On Lorentzian Quasi-Einstein Manifolds

Shaikh, Absos Ali; Kim, Young Ho; Hui, Shyamal Kumar

doi:10.4134/jkms.2011.48.4.669

Cited by 37 publications

(27 citation statements)

References 16 publications

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“…, for some scalar α. For k = 1 (resp., k = 0) the manifold is called quasi-Einstein (resp., Einstein) and the Ricci tensor locally takes the form [57,58,59]…”

Section: Preliminariesmentioning

confidence: 99%

Curvature properties of Melvin magnetic metric

Shaikh

Ali

Alkhaldi

et al. 2020

Journal of Geometry and Physics

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This paper aims to investigate the curvature restricted geometric properties admitted by Melvin magnetic spacetime metric, a warped product metric with 1-dimensional fibre.For this, we have considered a Melvin type static, cylindrically symmetric spacetime metric in Weyl form and it is found that such metric, in general, is generalized Roter type, Ein(3) and has pseudosymmetric Weyl conformal tensor satisfying the pseudosymmetric type condition R · R − Q(S, R) = L ′ Q(g, C). The condition for which it satisfies the Roter type condition has been obtained. It is interesting to note that Melvin magnetic metric is pseudosymmetric and pseudosymmetric due to conformal tensor. Moreover such metric is 2-quasi-Einstien, its Ricci tensor is Reimann compatible and Weyl conformal 2-forms are recurrent. The Maxwell tensor is also pseudosymmetric type. 0 * Corresponding author.

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“…, for some scalar α. For k = 1 (resp., k = 0) the manifold is called quasi-Einstein (resp., Einstein) and the Ricci tensor locally takes the form [57,58,59]…”

Section: Preliminariesmentioning

confidence: 99%

Curvature properties of Melvin magnetic metric

Shaikh

Ali

Alkhaldi

et al. 2020

Journal of Geometry and Physics

Self Cite

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“…Thus we have proved Theorem 4.1 Let M × F N be a warped product manifold with 1-dimensional base manifold (M ,ḡ) and non-Einstein (n − 1)-dimensional fiber ( N ,g), n 4. If (4) is satisfied on M × F N then on the set U we have (45) and fiber manifold ( N ,g) is a Riccipseudosymmetric manifold of constant type (see e.g. [32]), precisely R · S = εa 2 Q(g, S) .…”

Section: Warped Products With Non-einsteinian Fibrementioning

confidence: 99%

On generalized Robertson--Walker spacetimes satisfying some curvaturecondition

Arslan¹,

Deszcz²,

Ezentaş³

et al. 2014

Turk J Math

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We give necessary and sufficient conditions for warped product manifolds (M, g) , of dimension ⩾ 4 , with 1 -dimensional base, and in particular, for generalized Robertson-Walker spacetimes, to satisfy some generalized Einstein metric condition. Namely, the difference tensor R•C −C •R , formed from the curvature tensor R and the Weyl conformal curvature tensor C , is expressed by the Tachibana tensor Q(S, R) formed from the Ricci tensor S and R . We also construct suitable examples of such manifolds. They are quasi-Einstein, i.e. at every point of M rank (S − α g) ⩽ 1 , for some α ∈ R , or non-quasi-Einstein.

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“…Kim and S.K. Hui [17] have been studied on Lorentzian quasi Einstein spacetimes. Many authors have been generalized the notion of quasi Einstein manifolds, in different ways such as nearly quasi Einstein manifolds [5], generalized quasi Einstein manifolds [2], N.k/-quasi Einstein manifolds [4,20] and so on.…”

Section: Introductionmentioning

confidence: 99%

On ricci symmetric generalized quasi einstein spacetimes

Güler¹,

Demirbağ²

2015

Miskolc Math Notes

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Abstract. The object of the present paper is to prove the existence of a generalized quasi Einstein spacetime, briefly G.QE/ 4 , by constructing a non-trivial Lorentzian metric and to study such spacetime. First, we prove that every W 2 -Ricci pseudosymmetric G.QE/ 4 is an N.k/-quasi Einstein spacetime which can be considered as a model of perfect fluid, in general relativity. Then, we consider Ricci symmetric G.QE/ 4 and we prove that in such spacetime satisfying Einstein's field equations, the energy density and the isotropic pressure are constants. As a consequence of this result, the expansion scalar and the acceleration vector vanish and also the possible local cosmological structures of this spacetime obeying Einstein's field equations are of Petrov I, D or O.

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On Lorentzian Quasi-Einstein Manifolds

Cited by 37 publications

References 16 publications

Curvature properties of Melvin magnetic metric

Curvature properties of Melvin magnetic metric

On generalized Robertson--Walker spacetimes satisfying some curvaturecondition

On ricci symmetric generalized quasi einstein spacetimes

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