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In this paper, we introduce and characterize a pseudo-generalized Ricci-recurrent spacetimes and produce an example to verify the existence of such a spacetime. Then we demonstrate that a conformally flat generalized Ricci-recurrent spacetime with certain condition is a pseudo quasi-Einstein spacetime. Besides, it is proved that a pseudo-generalized Ricci-recurrent generalized Robertson–Walker spacetime represents a perfect fluid spacetime. Lastly, we study the impact of this spacetime under f (ℛ , T 2) and f (ℛ ∗) gravity scenario and deduce several energy conditions. [ABSTRACT FROM AUTHOR]

In general, a perfect fluid spacetime is not a generalized Robertson–Walker spacetime and the converse is also not true. In this paper, it is shown that if a perfect fluid spacetime satisfies the critical point equation, then either the spacetime becomes a generalized Robertson–Walker spacetime and represents dark era or the vorticity of the fluid vanishes as well as the spacetime is expansion free. Besides, we prove that if a generalized Robertson–Walker spacetime with constant scalar curvature satisfies the critical point equation, then the spacetime becomes a perfect fluid spacetime. Next, the existence of critical point equation is established by a non-trivial example. Finally, we discuss the critical point equation in f (r) -gravity. For the model f (r) = r − α (1 − e − r α ) (α = constant and r is the scalar curvature of the spacetime), various energy conditions in terms of the scalar curvature are examined and state that the Universe is in an accelerating phase and satisfies the weak, null, dominant, and strong energy conditions. [ABSTRACT FROM AUTHOR]

First, we illustrate that a conformally flat generalized Ricci recurrent spacetime is a perfect fluid spacetime. As a consequence, we prove that such a spacetime represents a dust matter fluid, stiff matter and dark energy era under certain restrictions on the Ricci scalar. Then we establish that such a spacetime is a Robertson–Walker spacetime. Also, we note that in a conformally flat generalized Ricci recurrent spacetime, the fluid is shear-free, vorticity-free and its flow vector is hypersurface orthogonal. We investigate a conformally flat generalized Ricci recurrent spacetime as a solution of curvature-squared gravity theory. In this study, various energy conditions in terms of the Ricci scalar are examined and state that the Universe is in an accelerating phase and satisfies the weak, null, dominant, and strong energy conditions. [ABSTRACT FROM AUTHOR]

First, we illustrate that a Ricci symmetric quasi-Einstein spacetime is a static spacetime and belongs to Petrov classification I , D or O. We investigate conformally flat quasi-Einstein spacetime as a solution of f (ℛ , G) -gravity theory and explain the physical meaning of the Friedmann–Robertson–Walker metric. For the models f (ℛ , G) = μ ℛ β 1 G β 2 , ( β 1 , β 2 and μ are constants) and f (ℛ , G) = ℛ 2 − 6 G , various energy conditions in terms of associated scalars are examined. These models satisfy the weak, null, and dominant energy conditions, while violating the strong energy condition, which is in good agreement with recent empirical investigations that show the Universe is currently in an accelerating phase. [ABSTRACT FROM AUTHOR]

In this paper, we investigate almost Schouten solitons and almost gradient Schouten solitons in spacetimes of general relativity. At first, it is proven that if a generalized Robertson–Walker spacetime permits an almost Schouten soliton, then it becomes a perfect fluid spacetime as well as the spacetime represents a dark matter era. Besides this, we investigate almost gradient Schouten solitons in generalized Robertson–Walker spacetimes. Moreover, a spacetime obeying almost Schouten solitons whose potential vector field is a non-homothetic conformal vector field is of Petrov type O or N. [ABSTRACT FROM AUTHOR]

In this paper, we characterize weakly Ricci-symmetric (shortly, (WRS) 4 ) spacetimes and their solutions in f (ℛ) -gravity. It is demonstrated that a (WRS) 4 spacetime represents a stiff matter fluid. In addition, we obtain that a conformally flat (WRS) 4 spacetime is a space of quasi-constant sectional curvature. Moreover, we establish that a Ricci symmetric (WRS) 4 spacetime represents a static spacetime. Finally, we investigate the effect of (WRS) 4 spacetime solutions in f (ℛ) -gravity. [ABSTRACT FROM AUTHOR]

The main aim of this article is to investigate a spacetime of quasi‐constant sectional curvature. At first, the existence of such a spacetime is established by several examples. We have shown that a spacetime of quasi‐constant sectional curvature agrees with the present state of the universe and it represents a Robertson Walker spacetime. Moreover, if the spacetime is Ricci semi‐symmetric or Ricci symmetric, then either the spacetime represents a spacetime of constant sectional curvature, or the spacetime represents phantom era. Also, we prove that a Ricci symmetric spacetime of quasi‐constant sectional curvature represents a static spacetime and the spacetime under consideration is of Petrov type I, D or O. Finally, we concentrate on a quasi‐constant sectional curvature spacetime solution in F(R)$\mathcal {F}(\mathcal {R})$‐gravity. As a result, various energy conditions are studied and analyzed our obtained outcomes in terms of a F(R)$\mathcal {F}(\mathcal {R})$‐gravity model. The main aim of this article is to investigate a spacetime of quasi‐constant sectional curvature. At first, the existence of such a spacetime is established by several examples. We have shown that a spacetime of quasi‐constant sectional curvature agrees with the present state of the universe and it represents a Robertson Walker spacetime. Moreover, if the spacetime is Ricci semi‐symmetric or Ricci symmetric, then either the spacetime represents a spacetime of constant sectional curvature, or the spacetime represents phantom era. Also, we prove that a Ricci symmetric spacetime of quasi‐constant sectional curvature represents a static spacetime and the spacetime under consideration is of Petrov type I, D or O. Finally, we concentrate on a quasi‐constant sectional curvature spacetime solution in F(R)$\mathcal{F}(\mathcal{R})\ $‐gravity. As a result, various energy conditions are studied and analyzed our obtained outcomes in terms of a F(R)$\mathcal{F}(\mathcal{R})\ $‐gravity model. [ABSTRACT FROM AUTHOR]

This article concerns the study of perfect fluid spacetimes equipped with different types of gradient solitons. It is shown that if a perfect fluid spacetime with Killing velocity vector admits a τ-Einstein soliton of gradient type, then the spacetime represents phantom regime, or ψ remains invariant under the velocity vector field ρ. Besides, we establish that in a perfect fluid spacetime with constant scalar curvature, if the Lorentzian metric is the gradient τ-Einstein soliton, then either the τ-Einstein gradient potential function is pointwise collinear with ρ, or the spacetime represents stiff matter fluid. Furthermore, we prove that under certain conditions, a perfect fluid spacetime turns into a generalized Robertson–Walker spacetime, as well as a static spacetime and such a spacetime is of Petrov type I, D or O. We also characterize perfect fluid spacetimes whose Lorentzian metric is equipped with gradient m -quasi Einstein solitons and that the perfect fluid spacetime has vanishing expansion scalar, or it represents dark energy era under certain restriction on the potential function. [ABSTRACT FROM AUTHOR]

In this study, we analyze generalized quasi-Einstein spacetimes endowed with Gray's decomposition, as well as generalized Robertson–Walker spacetimes. It is shown that the Ricci tensor of a generalized quasi-Einstein spacetime assumes the form of a perfect fluid in all Gray's subspaces under certain restrictions. Finally, it is established that a generalized quasi-Einstein generalized Robertson–Walker spacetime is a perfect fluid spacetime. [ABSTRACT FROM AUTHOR]

In this article, we present a new tensor that generalizes the conharmonic curvature tensor, termed the ψ-conharmonic curvature tensor. In the first part, we derive some fundamental geometrical properties of a pseudo ψ-conharmonically symmetric manifold (briefly,) and give some fascinating outcomes. Among others, we establish that an Einstein manifold is of zero scalar curvature. Besides, we investigate ψ-conharmonically flat perfect fluid and spacetimes, respectively. As a result, we acquire some significant theorems. Finally, we construct several non-trivial examples to establish the existence of. [ABSTRACT FROM AUTHOR]

In this study, we analyze mixed quasi-Einstein spacetimes endowed with Gray's decomposition, as well as generalized Robertson–Walker spacetimes. For mixed quasi-Einstein spacetimes, the shape of the Ricci tensor in all (n) -invariant subspaces can be identified using Gray's decomposition of the gradient of the Ricci tensor. In case one, such a spacetime is found to be static; in three cases, the Ricci tensor is found to be in the form of a  perfect fluid; and in the other three situations, the spacetime becomes a quasi-Einstein spacetime under certain restrictions on the associated vector fields. Finally, it is established that a mixed quasi-Einstein generalized  Robertson–Walker spacetime is a perfect fluid spacetime. [ABSTRACT FROM AUTHOR]

We prove that an almost pseudo symmetric spacetime with Codazzi type of tensor is perfect fluid whose velocity vector field B i is parallel. The energy density σ of such perfect fluid spacetime is constant and the state equation is obtained. Also, this spacetime is shown to be a static spacetime. Next, it is shown that such spacetime is a GRW spacetime with Einstein fiber. This kind of spacetime is investigated in f (R) theory of gravity; in this case, we find the forms of the isotropic pressure p and the energy density σ which are constant. Further, some energy conditions are studied in this spacetime. Finally, a concrete example of almost pseudo symmetric spacetimes is considered. [ABSTRACT FROM AUTHOR]

In this paper, we study generalized quasi-Einstein manifolds (Mn, g, V, λ) satisfying certain geometric conditions on its potential vector field V whenever it is harmonic, conformal, and parallel. First, we construct some integral formulas and obtain some triviality results. Then, we find some necessary conditions to construct a quasi-Einstein structure on (Mn, g, V, λ). Moreover, we prove that for any generalized Ricci soliton ( M ̄ = I × f M , g ̄ , ξ ̄ , λ) , where g ̄ is a generalized Robertson–Walker spacetime metric and the potential field ξ ̄ = h ∂ t + ξ is conformal, ( M ̄ , g ̄ ) can be considered as the model of perfect fluids in general relativity. Moreover, the fiber (M, g) also satisfies the quasi-Einstein metric condition. Therefore, the state equation of ( M ̄ = I × f M , g ̄ ) is presented. We also construct some explicit examples of generalized quasi-Einstein metrics by using a four-dimensional Walker metric. [ABSTRACT FROM AUTHOR]

In this research paper, we determine the Ricci–Yamabe soliton on a perfect fluid spacetime with a torse-forming vector field. Besides this, we evaluate a specific situation when the potential vector field ζ is of the gradient type, we deduce a Poisson and a Liouville equation from the soliton equation. In addition, we explore some harmonic significance of γ -Ricci–Yamabe soliton on perfect fluid spacetime with a harmonic potential function ψ . Finally, we discuss necessary and sufficient conditions for the 1-form γ , which is the g-dual of the vector field ζ on a perfect fluid spacetime to be a solution for the Schrödinger–Ricci equation. [ABSTRACT FROM AUTHOR]

Classification theorems of locally symmetric, second-order locally symmetric, Ricci symmetric, semi-symmetric and Ricci semi-symmetric generalized Robertson-Walker (in short, GRW) spacetime I f × N are given. Some implications for these symmetries of GRW space-times are considered. A summary of sufficient conditions on the warping function f and the fibre manifold N is given to verify each symmetry. [ABSTRACT FROM AUTHOR]

This study aims mainly at investigating the effects of concircular flatness and concircular symmetry of a warped product manifold on its fiber and base manifolds. Concircularly flat and concircularly symmetric warped product manifolds are investigated. The divergence-free concircular curvature tensor on warped product manifolds is considered. Finally, we apply some of these results to generalized Robertson–Walker and standard static space-times. [ABSTRACT FROM AUTHOR]

Recently, it is proven that generalized Robertson–Walker space-times in all orthogonal subspaces of Gray's decomposition except one (unrestricted) are perfect fluid space-times. GRW space-times in the unrestricted subspace are identified by having constant scalar curvature. Generalized quasi-Einstein GRW space-times have a constant scalar curvature. It is shown that generalized quasi-Einstein GRW space-times reduce to Einstein space-times or perfect fluid space-times. [ABSTRACT FROM AUTHOR]

This paper concerns with the characterization of a spacetime and modified gravity such as f (R , G) , f (R , T) and f (R , L m) -gravity equipped with the projective curvature tensor. We establish that a projectively flat perfect fluid spacetime represents dark energy era. Also, we prove that a projectively flat perfect fluid spacetime is either locally isometric to Minkowski spacetime or a de-Sitter spacetime. Furthermore, it is shown that a perfect fluid spacetime admitting harmonic projective curvature tensor becomes a generalized Robertson–Walker spacetime and is of Petrov type I , D or O. Lastly, we investigate the effect of projectively flat perfect fluid spacetime solutions in f (R , G) , f (R , T) and f (R , L m) -gravity, respectively. We also investigate the spacetime as a f (R , G) -gravity solution and use the flat Friedmann–Robertson–Walker metric to establish a relation among jerk, snap, and deceleration parameters. Numerous energy conditions are studied in terms of Ricci scalar with the model f (R , G) = exp (R) + α (6 G) β . For this model, the strong energy condition is violated but the weak, dominant and null energy conditions are fulfilled, which is in excellent accordance with current observational investigations that show the universe is now accelerating. [ABSTRACT FROM AUTHOR]

Generalized Robertson–Walker (GRW) spacetime is the generalization of the Robertson–Walker (RW) spacetime and a further generalization of GRW spacetime is the twisted spacetime. In this paper, we generalize the results of the paper [C. A. Mantica, Y. J. Suh and U. C. De, A note on generalized Robertson–Walker spacetimes, Int. J. Geom. Methods Mod. Phys.13 (2016), Article Id: 1650079, 9 pp., doi: 101142/s0219887816500791 ]. We prove that a Ricci simple Lorentzian manifold with vanishing quasi-conformal curvature tensor is a RW spacetime. Further, we prove that a Ricci simple Lorentzian manifold with harmonic quasi-conformal curvature tensor is a GRW spacetime. As a consequence, we obtain several corollaries. Finally, we have cited some examples of the obtained results. [ABSTRACT FROM AUTHOR]

The present paper is concerned with the study of pseudo Z -symmetric spacetimes (in short, (P Z S) 4) admitting harmonic conformal curvature tensor in F (R) -gravity. First, we prove that a (P Z S) 4 spacetime admitting harmonic conformal curvature tensor is conformally flat and this spacetime is a RW spacetime. For these manifolds, we determine the scale factor a (t) in a spatially flat RW spacetime and we obtain the scalar curvature of this spacetime. We find the properties of (P Z S) 4 spacetimes with harmonic conformal curvature tensor. After that, the isotropic pressure and the energy density are found for such a manifold in F (R) -gravity. At the end, choosing F (R) = R , we prove that the density parameter is 1 and this spacetime has the properties of type I (Big Rip). [ABSTRACT FROM AUTHOR]

The object of the present paper is to study a spacetime admitting conharmonic curvature tensor and some geometric properties related to this spacetime. It is shown that in a conharmonically flat spacetime with cyclic parallel Ricci tensor, the energy-momentum tensor is cyclic parallel and conversely. Finally, we prove that for a radiative perfect fluid spacetime if the energy-momentum tensor satisfying the Einstein's equations without cosmological constant is generalized recurrent, then the fluid has vanishing vorticity and the integral curves of the vector field U are geodesics. [ABSTRACT FROM AUTHOR]

The object of this paper is to study spacetimes admitting W2-curvature tensor. At first we prove that a W2-flat spacetime is conformally flat and hence it is of Petrov type O. Next, we prove that if the perfect fluid spacetime with vanishing W2-curvature tensor obeys Einstein's field equation without cosmological constant, then the spacetime has vanishing acceleration vector and expansion scalar and the perfect fluid always behaves as a cosmological constant. It is also shown that in a perfect fluid spacetime of constant scalar curvature with divergence-free W2-curvature tensor, the energy-momentum tensor is of Codazzi type and the possible local cosmological structure of such a spacetime is of type I, D or O. [ABSTRACT FROM AUTHOR]

The objective of the present research article is to examine the behavior of spacetime with content of matter having magnetism say relativistic magneto-fluid spacetime. We discuss some geometrical properties of relativistic magneto-fluid spacetime. Next, we assume the relativistic magneto-fluid spacetime with Codazzi type and Ricci cyclic type matter tensor and study various results. In addition, we turn up some curvature characteristics of relativistic magneto-fluid spacetime. Finally, we provide some physical perception of relativistic magneto-fluid spacetime. [ABSTRACT FROM AUTHOR]

Our aim is to characterize the Lorentzian manifolds endowed with a type of semi-symmetric non-metric connection. It is proven that a Lorentzian manifold with a type of semi-symmetric non-metric connection is a GRW space-time. To minimize the gap between the RW space-time and the GRW space-time, we establish a bridge between them. It is shown that a weakly Ricci symmetric GRW space-time is a stiff matter fluid. We also classify the space-times by using the Petrov's classification. Finally, we construct an example of a 4-dimensional Lorentzian manifold admitting a semi-symmetric non-metric connection whose associated vector field is a unit timelike concircular vector field. [ABSTRACT FROM AUTHOR]

The object of the present paper is to introduce and investigate the P -curvature tensor that generalizes projective, conharmonic, M -projective and the set of W i curvature tensors introduced by Pokhariyal and Mishra. It is proven that pseudo-Riemannian manifolds admitting a traceless P -curvature tensor are Einstein and those admitting flat P -curvature tensor has a constant curvature. Classification theorems for pseudo-Riemannian manifolds admitting a divergence-free P -curvature tensor are given in each subspace of Gray's decomposition of the covariant derivative of the Ricci tensor. Space–times having a flat P -curvature tensor or a divergence free P -curvature tensor are scrutinized. Finally, perfect fluid space–times admitting various features of the P -curvature tensor are considered. [ABSTRACT FROM AUTHOR]

A generalized Robertson-Walker (GRW) space-time is the generalization of the classical Robertson-Walker space-time. In the present paper, we show that a Ricci simple manifold with vanishing divergence of the conformal curvature tensor admits a proper concircular vector field and it is necessarily a GRW space-time. Further, we show that a stiff matter perfect fluid space-time or a mass-less scalar field with time-like gradient and with divergence-free Weyl tensor are GRW space-times. [ABSTRACT FROM AUTHOR]

The object of the present paper is to study spacetimes admitting pseudo-projective curvature tensor. At first we prove that a pseudo-projectively flat spacetime is Einstein and hence it is of constant curvature and the energy momentum tensor of such a spacetime satisfying Einstein's field equation with cosmological constant is covariant constant. Next, we prove that if the perfect fluid spacetime with vanishing pseudoprojective curvature tensor obeys Einstein's field equation without cosmological constant, then the spacetime has constant energy density and isotropic pressure, and the perfect fluid always behaves as a cosmological constant and also such a spacetime is infinitesimally spatially isotropic relative to the unit timelike vector field U. Moreover, it is shown that a pseudo-projectively flat spacetime satisfying Einstein's equation without cosmological constant for a purely electromagnetic distribution is an Euclidean space. We also prove that under certain conditions a perfect fluid spacetime with divergence-free pseudo-projective curvature is a Robertson-Walker spacetime and the possible local cosmological structure of such a spacetime is of type I, D or O. We also study dust-like fluid spacetime with vanishing pseudo-projective curvature tensor. [ABSTRACT FROM AUTHOR]

A perfect-fluid space-time of dimension n = 4, with (1) irrotational velocity vector field and (2) null divergence of the Weyl tensor, is a generalised Robertson-Walker space-time with an 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 spacetime. [ABSTRACT FROM AUTHOR]