• Title, Summary, Keyword: Tanaka-Webster connection

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TRANS-SASAKIAN MANIFOLDS WITH RESPECT TO GENERALIZED TANAKA-WEBSTER CONNECTION

  • Kazan, Ahmet;Karadag, H.Bayram
    • Honam Mathematical Journal
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    • v.40 no.3
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    • pp.487-508
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    • 2018
  • In this study, we use the generalized Tanaka-Webster connection on a trans-Sasakian manifold of type (${\alpha},{\beta}$) and obtain the curvature tensors of a trans-Sasakian manifold with respect to this connection. Also, we investigate some special curvature conditions of a trans-Sasakian manifold with respect to generalized Tanaka-Webster connection and finally, give an example for trans-Sasakian manifolds.

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 LIE DERIVATIVE OF REAL HYPERSURFACES IN ℂP2 AND ℂH2 WITH RESPECT TO THE GENERALIZED TANAKA-WEBSTER CONNECTION

  • PANAGIOTIDOU, KONSTANTINA;PEREZ, JUAN DE DIOS
    • Bulletin of the Korean Mathematical Society
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    • v.52 no.5
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    • pp.1621-1630
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    • 2015
  • In this paper the notion of Lie derivative of a tensor field T of type (1,1) of real hypersurfaces in complex space forms with respect to the generalized Tanaka-Webster connection is introduced and is called generalized Tanaka-Webster Lie derivative. Furthermore, three dimensional real hypersurfaces in non-flat complex space forms whose generalized Tanaka-Webster Lie derivative of 1) shape operator, 2) structure Jacobi operator coincides with the covariant derivative of them with respect to any vector field X orthogonal to ${\xi}$ are studied.

REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS WHOSE SHAPE OPERATOR IS OF CODAZZI TYPE IN GENERALIZED TANAKA-WEBSTER CONNECTION

  • Cho, Kyusuk;Lee, Hyunjin;Pak, Eunmi
    • Bulletin of the Korean Mathematical Society
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    • v.52 no.1
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    • pp.57-68
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    • 2015
  • In this paper, we give a non-existence theorem of Hopf hypersurfaces in complex two-plane Grassmannians $G_2(\mathbb{C}^{m+2})$, $m{\geq}3$, whose shape operator is of Codazzi type in generalized Tanaka-Webster connection $\hat{\nabla}^{(k)}$.

SOME RESULTS ON PROJECTIVE CURVATURE TENSOR IN SASAKIAN MANIFOLDS

  • Gautam, Umesh Kumar;Haseeb, Abdul;Prasad, Rajendra
    • Communications of the Korean Mathematical Society
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    • v.34 no.3
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    • pp.881-896
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    • 2019
  • In the present paper, we study certain curvature conditions satisfying by the projective curvature tensor in Sasakian manifolds with respect to the generalized-Tanaka-Webster connection. Finally, we give an example of a 3-dimensional Sasakian manifold with respect to the generalized-Tanaka-Webster connection.

KILLING STRUCTURE JACOBI OPERATOR OF A REAL HYPERSURFACE IN A COMPLEX PROJECTIVE SPACE

  • Perez, Juan de Dios
    • Journal of the Korean Mathematical Society
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    • v.58 no.2
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    • pp.473-486
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    • 2021
  • We prove non-existence of real hypersurfaces with Killing structure Jacobi operator in complex projective spaces. We also classify real hypersurfaces in complex projective spaces whose structure Jacobi operator is Killing with respect to the k-th generalized Tanaka-Webster connection.

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.

GEOMETRY OF CONTACT STRONGLY PSEUDO-CONVEX CR-MANIFOLDS

  • Cho, Jong-Taek
    • Journal of the Korean Mathematical Society
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    • v.43 no.5
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    • pp.1019-1045
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    • 2006
  • As a natural generalization of a Sasakian space form, we define a contact strongly pseudo-convex CR-space form (of constant pseudo-holomorphic sectional curvature) by using the Tanaka-Webster connection, which is a canonical affine connection on a contact strongly pseudo-convex CR-manifold. In particular, we classify a contact strongly pseudo-convex CR-space form $(M,\;\eta,\;\varphi)$ with the pseudo-parallel structure operator $h(=1/2L\xi\varphi)$, and then we obtain the nice form of their curvature tensors in proving Schurtype theorem, where $L\xi$ denote the Lie derivative in the characteristic direction $\xi$.