• Title, Summary, Keyword: Ricci tensor

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ON THE EXISTENCE OF SOME TYPES OF LP-SASAKIAN MANIFOLDS

  • Shaikh, Absos A.;Baishya, Kanak K.;Eyasmin, Sabina
    • Communications of the Korean Mathematical Society
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    • v.23 no.1
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    • pp.95-110
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    • 2008
  • The object of the present paper is to provide the existence of LP-Sasakian manifolds with $\eta$-recurrent, $\eta$-parallel, $\phi$-recurrent, $\phi$-parallel Ricci tensor with several non-trivial examples. Also generalized Ricci recurrent LP-Sasakian manifolds are studied with the existence of various examples.

RICCI SOLITONS ON RICCI PSEUDOSYMMETRIC (LCS)n-MANIFOLDS

  • Hui, Shyamal Kumar;Lemence, Richard S.;Chakraborty, Debabrata
    • Honam Mathematical Journal
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    • v.40 no.2
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    • pp.325-346
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    • 2018
  • The object of the present paper is to study some types of Ricci pseudosymmetric $(LCS)_n$-manifolds whose metric is Ricci soliton. We found the conditions when Ricci soliton on concircular Ricci pseudosymmetric, projective Ricci pseudosymmetric, $W_3$-Ricci pseudosymmetric, conharmonic Ricci pseudosymmetric, conformal Ricci pseudosymmetric $(LCS)_n$-manifolds to be shrinking, steady and expanding. We also construct an example of concircular Ricci pseudosymmetric $(LCS)_3$-manifold whose metric is Ricci soliton.

GRADIENT ALMOST RICCI SOLITONS WITH VANISHING CONDITIONS ON WEYL TENSOR AND BACH TENSOR

  • Co, Jinseok;Hwang, Seungsu
    • Journal of the Korean Mathematical Society
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    • v.57 no.2
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    • pp.539-552
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    • 2020
  • In this paper we consider gradient almost Ricci solitons with weak conditions on Weyl and Bach tensors. We show that a gradient almost Ricci soliton has harmonic Weyl curvature if it has fourth order divergence-free Weyl tensor, or it has divergence-free Bach tensor. Furthermore, if its Weyl tensor is radially flat, we prove such a gradient almost Ricci soliton is locally a warped product with Einstein fibers. Finally, we prove a rigidity result on compact gradient almost Ricci solitons satisfying an integral condition.

REAL HYPERSURFACES WITH ∗-RICCI TENSORS IN COMPLEX TWO-PLANE GRASSMANNIANS

  • Chen, Xiaomin
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.3
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    • pp.975-992
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    • 2017
  • In this article, we consider a real hypersurface of complex two-plane Grassmannians $G_2({\mathbb{C}}^{m+2})$, $m{\geq}3$, admitting commuting ${\ast}$-Ricci and pseudo anti-commuting ${\ast}$-Ricci tensor, respectively. As the applications, we prove that there do not exist ${\ast}$-Einstein metrics on Hopf hypersurfaces as well as ${\ast}$-Ricci solitons whose potential vector field is the Reeb vector field on any real hypersurfaces.

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.

Curvature Properties of 𝜂-Ricci Solitons on Para-Kenmotsu Manifolds

  • Singh, Abhishek;Kishor, Shyam
    • Kyungpook Mathematical Journal
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    • v.59 no.1
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    • pp.149-161
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    • 2019
  • In the present paper, we study curvature properties of ${\eta}$-Ricci solitons on para-Kenmotsu manifolds. We obtain some results of ${\eta}$-Ricci solitons on para-Kenmotsu manifolds satisfying $R({\xi},X).C=0$, $R({\xi},X).{\tilde{M}}=0$, $R({\xi},X).P=0$, $R({\xi},X).{\tilde{C}}=0$ and $R({\xi},X).H=0$, where $C,\;{\tilde{M}},\;P,\;{\tilde{C}}$ and H are a quasi-conformal curvature tensor, a M-projective curvature tensor, a pseudo-projective curvature tensor, and a concircular curvature tensor and conharmonic curvature tensor, respectively.

THE RICCI TENSOR OF REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS

  • Perez Juan De Dios;Suh Young-Jin
    • Journal of the Korean Mathematical Society
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    • v.44 no.1
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    • pp.211-235
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    • 2007
  • In this paper, first we introduce the full expression of the curvature tensor of a real hypersurface M in complex two-plane Grass-mannians $G_2(\mathbb{C}^{m+2})$ from the equation of Gauss and derive a new formula for the Ricci tensor of M in $G_2(\mathbb{C}^{m+2})$. Next we prove that there do not exist any Hopf real hypersurfaces in complex two-plane Grassmannians $G_2(\mathbb{C}^{m+2})$ with parallel and commuting Ricci tensor. Finally we show that there do not exist any Einstein Hopf hypersurfaces in $G_2(\mathbb{C}^{m+2})$.

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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RIGIDITY CHARACTERIZATION OF COMPACT RICCI SOLITONS

  • Li, Fengjiang;Zhou, Jian
    • Journal of the Korean Mathematical Society
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    • v.56 no.6
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    • pp.1475-1488
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    • 2019
  • In this paper, we firstly define the Ricci mean value along the gradient vector field of the Ricci potential function and show that it is non-negative on a compact Ricci soliton. Furthermore a Ricci soliton is Einstein if and only if its Ricci mean value is vanishing. Finally, we obtain a compact Ricci soliton $(M^n,g)(n{\geq}3)$ is Einstein if its Weyl curvature tensor and the Kulkarni-Nomizu product of Ricci curvature are orthogonal.