• Title, Summary, Keyword: Ricci soliton

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A NOTE ON ALMOST RICCI SOLITON AND GRADIENT ALMOST RICCI SOLITON ON PARA-SASAKIAN MANIFOLDS

  • De, Krishnendu;De, Uday Chand
    • Korean Journal of Mathematics
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    • v.28 no.4
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    • pp.739-751
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    • 2020
  • The object of the offering exposition is to study almost Ricci soliton and gradient almost Ricci soliton in 3-dimensional para-Sasakian manifolds. At first, it is shown that if (g, V, λ) be an almost Ricci soliton on a 3-dimensional para-Sasakian manifold M, then it reduces to a Ricci soliton and the soliton is expanding for λ=-2. Besides these, in this section, we prove that if V is pointwise collinear with ξ, then V is a constant multiple of ξ and the manifold is of constant sectional curvature -1. Moreover, it is proved that if a 3-dimensional para-Sasakian manifold admits gradient almost Ricci soliton under certain conditions then either the manifold is of constant sectional curvature -1 or it reduces to a gradient Ricci soliton. Finally, we consider an example to justify some results of our paper.

RICCI SOLITONS AND RICCI ALMOST SOLITONS ON PARA-KENMOTSU MANIFOLD

  • Patra, Dhriti Sundar
    • Bulletin of the Korean Mathematical Society
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    • v.56 no.5
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    • pp.1315-1325
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    • 2019
  • The purpose of this article is to study the Ricci solitons and Ricci almost solitons on para-Kenmotsu manifold. First, we prove that if a para-Kenmotsu metric represents a Ricci soliton with the soliton vector field V is contact, then it is Einstein and the soliton is shrinking. Next, we prove that if a ${\eta}$-Einstein para-Kenmotsu metric represents a Ricci soliton, then it is Einstein with constant scalar curvature and the soliton is shrinking. Further, we prove that if a para-Kenmotsu metric represents a gradient Ricci almost soliton, then it is ${\eta}$-Einstein. This result is also hold for Ricci almost soliton if the potential vector field V is pointwise collinear with the Reeb vector field ${\xi}$.

ALMOST HERMITIAN SUBMERSIONS WHOSE TOTAL MANIFOLDS ADMIT A RICCI SOLITON

  • Gunduzalp, Yilmaz
    • Honam Mathematical Journal
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    • v.42 no.4
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    • pp.733-745
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    • 2020
  • The object of the present paper is to study the almost Hermitian submersion from an almost Hermitian manifold admits a Ricci soliton. Where, we investigate any fibre of such a submersion is a Ricci soliton or Einstein. We also get necessary conditions for which the base manifold of an almost Hermitian submersion is a Ricci soliton or Einstein. Moreover, we obtain the harmonicity of an almost Hermitian submersion from a Ricci soliton to an almost Hermitian manifold.

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.

THE k-ALMOST RICCI SOLITONS AND CONTACT GEOMETRY

  • Ghosh, Amalendu;Patra, Dhriti Sundar
    • Journal of the Korean Mathematical Society
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    • v.55 no.1
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    • pp.161-174
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    • 2018
  • The aim of this article is to study the k-almost Ricci soliton and k-almost gradient Ricci soliton on contact metric manifold. First, we prove that if a compact K-contact metric is a k-almost gradient Ricci soliton, then it is isometric to a unit sphere $S^{2n+1}$. Next, we extend this result on a compact k-almost Ricci soliton when the flow vector field X is contact. Finally, we study some special types of k-almost Ricci solitons where the potential vector field X is point wise collinear with the Reeb vector field ${\xi}$ of the contact metric structure.

RICCI 𝜌-SOLITONS ON 3-DIMENSIONAL 𝜂-EINSTEIN ALMOST KENMOTSU MANIFOLDS

  • Azami, Shahroud;Fasihi-Ramandi, Ghodratallah
    • Communications of the Korean Mathematical Society
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    • v.35 no.2
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    • pp.613-623
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    • 2020
  • The notion of quasi-Einstein metric in theoretical physics and in relation with string theory is equivalent to the notion of Ricci soliton in differential geometry. Quasi-Einstein metrics or Ricci solitons serve also as solution to Ricci flow equation, which is an evolution equation for Riemannian metrics on a Riemannian manifold. Quasi-Einstein metrics are subject of great interest in both mathematics and theoretical physics. In this paper the notion of Ricci 𝜌-soliton as a generalization of Ricci soliton is defined. We are motivated by the Ricci-Bourguignon flow to define this concept. We show that if a 3-dimensional almost Kenmotsu Einstein manifold M is a 𝜌-soliton, then M is a Kenmotsu manifold of constant sectional curvature -1 and the 𝜌-soliton is expanding with λ = 2.

CERTAIN RESULTS ON CONTACT METRIC GENERALIZED (κ, µ)-SPACE FORMS

  • Huchchappa, Aruna Kumara;Naik, Devaraja Mallesha;Venkatesha, Venkatesha
    • Communications of the Korean Mathematical Society
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    • v.34 no.4
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    • pp.1315-1328
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    • 2019
  • The object of the present paper is to study ${\eta}$-recurrent ${\ast}$-Ricci tensor, ${\ast}$-Ricci semisymmetric and globally ${\varphi}-{\ast}$-Ricci symmetric contact metric generalized (${\kappa}$, ${\mu}$)-space form. Besides these, ${\ast}$-Ricci soliton and gradient ${\ast}$-Ricci soliton in contact metric generalized (${\kappa}$, ${\mu}$)-space form have been studied.

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 RICCI ALMOST SOLITONS ON TWO CLASSES OF ALMOST KENMOTSU MANIFOLDS

  • Wang, Yaning
    • Journal of the Korean Mathematical Society
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    • v.53 no.5
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    • pp.1101-1114
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    • 2016
  • Let ($M^{2n+1}$, ${\phi}$, ${\xi}$, ${\eta}$, g) be a (k, ${\mu}$)'-almost Kenmotsu manifold with k < -1 which admits a gradient Ricci almost soliton (g, f, ${\lambda}$), where ${\lambda}$ is the soliton function and f is the potential function. In this paper, it is proved that ${\lambda}$ is a constant and this implies that $M^{2n+1}$ is locally isometric to a rigid gradient Ricci soliton ${\mathbb{H}}^{n+1}(-4){\times}{\mathbb{R}}^n$, and the soliton is expanding with ${\lambda}=-4n$. Moreover, if a three dimensional Kenmotsu manifold admits a gradient Ricci almost soliton, then either it is of constant sectional curvature -1 or the potential vector field is pointwise colinear with the Reeb vector field.