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The aim of the present article is to characterize (almost) coKähler manifolds admitting η$$ \eta $$‐Ricci‐Bourguignon solitons. Among others, it is proved that if a three‐dimensional compact almost coKähler manifold admits η$$ \eta $$‐Ricci‐Bourguignon soliton and the potential vector field is pointwise collinear with the Reeb vector field, then the manifold is K‐almost coKähler and η$$ \eta $$‐Einstein. Gradient η$$ \eta $$‐Ricci‐Bourguignon solitons on such manifolds are also considered. Finally, we construct an example of almost coKähler manifold to verify a result. [ABSTRACT FROM AUTHOR]

In this article, generalized Robertson Walker spacetimes are investigated in light of perfect fluid spacetimes. First, we establish that a perfect fluid spacetime with nonvanishing vorticity whose associated scalars are constant along the velocity vector field becomes a generalized Robertson Walker spacetime. Among others, it is also shown that a Ricci parallel perfect fluid spacetime is either a generalized Robertson Walker spacetime or a static spacetime. Finally, we acquire that in a conformally semisymmetric generalized Robertson Walker spacetime of dimension 4, the scalar curvature vanishes and the spacetime is locally isometric to the Minkowski spacetime, provided the electric part of the Weyl tensor vanishes. Moreover, it is established that the last result also holds in a conformally recurrent generalized Robertson Walker spacetime. [ABSTRACT FROM AUTHOR]

A generalized Robertson–Walker spacetime is not, in general, a perfect fluid spacetime and the converse is not, in general, true. In this paper, we show that if a perfect fluid spacetime admits an η – ρ -Einstein soliton, then the integral curves generated by the velocity vector field u are geodesics and the acceleration vector vanishes. Also, we show that if a perfect fluid spacetime with Killing velocity vector field admits a gradient η – ρ -Einstein soliton, then the spacetime represents either dark matter era or, the acceleration vector vanishes. Next, we show that if a generalized Robertson–Walker spacetime admits an η – ρ -Einstein soliton, then it becomes a perfect fluid spacetime. Finally, we prove that in a dark matter fluid with vanishing vorticity satisfying gradient η – ρ -Einstein soliton in f (ℛ) -gravity with constant Ricci scalar, either the energy density and isotropic pressure are constant or, the potential function ψ remains invariant under the velocity vector u. Also, for two models f (ℛ) = e α ℛ + α log ℛ and f (ℛ) = ℛ n (α = constant and ℛ is the Ricci scalar of the spacetime), 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 and dominant energy conditions, but violate the strong energy condition. [ABSTRACT FROM AUTHOR]

The prime object of this paper is to study the perfect fluid spacetimes obeying f (ℛ) -gravity, when η -Ricci solitons, gradient η -Ricci solitons, gradient Einstein solitons and gradient m -quasi Einstein solitons are its metrics. At first, the existence of the η -Ricci solitons is proved by a non-trivial example. We establish conditions for which the η -Ricci solitons are expanding, steady or shrinking. Besides, in the perfect fluid spacetimes obeying f (ℛ) -gravity, when the potential vector field of η -Ricci soliton is of gradient type, we acquire a Poisson equation. Moreover, we investigate gradient η -Ricci solitons, gradient Einstein solitons and gradient m -quasi Einstein solitons in f (ℛ) -gravity, respectively. As a result, we establish some significant theorems about dark matter era. [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 the η -Einstein cosymplectic manifolds with the gradient Einstein solitons and the conformal vector fields. It is proven that if an η -Einstein cosymplectic manifold M 2 n + 1 of dimension 2 n + 1 with n > 1 admits a gradient Einstein soliton, then either M 2 n + 1 is Ricci flat or the gradient of Einstein potential function is pointwise collinear with the Reeb vector field of M 2 n + 1 . Also, we prove that if M 2 n + 1 admits a conformal vector field W , then either M 2 n + 1 is Ricci flat or W is homothetic. We also establish that a conformal vector field W on M 2 n + 1 is a strict infinitesimal contact transformation, provided the scalar curvature of M 2 n + 1 is constant. Finally, the non-trivial examples of cosymplectic manifolds admitting the gradient Einstein solitons are given. [ABSTRACT FROM AUTHOR]

In the present article, our aim is to characterize Bach flat paraSasakian manifolds. It is established that a Bach flat paraSasakian manifold of dimension greater than three is of constant scalar curvature. Next, we prove that if the metric of a Bach flat paraSasakian manifold is a Yamabe soliton, then the soliton field becomes a Killing vector field. Finally, it is shown that a 3-dimensional Bach flat paraSasakian manifold is locally isometric to the hyperbolic space H2n+1(1). [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 article, we presumed that a perfect fluid is the source of the gravitational field while analyzing the solutions to the Einstein field equations. With this new and creative approach, here we study k -almost Yamabe solitons and gradient k -almost Yamabe solitons. First, two examples are constructed to ensure the existence of gradient k -almost Yamabe solitons. Then we show that if a perfect fluid spacetime admits a k -almost Yamabe soliton, then its potential vector field is Killing if and only if the divergence of the potential vector field vanishes. Besides, we prove that if a perfect fluid spacetime permits a k -almost Yamabe soliton (g, k, ρ, λ), then the integral curves of the vector field ρ are geodesics, the spacetime becomes stationary and the isotopic pressure and energy density remain invariant under the velocity vector field ρ. Also, we establish that if the potential vector field is pointwise collinear with the velocity vector field and ρ(a) = 0 where a is a scalar, then either the perfect fluid spacetime represents a phantom era, or the potential function Φ is invariant under the velocity vector field ρ. Finally, we prove that if a perfect fluid spacetime permits a gradient k -almost Yamabe soliton (g, k,DΦ, λ) and R, λ, k are invariant under ρ, then the vorticity of the fluid vanishes. [ABSTRACT FROM AUTHOR]

The aim of this paper is to characterize almost co-Kähler manifolds and co-Kähler three-manifolds whose metrices are the gradient ρ -Einstein solitons. At first we prove that a proper (κ ̃ , μ ̃) -almost co-Kähler manifold with κ ̃ < 0 does not admit gradient ρ -Einstein soliton. It is also shown that if a proper -Einstein almost co-Kähler manifold with constant coefficients admits a gradient ρ -Einstein soliton, then either the manifold is a K -almost co-Kähler manifold or the soliton is trivial. Next, we prove that in case of co-Kähler three-manifold the manifold is of constant scalar curvature. Moreover, either the manifold is flat or the gradient of the potential function is collinear with the Reeb vector field ξ. Finally, we construct two examples to illustrate our results. [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]

In this paper, we characterize almost co-Kähler manifolds with a conformal vector field. It is proven that if an almost co-Kähler manifold has a conformal vector field that is collinear with the Reeb vector field, then the manifold is a K-almost co-Kähler manifold. It is also shown that if a (κ, μ)-almost co-Kähler manifold admits a Killing vector field V, then either the manifold is K-almost co-Kähler or the vector field V is an infinitesimal strict contact transformation, provided that the (1,1) tensor h remains invariant under the Killing vector field. [ABSTRACT FROM AUTHOR]

The aim of this paper is to characterize paracontact metric manifolds admitting Yamabe and quasi-Yamabe solitons. Several results of such solitons are proved. In particular, we classify Yamabe and quasi-Yamabe solitons on (k , μ) -paracontact metric manifolds. [ABSTRACT FROM AUTHOR]

In this paper, we study almost coKähler manifolds admitting k -almost Yamabe solitons (g , V , k , λ). First, we obtain a classification of almost coKähler (κ , μ) -manifolds admitting nontrivial closed k -almost Yamabe solitons. Next, we consider an almost α -coKähler manifold admitting a nontrivial k -almost Yamabe soliton and prove that it is locally the Riemannian product of an almost Kähler manifold with the real line if the potential vector field V is collinear with the Reeb vector field. For the potential vector field V being orthogonal to the Reeb vector field, we also obtain two results. [ABSTRACT FROM AUTHOR]

This paper delivers the characterization of a pseudo B-symmetric spacetimes and we illustrate that a pseudo B-symmetric spacetime admitting Codazzi type of B-tensor represents a perfect fluid spacetime and if this spacetime admits the time-like convergence criterion, then the pseudo B-symmetric spacetime fulfills cosmic strong energy criterion and contains pure matter. Besides, we find in a pseudo B-symmetric spacetime with Codazzi type of B-tensor the electric part of the Weyl tensor vanishes and has Riemann and Weyl compatible vector fields. Furthermore, it is established that the chosen spacetime with Codazzi type of B-tensor is conformally flat and represents a Robertson–Walker spacetime. Also, we calculate the scale factor Ψ(t) for these spacetimes in a spatially flat Robertson–Walker spacetime. Finally, we study the impact of this spacetime under f(R) gravity scenario and deduce several energy conditions by considering a new model f(ℛ) = e(αℛ) −ln(βℛ) in which α and β are positive constants. [ABSTRACT FROM AUTHOR]

We characterize almost co-Kähler manifolds with gradient Yamabe, gradient Einstein and quasi-Yamabe solitons. It is proved that if the metric of a (κ , μ) -almost co-Kähler manifold M 2 n + 1 is a gradient Yamabe soliton, then M 2 n + 1 is either K -almost co-Kähler or N (κ) -almost co-Kähler or the metric of M 2 n + 1 is a trivial gradient Yamabe soliton. A (κ , μ) -almost co-Kähler manifold with gradient Einstein soliton is K -almost co-Kähler. Finally, it is shown that an almost co-Kähler manifold admitting a quasi-Yamabe soliton, whose soliton vector is pointwise collinear with the Reeb vector field of the manifold, is K -almost co-Kähler. Consequently, some results of almost co-Kähler manifolds are deduced. [ABSTRACT FROM AUTHOR]

The object of the present paper is to study almost Ricci solitons and gradient almost Ricci solitons in 3-dimensional LP-Sasakian manifolds. We prove that if (g, V, δ) is an almost Ricci soliton on a 3-dimensional LP-Sasakian manifold M³, then it reduces to a Ricci soliton and the soliton is shrinking for δ=2. Furthermore, if the scalar curvature is constant on M³, then the potential vector field is Killing. Also, if the manifold admits a gradient almost Ricci soliton (f, ρ, δ), then the manifold is locally isometric to the unit sphere Sn(1). [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study Ricci semisymmetric paracontact metric manifolds satisfying ∇ ξ ⁡ h = 0 {\nabla_{\xi}h=0} and such that the sectional curvature of the plane section containing ξ equals a non-zero constant c. Also, we study paracontact metric manifolds satisfying the curvature condition Q ⋅ R = 0 {Q\cdot R=0} , where Q and R are the Ricci operator and the Riemannian curvature tensor, respectively, and second order symmetric parallel tensors in paracontact metric manifolds under the same conditions. Several consequences of these results are discussed. [ABSTRACT FROM AUTHOR]

The object of the present paper is to study mixed quasi-Einstein manifolds. Some geometric properties of mixed quasi-Einstein manifolds have been studied. We also discuss spacetime with space-matter tensor and some properties related to it. Finally, we construct an example of a mixed quasi-Einstein spacetime. [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 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]

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]

We characterize the three-dimensional Riemannian manifolds equipped with a semi-symmetric metric ρ -connection under the assumption that the Riemannian metric is a Yamabe soliton. It is shown that a three-dimensional Riemannian manifold endowed with a semi-symmetric ρ -connection, whose metric is Yamabe soliton, is a manifold of constant sectional curvature − 1 and the Yamabe soliton is expanding with soliton constant − 6. Finally, we give an example of a three-dimensional Riemannian manifold and validate our findings. [ABSTRACT FROM AUTHOR]