• Title, Summary, Keyword: $N({\kappa})$-contact metric manifolds

### On N(κ)-Contact Metric Manifolds Satisfying Certain Curvature Conditions

• De, Avik;Jun, Jae-Bok
• Kyungpook Mathematical Journal
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• v.51 no.4
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• pp.457-468
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• 2011
• We consider pseudo-symmetric and Ricci generalized pseudo-symmetric N(${\kappa}$) contact metric manifolds. We also consider N(${\kappa}$)-contact metric manifolds satisfying the condition $S{\cdot}R$ = 0 where R and S denote the curvature tensor and the Ricci tensor respectively. Finally we give some examples.

### CERTAIN SEMISYMMETRY PROPERTIES OF (𝜅, 𝜇)-CONTACT METRIC MANIFOLDS

• De, Uday Chand;Jun, Jae-Bok;Samui, Srimayee
• Bulletin of the Korean Mathematical Society
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• v.53 no.4
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• pp.1237-1247
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• 2016
• The object of the present paper is to characterize $({\kappa}$, ${\mu}$)-contact metric manifolds whose concircular curvature tensor satisfies certain semisymmetry conditions. We also verify that the result holds by a concrete example.

### ON WEAKLY EINSTEIN ALMOST CONTACT MANIFOLDS

• Chen, Xiaomin
• Journal of the Korean Mathematical Society
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• v.57 no.3
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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
• Communications of the Korean Mathematical Society
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• v.35 no.2
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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 ��2n+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 ��n+1 and a sphere ��n(4) of constant curvature +4.

### ON THE CONCIRCULAR CURVATURE TENSOR OF A CONTACT METRIC MANIFOLD

• BLAIR, D. E.;KIM, JEONG-SIK;TRIPATHI, MUKUT MANI
• Journal of the Korean Mathematical Society
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• v.42 no.5
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• pp.883-892
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• 2005
• We classify N($\kappa$)-contact metric manifolds which satisfy $Z(\xi,\;X)\cdotZ\;=\;0,\;Z(\xi,\;X)\cdotR\;=\;0\;or\;R(\xi,\;X)\cdotZ\;=\;0$.
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