• Title, Summary, Keyword: Jacobi operator

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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.

ON THE STRUCTURE JACOBI OPERATOR AND RICCI TENSOR OF REAL HYPERSURFACES IN NONFLAT COMPLEX SPACE FORMS

  • Kim, Soo-Jin
    • Honam Mathematical Journal
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    • v.32 no.4
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    • pp.747-761
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    • 2010
  • It is known that there are no real hypersurfaces with parallel structure Jacobi operator $R_{\xi}$ (cf.[16], [17]). In this paper we investigate real hypersurfaces in a nonflat complex space form using some conditions of the structure Jacobi operator $R_{\xi}$ which are weaker than ${\nabla}R_{\xi}$ = 0. Under further condition $S\phi={\phi}S$ for the Ricci tensor S we characterize Hopf hypersurfaces in a complex space form.

REAL HYPERSURFACES OF TYPE A IN COMPLEX TWO-PLANE GRASSMANNIANS RELATED TO THE NORMAL JACOBI OPERATOR

  • Jeong, Im-Soon;Suh, Young-Jin;Tripathi, Mukut Mani
    • Bulletin of the Korean Mathematical Society
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    • v.49 no.2
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    • pp.423-434
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    • 2012
  • In this paper we give a characterization of real hypersurfaces of type (A) in a complex two-plane Grassmannian $G_2(\mathbb{C}^{m+2})$ which is a tube over a totally geodesic $G_2(\mathbb{C}^{m+1})$ in $G_2(\mathbb{C}^{m+2})$, in terms of two commuting conditions related to the normal Jacobi operator and the shape operator.

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.

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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HOPF HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS WITH LIE PARALLEL NORMAL JACOBI OPERATOR

  • Jeong, Im-Soon;Lee, Hyun-Jin;Suh, Young-Jin
    • 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.