• Title/Summary/Keyword: quasi-Einstein

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ON A CLASS OF N(κ)-QUASI EINSTEIN MANIFOLDS

  • De, Avik;De, Uday Chand;Gazi, Abul Kalam
    • Communications of the Korean Mathematical Society
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    • v.26 no.4
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    • pp.623-634
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    • 2011
  • The object of the present paper is to study N(${\kappa}$)-quasi Einstein manifolds. Existence of N(${\kappa}$)-quasi Einstein manifolds are proved. Physical example of N(${\kappa}$)-quasi Einstein manifold is also given. Finally, Weyl-semisymmetric N(${\kappa}$)-quasi Einstein manifolds have been considered.

𝒵 Tensor on N(k)-Quasi-Einstein Manifolds

  • Mallick, Sahanous;De, Uday Chand
    • Kyungpook Mathematical Journal
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    • v.56 no.3
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    • pp.979-991
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    • 2016
  • The object of the present paper is to study N(k)-quasi-Einstein manifolds. We study an N(k)-quasi-Einstein manifold satisfying the curvature conditions $R({\xi},X){\cdot}Z=0$, $Z(X,{\xi}){\cdot}R=0$, and $P({\xi},X){\cdot}Z=0$, where R, P and Z denote the Riemannian curvature tensor, the projective curvature tensor and Z tensor respectively. Next we prove that the curvature condition $C{\cdot}Z=0$ holds in an N(k)-quasi-Einstein manifold, where C is the conformal curvature tensor. We also study Z-recurrent N(k)-quasi-Einstein manifolds. Finally, we construct an example of an N(k)-quasi-Einstein manifold and mention some physical examples.

Super Quasi-Einstein Manifolds with Applications to General Relativity

  • Mallick, Sahanous
    • Kyungpook Mathematical Journal
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    • v.58 no.2
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    • pp.361-375
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    • 2018
  • The object of the present paper is to study super quasi-Einstein manifolds. Some geometric properties of super quasi-Einstein manifolds have been studied. We also discuss $S(QE)_4$ spacetime with space-matter tensor and some properties related to it. Finally, we construct an example of a super quasi-Einstein spacetime.

QUASI-CONFORMAL CURVATURE TENSOR ON N (k)-QUASI EINSTEIN MANIFOLDS

  • Hazra, Dipankar;Sarkar, Avijit
    • Korean Journal of Mathematics
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    • v.29 no.4
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    • pp.801-810
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    • 2021
  • This paper deals with the study of N (k)-quasi Einstein manifolds that satisfies the certain curvature conditions 𝒞*·𝒞* = 0, 𝓢·𝒞* = 0 and ${\mathcal{R}}{\cdot}{\mathcal{C}}_*=f{\tilde{Q}}(g,\;{\mathcal{C}}_*)$, where 𝒞*, 𝓢 and 𝓡 denotes the quasi-conformal curvature tensor, Ricci tensor and the curvature tensor respectively. Finally, we construct an example of N (k)-quasi Einstein manifold.

GENERALIZED m-QUASI-EINSTEIN STRUCTURE IN ALMOST KENMOTSU MANIFOLDS

  • Mohan Khatri;Jay Prakash Singh
    • Bulletin of the Korean Mathematical Society
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    • v.60 no.3
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    • pp.717-732
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    • 2023
  • The goal of this paper is to analyze the generalized m-quasi-Einstein structure in the context of almost Kenmotsu manifolds. Firstly we showed that a complete Kenmotsu manifold admitting a generalized m-quasi-Einstein structure (g, f, m, λ) is locally isometric to a hyperbolic space ℍ2n+1(-1) or a warped product ${\tilde{M}}{\times}{_{\gamma}{\mathbb{R}}$ under certain conditions. Next, we proved that a (κ, µ)'-almost Kenmotsu manifold with h' ≠ 0 admitting a closed generalized m-quasi-Einstein metric is locally isometric to some warped product spaces. Finally, a generalized m-quasi-Einstein metric (g, f, m, λ) in almost Kenmotsu 3-H-manifold is considered and proved that either it is locally isometric to the hyperbolic space ℍ3(-1) or the Riemannian product ℍ2(-4) × ℝ.

Generalized Quasi-Einstein Metrics and Contact Geometry

  • Biswas, Gour Gopal;De, Uday Chand;Yildiz, Ahmet
    • Kyungpook Mathematical Journal
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    • v.62 no.3
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    • pp.485-495
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    • 2022
  • The aim of this paper is to characterize K-contact and Sasakian manifolds whose metrics are generalized quasi-Einstein metric. It is proven that if the metric of a K-contact manifold is generalized quasi-Einstein metric, then the manifold is of constant scalar curvature and in the case of a Sasakian manifold the metric becomes Einstein under certain restriction on the potential function. Several corollaries have been provided. Finally, we consider Sasakian 3-manifold whose metric is generalized quasi-Einstein metric.

ON LORENTZIAN QUASI-EINSTEIN MANIFOLDS

  • Shaikh, Absos Ali;Kim, Young-Ho;Hui, Shyamal Kumar
    • Journal of the Korean Mathematical Society
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    • v.48 no.4
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    • pp.669-689
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    • 2011
  • The notion of quasi-Einstein manifolds arose during the study of exact solutions of the Einstein field equations as well as during considerations of quasi-umbilical hypersurfaces. For instance, the Robertson-Walker spacetimes are quasi-Einstein manifolds. The object of the present paper is to study Lorentzian quasi-Einstein manifolds. Some basic geometric properties of such a manifold are obtained. The applications of Lorentzian quasi-Einstein manifolds to the general relativity and cosmology are investigated. Theories of gravitational collapse and models of Supernova explosions [5] are based on a relativistic fluid model for the star. In the theories of galaxy formation, relativistic fluid models have been used in order to describe the evolution of perturbations of the baryon and radiation components of the cosmic medium [32]. Theories of the structure and stability of neutron stars assume that the medium can be treated as a relativistic perfectly conducting magneto fluid. Theories of relativistic stars (which would be models for supermassive stars) are also based on relativistic fluid models. The problem of accretion onto a neutron star or a black hole is usually set in the framework of relativistic fluid models. Among others it is shown that a quasi-Einstein spacetime represents perfect fluid spacetime model in cosmology and consequently such a spacetime determines the final phase in the evolution of the universe. Finally the existence of such manifolds is ensured by several examples constructed from various well known geometric structures.