• 제목/요약/키워드: (${\kappa},{\mu}$)-contact metric manifold

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

  • Sarkar, Avijit;Bhakta, Pradip
    • Korean Journal of Mathematics
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    • 제28권4호
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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.

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

  • Huchchappa, Aruna Kumara;Naik, Devaraja Mallesha;Venkatesha, Venkatesha
    • 대한수학회논문집
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    • 제34권4호
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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.

ON WEAKLY EINSTEIN ALMOST CONTACT MANIFOLDS

  • Chen, Xiaomin
    • 대한수학회지
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    • 제57권3호
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    • pp.707-719
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    • 2020
  • In this article we study almost contact manifolds admitting weakly Einstein metrics. We first prove that if a (2n + 1)-dimensional Sasakian manifold admits a weakly Einstein metric, then its scalar curvature s satisfies -6 ⩽ s ⩽ 6 for n = 1 and -2n(2n + 1) ${\frac{4n^2-4n+3}{4n^2-4n-1}}$ ⩽ s ⩽ 2n(2n + 1) for n ⩾ 2. Secondly, for a (2n + 1)-dimensional weakly Einstein contact metric (κ, μ)-manifold with κ < 1, we prove that it is flat or is locally isomorphic to the Lie group SU(2), SL(2), or E(1, 1) for n = 1 and that for n ⩾ 2 there are no weakly Einstein metrics on contact metric (κ, μ)-manifolds with 0 < κ < 1. For κ < 0, we get a classification of weakly Einstein contact metric (κ, μ)-manifolds. Finally, it is proved that a weakly Einstein almost cosymplectic (κ, μ)-manifold with κ < 0 is locally isomorphic to a solvable non-nilpotent Lie group.

GRADIENT EINSTEIN-TYPE CONTACT METRIC MANIFOLDS

  • Kumara, Huchchappa Aruna;Venkatesha, Venkatesha
    • 대한수학회논문집
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    • 제35권2호
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    • pp.639-651
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    • 2020
  • Consider a gradient Einstein-type metric in the setting of K-contact manifolds and (κ, µ)-contact manifolds. First, it is proved that, if a complete K-contact manifold admits a gradient Einstein-type metric, then M is compact, Einstein, Sasakian and isometric to the unit sphere 𝕊2n+1. Next, it is proved that, if a non-Sasakian (κ, µ)-contact manifolds admits a gradient Einstein-type metric, then it is flat in dimension 3, and for higher dimension, M is locally isometric to the product of a Euclidean space 𝔼n+1 and a sphere 𝕊n(4) of constant curvature +4.