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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]

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]

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. With this new and innovative approach, we investigate some curvature properties of perfect fluid spacetimes. Firstly, we provide a necessary condition for a Lorentzian manifold to be a perfect fluid spacetime. We prove that a conformally flat perfect fluid space-time is a spacetime of quasi constant sectional curvature. Also, we acquire that a conharmonically flat perfect fluid spacetime represents the radiation era. Next, we show that if a stiff matter fluid obeys Yang's equations, then the vorticity of the fluid vanishes. Moreover, we find that if the Ricci tensor is Killing, then the perfect fluid spacetime is (i) expansion-free and shear-free, and its flow is geodesic, however, not necessarily vorticity-free, and (ii) σ and p are constant. Besides, we establish that Ricci semi-symmetric or Ricci symmetric perfect fluid spacetimes represent either dark matter era, or phantom era. Finally, we acquire that if a perfect fluid is Ricci symmetric, then either the spacetime represents a dark matter era, or a static spacetime. [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]

An almost Yamabe soliton is a generalization of the Yamabe soliton. In this article, we deduce some results regarding almost gradient Yamabe solitons. More specifically, we show that a compact almost gradient Yamabe soliton having non-negative Ricci curvature is trivial. Again, we prove that an almost gradient Yamabe soliton with a non-negative potential function and scalar curvature bound admitting an integral condition is trivial. Moreover, we give a characterization of a compact almost gradient Yamabe solitons. [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 Ricci–Yamabe solitons (RYSs) and gradient Ricci–Yamabe solitons (gradient RYSs) in generalized Robertson–Walker (GRW) spacetimes. At first, we prove that if a GRW spacetime admits a RYS, then it becomes a perfect fluid spacetime (PFS) and the divergence of the Weyl tensor vanishes. Also, a GRW spacetime admitting a RYS is of Petrov type I, D or O and in case of four dimension, the spacetime turns into a Robertson–Walker spacetime. Next, we show that if a GRW spacetime of constant scalar curvature admits a gradient RYS, then it becomes a PFS and the divergence of the Weyl tensor vanishes. [ABSTRACT FROM AUTHOR]

The current work first explores projective collineations on pseudo-Riemannian manifolds. Projective collineations, curvature collineations, and Ricci curvature collineations are examined in relation to one another. On warped product manifolds, the projective collineations of the form ζ = ζ 1 + ζ 2 are investigated. We scrutinize various inheritance aspects in projective collineations from warped product manifolds to its factor manifolds. This provides, for example, a partially negative solution to Besse's problem regarding the existence of Einstein warped product manifolds. Finally, Pseudo-Ricci symmetric space-times admitting projective collineations are investigated. [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 current article is about almost Schouten solitons and gradient Schouten solitons on almost cosymplectic and α -almost cosymplectic manifolds. Amongothers, we prove the non existence of almost Schouten solitons on a compact (k , μ) -almost cosymplectic manifold with k < 0 . Next, we characterize (k , μ) -almost cosymplectic manifolds obeying gradient Schouten solitons. Furthermore, it is shown that if a compact α -almost cosymplectic manifold admits an almost Schouten soliton, then the manifold becomes an almost cosymplectic manifold. Then we prove that a 3-dimensional cosymplectic manifold admitting a gradient Schouten soliton is either flat or the scalar curvature is constant. Finally, we build an example to demonstrate our outcome. [ABSTRACT FROM AUTHOR]

In this work, we have characterized the frame bundle F M admitting metallic structures on almost quadratic ϕ -manifolds ϕ 2 = p ϕ + q I − q η ⊗ ζ , where p is an arbitrary constant and q is a nonzero constant. The complete lifts of an almost quadratic ϕ -structure to the metallic structure on F M are constructed. We also prove the existence of a metallic structure on F M with the aid of the J ˜ tensor field, which we define. Results for the 2-Form and its derivative are then obtained. Additionally, we derive the expressions of the Nijenhuis tensor of a tensor field J ˜ on F M . Finally, we construct an example of it to finish. [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 aims to classify the Einstein-type metrics on Kenmotsu and almost Kenmotsu manifolds. In Kenmotsu case, we find that it is T-Einstein. Also, if the manifold is complete and the scalar curvature remains invariant along the Reeb vector field, then either, it is isometric to the hyperbolic space H 2 n + 1 (1) or, the warped product M ~ × γ R , provided ζ ψ ≠ ψ . Next, we investigate non-Kenmotsu (κ , μ) ′ -almost Kenmotsu manifolds obeying the Einstein-type metrics and give some classification. Finally, we establish that if (ψ , g) is a non-trivial solution of Einstein-type metrics with smooth function ψ which is constant along the Reeb vector field on almost Kenmotsu 3-H-manifold, then either, it is locally isometric to the hyperbolic space H 3 (1) or, the Riemannian product H 2 (4) × R . Finally, we construct several non-trivial examples to verify our main results. [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 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]

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 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]

A conformally flat GRW space-time is a perfect fluid RW space-time. In this note, we investigated the influence of many differential curvature conditions, such as the existence of recurrent and semi-symmetric curvature tensors. In each case, the form of the Ricci curvature tensor, the energy–momentum tensor, the energy density, the pressure of the fluid, and the equation of state are determined and interpreted. For example, it is demonstrated that a Ricci semi-symmetric RW space-time reduces to Einstein space-time or a Ricci recurrent RW space-time, and the perfect fluid space-time is referred to as Yang pure space-time or dark matter era. [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]

The objective of this paper is to explore the complete lifts of a quarter-symmetric metric connection from a Sasakian manifold to its tangent bundle. A relationship between the Riemannian connection and the quarter-symmetric metric connection from a Sasakian manifold to its tangent bundle was established. Some theorems on the curvature tensor and the projective curvature tensor of a Sasakian manifold with respect to the quarter-symmetric metric connection to its tangent bundle were proved. Finally, locally ϕ -symmetric Sasakian manifolds with respect to the quarter-symmetric metric connection to its tangent bundle were studied. [ABSTRACT FROM AUTHOR]

The purpose of the paper is to characterize gradient (m , ρ) -quasi Einstein solitons within the framework of two classes of almost Kenmotsu Manifolds. Finally, we consider an example to justify a result of our paper. [ABSTRACT FROM AUTHOR]

In this paper, Yamabe solitons, gradient Yamabe solitons and gradient ρ -Einstein solitons have been investigated on 3-dimensional f -Kenmotsu manifolds. Moreover, we characterize 3-dimensional f -Kenmotsu manifolds admitting *-Ricci solitons and gradient *-Ricci solitons, respectively. [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 article, we classify h-almost Ricci–Yamabe solitons in paracontact geometry. In particular, we characterize para-Kenmotsu manifolds satisfying h-almost Ricci–Yamabe solitons and 3-dimensional para-Kenmotsu manifolds obeying h-almost gradient Ricci–Yamabe solitons. Then, we classify para-Sasakian manifolds and para-cosymplectic manifolds admitting h-almost Ricci–Yamabe solitons and h-almost gradient Ricci–Yamabe solitons, respectively. Finally, we construct an example to illustrate our result. [ABSTRACT FROM AUTHOR]

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. [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 paper, warped product mixed generalized quasi-Einstein manifolds MG(QE) n are studied. It is shown that a Ricci recurrent MG(QE) n manifold is a product manifold whose factors are Ricci recurrent. Sufficient conditions for a MG(QE) n to be a G(QE) n are derived. Then, we classify three types of warped product MG(QE) n . For example, it is proved that the fiber manifold is Einstein in the first type, (QE) n in the second type and MG(QE) n in the third type. [ABSTRACT FROM AUTHOR]

We characterize the three-dimensional Riemannian manifolds endowed with a semi-symmetric metric ρ-connection if its Riemannian metrics are Ricci and gradient Ricci solitons, respectively. It is proved that if a three-dimensional Riemannian manifold equipped with a semi-symmetric metric ρ-connection admits a Ricci soliton, then the manifold possesses the constant sectional curvature −1 and the soliton is expanding with λ = −2. Next, we study the gradient Ricci solitons in such a manifold. Finally, we construct a non-trivial example of a three-dimensional Riemannian manifold endowed with a semi-symmetric metric ρ-connection admitting a Ricci soliton and validate our some results. [ABSTRACT FROM AUTHOR]

In this paper, we classify 3-dimensional Riemannian manifolds endowed with a special type of vector field if the Riemannian metrices are Ricci-Yamabe solitons and gradient Ricci-Yamabe solitons, respectively. Finally, we construct an example to illustrate our result. [ABSTRACT FROM AUTHOR]

In the first part of the article, the definition of pseudo-symmetric spacetime is given. In the second part, the conditions under which a four-dimensional pseudo-symmetric spacetime has Riemann compatible and Weyl compatible vector fields are obtained, and the spacetime is examined considering the case that this spacetime is a generalized Robertson–Walker spacetime. In the third part of the article, some investigations on four-dimensional pseudo-symmetric spacetimes admitting F(R)-gravity are made. [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]

The purpose of the present paper is to investigate the almost quasi-Yamabe soliton and gradient almost quasi-Yamabe solitons under the framework of three-dimensional normal almost paracontact metric manifolds. [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]

In this paper, we study almost quasi-Yamabe solitons and gradient almost quasi-Yamabe solitons in 3 -dimensional f -Kenmotsu manifolds. [ABSTRACT FROM AUTHOR]

The W 2 -curvature tensor is an important geometric invariant with relativistic significance, introduced in the early 1970s by Pokhariyal and Mishra, which can be identified in class 4 in the classification of skew-symmetric operators. In this work, we investigate 4-dimensional space-times admitting W 2 -curvature tensor in f (R , G) modified theory of gravity. It is proved that the W 2 -curvature flat perfect fluid space-times obeying f (R , G) gravity represent inflation. Also, it is shown that the isotropic pressure and the energy density of such space-times are constant. It is to be noted that in such space-times the considered energy conditions are consistently satisfied if the scalar curvature is positive. Next, we study perfect fluid space-times admitting divergence free W 2 -curvature tensor in f (R , G) gravity. Amongst other results, we prove that if the energy-momentum tensor of such space-times is bi-recurrent, then either the space-times represent inflation or their isotropic pressure and energy density are constants. [ABSTRACT FROM AUTHOR]

The purpose of the offering exposition is to characterize gradient Yamabe, gradient Einstein and gradient m -quasi Einstein solitons within the framework of 3-dimensional para-Kenmotsu manifolds. Finally, we consider an example to prove the result obtained in previous section. [ABSTRACT FROM AUTHOR]

If a three-dimensional N (k) -contact metric manifold M admits a Yamabe soliton of type (M , g , V) , then the manifold has a constant scalar curvature and the flow vector field V is Killing. Furthermore, either M has a constant curvature k or the flow vector field V is a strict contact infinitesimal transformation. Also, we prove that if the metric of a three-dimensional N (k) -contact metric manifold M admits a gradient Yamabe soliton, then either the manifold is flat or the scalar curvature is constant. Moreover, either the potential function is constant or the manifold is of constant sectional curvature k. Finally, we have given an example to verify our result. [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]

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]

We prove that a contact metric manifold does not admit a proper quasi-Yamabe soliton ( M , g , ξ , λ , μ ). Next we prove that if a contact metric manifold admits a quasi-Yamabe soliton ( M , g , V , λ , μ ) whose soliton field is pointwise collinear with the Reeb vector field, then the scalar curvature is constant, and the quasi-Yamabe soliton reduces to Yamabe soliton. Finally, it is shown that if a contact metric manifold admits a quasi-Yamabe soliton whose soliton field is the V-Ric vector field, then the Ricci operator Q commutes with the (1, 1) tensor ϕ . As a consequence of the main result we obtain several corollaries. [ABSTRACT FROM AUTHOR]

The authors consider a quarter-symmetric metric connection in a P-Sasakian manifold and study the second order parallel tensor in a P-Sasakian manifold with respect to the quarter-symmetric metric connection. Then Ricci semisymmetric P-Sasakian manifold with respect to the quarter-symmetric metric connection is considered. Next the authors study ξ-concircularly flat P-Sasakian manifolds and concircularly semisymmetric P-Sasakian manifolds with respect to the quarter-symmetric metric connection. Furthermore, the authors study P-Sasakian manifolds satisfying the condition Z ~ (ξ , Y) ⋅ S ~ = 0 , where Z ~ , S ~ are the concircular curvature tensor and Ricci tensor respectively with respect to the quarter-symmetric metric connection. Finally, an example of a 5-dimensional P-Sasakian manifold admitting quarter-symmetric metric connection is constructed. [ABSTRACT FROM AUTHOR]