN A NONFLAT COMPLEX SPACE FORM WHOSE STRUCTURE JACOBI OPERATOR IS ξ-PARALLEL

  • Kim, Nam-Gil
    • Honam Mathematical Journal
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    • v.31 no.2
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    • pp.185-201
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    • 2009
  • Let M be a real hypersurface with almost contact metric structure $({\phi},{\xi},{\eta},g)$ of a nonflat complex space form whose structure Jacobi operator $R_{\xi}=R({\cdot},{\xi}){\xi}$ is ${\xi}$-parallel. In this paper, we prove that the condition ${\nabla}_{\xi}R_{\xi}=0$ characterize the homogeneous real hypersurfaces of type A in a complex projective space $P_n{\mathbb{C}}$ or a complex hyperbolic space $H_n{\mathbb{C}}$ when $g({\nabla}_{\xi}{\xi},{\nabla}_{\xi}{\xi})$ is constant.

REEB FLOW INVARIANT UNIT TANGENT SPHERE BUNDLES

  • Cho, Jong Taek;Chun, Sun Hyang
    • 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.