• Title/Summary/Keyword: conformal gradient Ricci soliton

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SOME RESULTS IN η-RICCI SOLITON AND GRADIENT ρ-EINSTEIN SOLITON IN A COMPLETE RIEMANNIAN MANIFOLD

  • Mondal, Chandan Kumar;Shaikh, Absos Ali
    • Communications of the Korean Mathematical Society
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    • v.34 no.4
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    • pp.1279-1287
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    • 2019
  • The main purpose of the paper is to prove that if a compact Riemannian manifold admits a gradient ${\rho}$-Einstein soliton such that the gradient Einstein potential is a non-trivial conformal vector field, then the manifold is isometric to the Euclidean sphere. We have showed that a Riemannian manifold satisfying gradient ${\rho}$-Einstein soliton with convex Einstein potential possesses non-negative scalar curvature. We have also deduced a sufficient condition for a Riemannian manifold to be compact which satisfies almost ${\eta}$-Ricci soliton.

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.

CERTAIN SOLITONS ON GENERALIZED (𝜅, 𝜇) CONTACT METRIC MANIFOLDS

  • Sarkar, Avijit;Bhakta, Pradip
    • Korean Journal of Mathematics
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    • v.28 no.4
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    • pp.847-863
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    • 2020
  • The aim of the present paper is to study some solitons on three dimensional generalized (𝜅, 𝜇)-contact metric manifolds. We study gradient Yamabe solitons on three dimensional generalized (𝜅, 𝜇)-contact metric manifolds. It is proved that if the metric of a three dimensional generalized (𝜅, 𝜇)-contact metric manifold is gradient Einstein soliton then ${\mu}={\frac{2{\kappa}}{{\kappa}-2}}$. It is shown that if the metric of a three dimensional generalized (𝜅, 𝜇)-contact metric manifold is closed m-quasi Einstein metric then ${\kappa}={\frac{\lambda}{m+2}}$ and 𝜇 = 0. We also study conformal gradient Ricci solitons on three dimensional generalized (𝜅, 𝜇)-contact metric manifolds.

RICCI-BOURGUIGNON SOLITONS AND FISCHER-MARSDEN CONJECTURE ON GENERALIZED SASAKIAN-SPACE-FORMS WITH 𝛽-KENMOTSU STRUCTURE

  • Sudhakar Kumar Chaubey;Young Jin Suh
    • Journal of the Korean Mathematical Society
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    • v.60 no.2
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    • pp.341-358
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    • 2023
  • Our aim is to study the properties of Fischer-Marsden conjecture and Ricci-Bourguignon solitons within the framework of generalized Sasakian-space-forms with 𝛽-Kenmotsu structure. It is proven that a (2n + 1)-dimensional generalized Sasakian-space-form with 𝛽-Kenmotsu structure satisfying the Fischer-Marsden equation is a conformal gradient soliton. Also, it is shown that a generalized Sasakian-space-form with 𝛽-Kenmotsu structure admitting a gradient Ricci-Bourguignon soliton is either ψ∖Tk × M2n+1-k or gradient 𝜂-Yamabe soliton.

*-CONFORMAL RICCI SOLITONS ON ALMOST COKÄHLER MANIFOLDS

  • Tarak Mandal;Avijit Sarkar
    • Communications of the Korean Mathematical Society
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    • v.38 no.3
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    • pp.865-880
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    • 2023
  • The main intention of the current paper is to characterize certain properties of *-conformal Ricci solitons on non-coKähler (𝜅, 𝜇)-almost coKähler manifolds. At first, we find that there does not exist *-conformal Ricci soliton if the potential vector field is the Reeb vector field θ. We also prove that the non-coKähler (𝜅, 𝜇)-almost coKähler manifolds admit *-conformal Ricci solitons if the potential vector field is the infinitesimal contact transformation. It is also studied that there does not exist *-conformal gradient Ricci solitons on the said manifolds. An example has been constructed to verify the obtained results.