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This article 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 $\varPsi (t)$ 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 $f\left(\mathcal{R}\right)= e^{(\alpha \mathcal{R})}-ln(\beta \mathcal{R})$ in which $\alpha$ and $\beta$ are positive constants., Comment: 19 pages, 4 figures. arXiv admin note: text overlap with arXiv:2406.16113

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 $f(\mathcal{R},T^2)$ and $f\left(\mathcal{R}^{\ast}\right)$ gravity scenario and deduce several energy conditions., Comment: arXiv admin note: substantial text overlap with arXiv:2406.16113

In this article we introduce and characterize a pseudo generalized Ricci-recurrent spacetimes. At first, we produce an example to justify the existence of such a spacetime. Then, it is provided that a pseudo generalized Ricci-recurrent generalized Robertson-Walker spacetime represents a perfect fluid spacetime and a pseudo generalized Ricci-recurrent perfect fluid spacetime represents either a dark energy epoch of the Universe or, the velocity vector field is parallel, conservative, acceleration-free, vorticity-free, and shear-free and becomes a static spacetime. Lastly, we study the impact of this spacetime under $f(\mathcal{R})$ gravity scenario and deduce several energy conditions.

In this article, we characterize a Lorentzian manifold $\mathcal{M}$ with a semi-symmetric metric connection. At first, we consider a semi-symmetric metric connection whose curvature tensor vanishes and establish that if the associated vector field is a unit time-like torse-forming vector field, then $\mathcal{M}$ becomes a perfect fluid spacetime. Moreover, we prove that if $\mathcal{M}$ admits a semi-symmetric metric connection whose Ricci tensor is symmetric and torsion tensor is recurrent, then $\mathcal{M}$ represents a generalized Robertson-Walker spacetime. Also, we show that if the associated vector field of a semi-symmetric metric connection whose curvature tensor vanishes is a $f-$ Ric vector field, then the manifold is Einstein and if the associated vector field is a torqued vector field, then the manifold becomes a perfect fluid spacetime. Finally, we apply this connection to investigate Ricci solitons.

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 permit a $k$-almost yamabe soliton ($g,k,\rho,\lambda$), then the integral curves of the vector field $\rho$ are geodesics, the spacetime becomes stationary and the isotopic pressure and energy density remain invariant under the velocity vector field $\rho$. Also, we establish that if the potential vector field is pointwise collinear with the velocity vector field and $\rho(a)=0$ where a is a scalar, then either the perfect fluid spacetime represents phantom era, or the potential function $\Phi$ is invariant under the velocity vector field $\rho$. Finally, we prove that if a perfect fluid spacetime permits a gradient $k$-almost yamabe soliton ($g,k,D\Phi,\lambda$) and $R, \lambda, k$ are invariant under $\rho$, then the vorticity of the fluid vanishes.

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 $\mathcal{F}(\mathcal{R})$-gravity. As a result, various energy conditions are studied and analysed our obtained outcomes in terms of a $\mathcal{F}(\mathcal{R})$-gravity model., Comment: arXiv admin note: text overlap with arXiv:2402.15354

This article concerns with the characterization of a spacetime and modified gravity, such as $f\left(R,G\right)$, $f\left(R,T\right)$ and $f\left(R,L_{m}\right)$-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 permitting 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 $f\left(R,G\right)$, $f\left(R,T\right)$ and $f\left(R,L_{m}\right)$-gravity, respectively. We also investigate the spacetime as a $f\left(R,G\right)$-gravity solution of 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 $f\left(R,G\right)=\exp(R)+\alpha \left(6G\right)^{\beta}$. 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.

This article carries out the investigation of a three-dimensional Riemannian manifold $N^3$ endowed with a semi-symmetric type non-metric connection. Firstly, we construct a non-trivial example to prove the existence of a semi-symmetric type non-metric connection on $N^{3}$. It is established that a $N^3$ with the semi-symmetric type non-metric connection, whose metric is a gradient Ricci soliton, is a manifold of constant sectional curvature with respect to the semi-symmetric type non-metric connection. Moreover, we prove that if the Riemannian metric of $N^3$ with the semi-symmetric type non-metric connection is a gradient Yamabe soliton, then either $N^{3}$ is a manifold of constant scalar curvature or the gradient Yamabe soliton is trivial with respect to the semi-symmetric type non-metric connection. We also characterize the manifold $N^3$ with a semi-symmetric type non-metric connection whose metrics are Einstein solitons and $m$-quasi Einstein solitons of gradient type, respectively.

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.

In this article, we examine gradient type Ricci solitons and $(m,\tau)$-quasi Einstein solitons in generalized Robertson-Walker ($GRW$) spacetimes. Besides, we demonstrate that in this scenario the $GRW$ spacetime presents the Robertson-Walker ($RW$) spacetime and the perfect fluid ($PF$) spacetime presents the phantom era. Consequently, we show that if a $GRW$ spacetime permits a gradient $\tau$- Einstein solitons, then it also represents a $PF$ spacetime under certain condition.

In this paper, first we consider that the conformal vector field $\mathbf{X}$ is identical with the Reeb vector field $\varsigma$ and next, assume that $\mathbf{X}$ is pointwise collinear with %the Reeb vector field $\varsigma$, in both cases it is shown that the manifold $\mathbf{N}^{2m+1}$ becomes a Kenmotsu manifold and $\mathbf{N}^{2m+1}$ is locally a warped product $\mathbf{N}' \times_{f} \mathbf{M}^{2m}$, where $\mathbf{M}^{2m}$ is an almost K\"ahler manifold, $\mathbf{N}'$ is an open interval with coordinate t, and $f = ce^{t}$ for some positive constant c. Beside these, we prove that if a $(\verb"k",\boldsymbol{\mu})'$-almost Kenmotsu manifold admits a Killing vector field $\mathbf{X}$, then either it is locally a warped product of an almost K\"ahler manifold and an open interval or $\mathbf{X}$ is a strict infinitesimal contact transformation. Furthermore, we also investigate $\boldsymbol{\eta}$-Ricci-Yamabe soliton with conformal vector fields on $(\verb"k",\boldsymbol{\mu})'$-almost Kenmotsu manifolds and finally, we construct an example.

The present paper aims to investigate $(m,\rho)$-quasi-Einstein metrices on almost co-K\"ahler manifolds $\mathcal{M}$. It is proven that if a $(\kappa,\mu)$-almost co-K\"ahler manifold with $\kappa<0$ is $(m,\rho)$-quasi-Einstein manifold, then $\mathcal{M}$ represents a $N(\kappa)$-almost co-K\"ahler manifold and the manifold is locally isomorphic to a solvable non-nilpotent Lie group. Next, we study the three dimensional case and get the above mentioned result along with the manifold $\mathcal{M}^3$ becoming an $\eta$-Einstein manifold. We also show that there does not exist $(m,\rho)$-quasi-Einstein structure on a compact $(\kappa,\mu)$-almost co-K\"ahler manifold of dimension greater than three with $\kappa<0$. Further, we prove that an almost co-K\"ahler manifold satisfying $\eta$-Einstein condition with constant coefficients reduces to a $K$-almost co-K\"ahler manifold, provided $ma_{1} \neq (2n-1)b_{1}$ and $m \neq 1$. We also characterize perfect fluid spacetime whose Lorentzian metric is equipped with $(m, \rho)$-quasi Einstein solitons and acquired that the perfect fluid spacetime has vanishing vorticity, or it represents dark energy era under certain restriction on the potential function. Finally, we construct an example of an almost co-K\"ahler manifold with $(m,\rho)$-quasi-Einstein solitons.

The purpose of the article is to characterize \textbf{gradient $(m,\rho)$-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.

The aim of the present paper is to obtain the condition under which a pseudosymmetric spacetime to be a perfect fluid spacetime. It is proven that a pseudosymmetric generalized Robertson-Walker spacetime is a perfect fluid spacetime. Moreover, we establish that a conformally flat pseudosymmetric spacetime is a generalized Robertson-Walker spacetime. Next, it is shown that a pseudosymmetric dust fluid with constant scalar curvature satisfying Einstein's field equations without cosmological constant is vacuum. Finally, we construct a non-trivial example of pseudosymmetric spacetime.

This article deals with the investigation of perfect fluid spacetimes endowed with concircular vector field. It is shown that in a perfect fluid spacetime with concircular vector field, the velocity vector field annihilates the conformal curvature tensor and in dimension 4, a perfect fluid spacetime is a generalized Robertson-Walker spacetime with Einstein fibre. Moreover, we prove that if a perfect fluid spacetime equipped with concircular vector field admits a second order symmetric parallel tensor, then either the state equation of the perfect fluid spacetime is characterized by $p=\frac{3-n}{n-1}\sigma$ , or the tensor is a constant multiple of the metric tensor. We also characterize the perfect fluid spacetimes with concircular vector field whose Lorentzian metrics are Ricci soliton, gradient Ricci soliton, gradient Yamabe solitons and gradient $m$-quasi Einstein solitons, respectively., Comment: More authors are added in the new version and many mistakes are considered and corrected

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 $M^3$. Before all else, it is proved that if the metric of $M^3$ is Riemann soliton with divergence-free potential vector field $Z$, then the manifold is quasi-Sasakian and is of constant sectional curvature -$\lambda$, provided $\alpha,\beta =$ constant. Other than this, it is shown that if the metric of $M^3$ is \emph{ARS} and $Z$ is pointwise collinear with $\xi $ and has constant divergence, then $Z$ is a constant multiple of $\xi $ and the \emph{ARS} reduces to a Riemann soliton, provided $\alpha,\;\beta =$constant. Additionally, it is established that if $M^3$ with $\alpha,\; \beta =$ constant admits a gradient \emph{ARS} $(\gamma,\xi,\lambda)$, then the manifold is either quasi-Sasakian or is of constant sectional curvature $-(\alpha^2-\beta^2)$. At long last, we develop an example of $M^3$ conceding a Riemann soliton.