68 research outputs found

    A note on Almost Riemann Soliton and gradient almost Riemann soliton

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    The quest of the offering article is to investigate \emph{almost Riemann soliton} and \emph{gradient almost Riemann soliton} in a non-cosymplectic normal almost contact metric manifold M3M^3. Before all else, it is proved that if the metric of M3M^3 is Riemann soliton with divergence-free potential vector field ZZ, then the manifold is quasi-Sasakian and is of constant sectional curvature -λ\lambda, provided α,β=\alpha,\beta = constant. Other than this, it is shown that if the metric of M3M^3 is \emph{ARS} and ZZ is pointwise collinear with ξ\xi and has constant divergence, then ZZ is a constant multiple of ξ\xi and the \emph{ARS} reduces to a Riemann soliton, provided α,  β=\alpha,\;\beta =constant. Additionally, it is established that if M3M^3 with α,  β=\alpha,\; \beta = constant admits a gradient \emph{ARS} (γ,ξ,λ)(\gamma,\xi,\lambda), then the manifold is either quasi-Sasakian or is of constant sectional curvature −(α2−β2)-(\alpha^2-\beta^2). At long last, we develop an example of M3M^3 conceding a Riemann soliton

    A condition for a perfect-fluid space-time to be a generalized Robertson-Walker space-time

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    A perfect-fluid space-time of dimension n>3 with 1) irrotational velocity vector field, 2) null divergence of the Weyl tensor, is a generalised Robertson-Walker space-time with Einstein fiber. Condition 1) is verified whenever pressure and energy density are related by an equation of state. The contraction of the Weyl tensor with the velocity vector field is zero. Conversely, a generalized Robertson-Walker space-time with null divergence of the Weyl tensor is a perfect-fluid space-time.Comment: 7 pages. Misprint corrected in Sect II

    On Almost Pseudo-Z-symmetric Manifolds

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    summary:The object of the present paper is to study almost pseudo-Z-symmetric manifolds. Some geometric properties have been studied. Next we consider conformally flat almost pseudo-Z-symmetric manifolds. We obtain a sufficient condition for an almost pseudo-Z-symmetric manifold to be a quasi Einstein manifold. Also we prove that a totally umbilical hypersurface of a conformally flat A(PZS)nA(PZS)_{n} (n>3n>3) is a manifold of quasi constant curvature. Finally, we give an example to verify the result already obtained in Section 5

    On Φ-recurrent N(k)-contact Metric Manifolds

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    In this paper we prove that a &#934;-recurrent N(k)-contact metric manifold is an &#951;-Einstein manifold with constant coefficients. Next, we prove that a 3-dimensional &#934;-recurrent N(k)-contact metric manifold is of constant curvature. The existence of a &#934;-recurrent N(k)-contact metric manifold is also proved.</p

    Some curvature properties of perfect fluid spacetimes

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    In this paper we assume that a perfect fluid is the source of the gravitational field while analyzing the solutions to the Einstein field equations

    Characterizations of Perfect fluid spacetimes obeying f(R)f(\mathcal{R})-gravity equipped with different gradient solitons

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    The prime object of this article is to study the perfect fluid spacetimes obeying f(R)f(\mathcal{R})-gravity, when η\eta-Ricci solitons, gradient η\eta-Ricci solitons, gradient Einstein Solitons and gradient mm-quasi Einstein solitons are its metrics. At first, the existence of the η\eta-Ricci solitons is proved by a non-trivial example. We establish conditions for which the η\eta-Ricci solitons are expanding, steady or shrinking. Besides, in the perfect fluid spacetimes obeying f(R)f(\mathcal{R})-gravity, when the potential vector field of η\eta-Ricci soliton is of gradient type, we acquire a Poisson equation. Moreover, we investigate gradient η\eta-Ricci solitons, gradient Einstein Solitons and gradient mm-quasi Einstein solitons in f(R)f(\mathcal{R})-gravity, respectively. As a result, we establish some significant theorems about dark matter era.Comment: 1
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