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\begin{document}
\date{}
\title{Characterization of  generalized quasi-Einstein manifolds and modified gravity }
\author{Uday Chand DE$^{1} $ and Hülya BAĞDATLI YILMAZ $^{2}$\\ \\
	$ ^{1} $ Calcutta University, 	Department of Pure Mathematics,\\ West Bengal/India\\ 
	\\$ ^{2} $Marmara University, Faculty of Sciences,\\ Department of Mathematics, \\ Istanbul/Turkey \\  \\
	 E-mails: uc\_de@yahoo.com$^{1} $; hbagdatli@marmara.edu.tr$^{2} $\\
 }
\maketitle

\begin{abstract}
In this work, a detailed examination of a specific case of a generalized quasi-Einstein manifold $ (GQE)_{n} $ is provided. It begins by exploring generalized quasi-Einstein spacetimes under certain conditions. The analysis then focuses on cases that admit a parallel time-like vector field. Among the findings, it is demonstrated that such spacetimes can be categorized as generalized Robertson-Walker spacetimes, Robertson-Walker spacetimes, and quasi-constant curvature spacetimes. Additionally, the physical implications of these results are discussed. It is also investigated $ (GQE)_{4} $ spacetimes, which accept $ \mathcal{F(\mathfrak{R})} $-gravity and feature a parallel unit timelike vector field. Finally, various energy conditions are analyzed based on the results related to $ \mathcal{F(\mathfrak{R})} $-gravity.
\end{abstract}

\begin{MSC 2020} 53B30; 53C25; 53C50; 53Z05
\end{MSC 2020}

\begin{keywords} \textit{Generalized quasi-Einstein manifold, Perfect fluid spacetime,  GRW spacetime, RW spacetime, $ \mathcal{F(\mathfrak{R})} $-gravity.}
\end{keywords}


\section{Introduction}
Complete Riemannian manifolds $(\mathcal{M}^{n}, \mathfrak{g})$, where $n \geq 3$, are classified as generalized quasi-Einstein manifolds if they include smooth functions $\mu$, $\lambda$, and $\mathfrak{f}$ that are defined on $\mathcal{M}$ and satisfy the following equation:

\begin{equation}
Ric + \nabla^{2}\mathfrak{f} - \mu \, d\mathfrak{f} \otimes d\mathfrak{f} - \lambda \mathfrak{g} = 0. \label{eq1}
\end{equation}
In this equation, $Ric $ denotes the Ricci operator, while  $\nabla^{2}\mathfrak{f} $ represents the Hessian of the function  $\mathfrak{f} $. In \cite{bibC2012}, it is established that such a manifold, which is defined by a harmonic Weyl tensor and a radial Weyl curvature that vanishes, can be locally represented as a special warped product.

Several notable types characterize natural examples of generalized quasi-Einstein manifolds. These include Einstein manifolds, in which both $ \mathfrak{f} $ and $ \lambda $ are constants, and gradient Ricci solitons that exhibit a constant $ \lambda $ with $ \mu = 0 $. Additionally, gradient Ricci almost solitons are considered in cases where $ \mu = 0$ \cite{bibPRS10}. Quasi-Einstein manifolds arise when both $ \mu $ and $ \lambda $ are constants \cite{bibCSW2011}, \cite{bibCMMR13}. 

The $m$-quasi Einstein metric is introduced by Case et al. \cite{bibCSW2011}. In this case, $ \mu=\frac{1}{m} $ for a positive integer $ m , (0<m\leqslant+\infty )$ and $ \lambda\in \mathbb{R} $.  In \cite{bibBR2014} and \cite{bibD2015} delved into a specific case of this result. When $ m = 2 $, the $m$-quasi Einstein metric is recognized as a vacuum near-horizon geometry \cite{bibG25}, and (\ref{eq1}) reduces to the vacuum near-horizon geometry equation of spacetime with $ \lambda $ playing the role of cosmological constant. 

In this work, we aim to explore a particular case of (\ref{eq1}). Specifically, we will use the following definition:

A complete Riemannian manifold $(\mathcal{M}^{n}, \mathfrak{g})$, defined for dimensions $n \geq 3$, is classified as a generalized quasi-Einstein manifold, denoted as $(GQE)_{n}$, if there are non-zero constants $\mu$ and $\lambda$ present, along with a smooth function $\mathfrak{f}$ defined on $\mathcal{M}$ that fulfilling 
\begin{equation*}
	Ric + \nabla^{2}\mathfrak{f} - \mu \, d\mathfrak{f} \otimes d\mathfrak{f} - \lambda \mathfrak{g} = 0.
\end{equation*}

In the local coordinate system, it becomes
\begin{equation}
\mathfrak{R}_{ij} + \mathfrak{f}_{ij} - \mu \,  \mathfrak{f}_{i}  \mathfrak{f}_{j} - \lambda \mathfrak{g}_{ij} = 0,\label{eq2}
\end{equation}
in here $  \mathfrak{f}_{i}=\nabla_{i}\mathfrak{f} $ and $  \mathfrak{f}_{ij}=\nabla_{i}\nabla_{j}\mathfrak{f} $.

A Lorentzian manifold of dimension $n$ is an exclusive category of semi-Riemannian manifold, which holds significant relevance in the domains of cosmology and relativity due to its capacity to classify vectors based on their causal characteristics. A Lorentzian manifold that possesses a globally time-like vector field is categorized as a spacetime.

In the study of spacetime, a perfect fluid (PF) spacetime is characterized by the condition that the Ricci tensor is non-zero and adheres to the following equation:
\begin{equation}
\mathfrak{R}_{ij} = \eta \mathfrak{g}_{ij} + \tau \mathfrak{u}_{i} \mathfrak{u}_{j}, \label{eq3}
\end{equation}
where $ \eta $ and $ \tau $ are scalars. In this equation, $ \mathfrak{u} $ represents a unit time-like vector, meaning that $ \mathfrak{u}_{i} \mathfrak{u}^{i} = -1 $. (\cite{bib17}, \cite{36}).

Recently, researchers have explored geometric flows in the quest to identify a cosmological model that behaves like a PF spacetime. Blaga \cite{bibB2020} examined regular and Einstein solitons within a PF spacetime framework. Meanwhile, Venkatesha and Kumar \cite{bibVK19} investigated regular solutions for perfect fluid that follows a torse forming vector field.

A generalized Robertson-Walker (GRW) spacetime is defined as an $ n $- dimensional Lorentzian manifold, characterized by the following metric, which serves as a fundamental framework for understanding the geometric structure of space-time:

\begin{equation*}
	d\mathfrak{s}^2 = -d\mathfrak{t}^2 +\mathfrak{a}^2(\mathfrak{t}) \tilde{\mathfrak{g}}_{ij}(\mathfrak{x}) d\mathfrak{x}^i d\mathfrak{x}^j,
\end{equation*}
here $ \tilde{\mathfrak{g}}_{ij}(\mathfrak{x}) $ represents the metric tensor of a Riemannian submanifold and $\mathfrak{a} > 0$ is a scale function or smooth warping. This structure is not simply a minor extension; it constitutes a significant and comprehensive enhancement of the Robertson-Walker (RW) spacetime, making it essential for the study of large-scale cosmology. Both GRW and RW spacetimes play crucial roles in cosmology by effectively illustrating the temporal evolution of a homogeneous and isotropic universe (\cite{bibA1995}, \cite{bib17}, \cite{bib031}, \cite{bib2017}, \cite{16}, \cite{bib98}, etc....).


The theory of general relativity provides a comprehensive framework for understanding the distribution of matter within spacetime through the energy-momentum tensor (EMT), denoted as \(\mathfrak{T}_{ij}\). The geometric structure of spacetime is profoundly influenced by the Ricci tensor, which is interconnected with the EMT via Einstein's field equations (EFEs).

 The characteristics of PF are defined by its rest frame mass density and isotropic pressure. Consequently, the EMT $\mathfrak{T}_{ij}$ in a PF spacetime can be formulated as follows:
\begin{equation}
\mathfrak{T}_{ij} =\mathfrak{p} \mathfrak{g}_{ij} + (\mathfrak{p} + \sigma)  \mathfrak{u}_{i}  \mathfrak{u}_{j}. \label{eq4}
\end{equation}
In the foregoing equation, let $\mathfrak{p} $ denote the isotropic pressure, while $\sigma $ signifies the energy density. A perfect fluid is classified as isentropic if it depends on a barotropic equation of state (EoS) $\mathfrak{p} = \mathfrak{p}(\sigma)$. Several specific cases can further delineate the nature of this spacetime:  $ \sigma = 3\mathfrak{p} $ corresponds to the radiation era;  $\mathfrak{p} = \sigma $ characterizes a stiff matter fluid; $\mathfrak{p} + \sigma = 0 $ is associated with the dark energy era ( \cite{bibDS1999}, \cite{bib25}, etc....).

%Moreover, the equation $ p = \omega \sigma $ introduces the EOS parameter $ \omega $, which plays a vital role in shaping the evolution of energy density and the expansion of the universe. The universe is regarded as being in an accelerating phase when $ \omega < -\frac{1}{3} $, in a quintessence phase when $ -1 < \omega < 0 $, and in a phantom phase when $ \omega < -1 $.

The EFEs, excluding the cosmological constant, are given by:
\begin{equation}
\mathfrak{R}_{ij} - \frac{\mathfrak{R}}{2} \mathfrak{g}_{ij} = \kappa \mathfrak{T}_{ij}, \label{eq5}
\end{equation}
in which \(\kappa\) indicates the gravitational constant and $\mathfrak{R}$ represents the scalar curvature, $\mathfrak{R}=\mathfrak{R}_{ij}\mathfrak{g}^{ij}$. In the field of cosmology, the scalar curvature plays a pivotal role in elucidating the structure of the universe. Under conditions of vacuum, the EFEs can be simplified to $\mathfrak{R}_{ij} = 0$. This represents a non-linear differential equation, which leads to the conclusion that the scalar curvature $ \mathfrak{R}  $ is also zero.

Utilizing (\ref{eq3}) and (\ref{eq4}) in (\ref{eq5}), we deduce that
$$\eta= \frac{\kappa(\sigma-\mathfrak{p})}{n-2} \quad \text{and} \quad \tau=\kappa(\sigma+\mathfrak{p}),$$
where $\mathfrak{R}=n\eta-\tau  $ obtained by multiplying (\ref{eq3}) by $ \mathfrak{g}^{ij} $ is used.

The (0,4) Weyl tensor $\mathcal{C}_{hijk}$ on an  $n$-dimensional semi-Riemannian manifold is given by \cite{20}
\begin{equation}
	\mathcal{C}_{hijk}=\mathfrak{R}_{hijk}-\frac{1}{n-2}\left\lbrace \mathfrak{R}_{hk}\mathfrak{g}_{ij}-\mathfrak{R}_{hj}\mathfrak{g}_{ik}+\mathfrak{R}_{ij}\mathfrak{g}_{hk}-\mathfrak{R}_{ik}\mathfrak{g}_{hj}\right\rbrace \\ \label{eq6}
\end{equation}
\begin{equation*}
	+\frac{\mathfrak{R}}{(n-1)(n-2)}\left\lbrace \mathfrak{g}_{hk}\mathfrak{g}_{ij}-\mathfrak{g}_{hj}\mathfrak{g}_{ik}\right\rbrace, 
\end{equation*}
$\mathfrak{R}_{jkl}^{i}$ being the curvature tensor of type  $(1,3)$ and $\mathfrak{R}_{hjkl}=\mathfrak{g}_{ih}\mathfrak{R}_{jkl}^{i}. $  
\\

The divergence of the Weyl tensor is expressed as follows: \cite{bibP2001}
\begin{equation}
 \nabla_{h}\mathcal{C}^{h}_{ijk} = \frac{n-3}{n-2}\left[ \nabla_{j}\mathfrak{R}_{ik}-\nabla_{i}\mathfrak{R}_{jk}-\frac{1}{2(n-1)}\left( \mathfrak{g}_{ik}\nabla_{j}\mathfrak{R}-\mathfrak{g}_{jk}\nabla_{i}\mathfrak{R}\right) \right]. \label{eq06}
\end{equation}

The foundation of $\mathcal{F(\mathfrak{R})}$-gravity theories lies in the integration of higher-order curvature terms into the Einstein-Hilbert Lagrangian density \cite{bibS1980}. This modification aims to address quantum effects that may arise in the early universe. These gravitational theories, which incorporate correction terms to the Einstein-Hilbert Lagrangian density, are commonly referred to as nonlinear gravitational theories, variational gravitational theories, and higher-order gravitational theories (\cite{bibCRM1998}, \cite{bibCRM98}, \cite{bibS1998}, etc....).

The general relativity theory is fundamentally based on a metric framework. In contrast, $\mathcal{F(\mathfrak{R})}$-gravity theories allow for the characterization of spacetime through both metric and affine structures. This distinction gives rise to two primary approaches within $\mathcal{F(\mathfrak{R})}$-gravity theories: the Palatini formalism, which focuses on the affine structure, and the metric formalism. Moreover, $\mathcal{F(\mathfrak{R})}$-gravity theories can be examined through scalar gravity theories, such as the Brans–Dicke theory, which emphasizes the role of scalar fields in gravitational interactions \cite{bibBD1961}.

In the next parts of this work, we will study $ (GQE)_{n} $ manifolds. In section $ 2 $, we will commence by examining the characteristics of $(GQE)_{n}$ spacetimes under defined conditions. Following this initial exploration, we will shift our focus to $(GQE)_{n}$ spacetimes with a parallel unit time-like vector field. In section $ 3 $, we will examine $(GQE)_{4}$ spacetimes endowed with a parallel unit timelike vector field that admits $\mathcal{F(\mathfrak{R})}$-gravity. Finally, we will analyze the various energy conditions based on the outcomes we have achieved within the framework of $\mathcal{F(\mathfrak{R})}$-gravity.

\section{Generalized quasi-Einstein spacetimes  } 
Let's now deal with $(GQE)_{n}$ spacetimes.

Covariant differentiation of (\ref{eq2}) yields
	\begin{equation}
		\nabla_{l}\mathfrak{R}_{ij} +\nabla_{l}\mathfrak{f}_{ij}-\mu\left[ (\nabla_{l}\mathfrak{f}_{i})\mathfrak{f}_{j}+ \mathfrak{f}_{i}(\nabla_{l}\mathfrak{f}_{j})\right]=0.  \label{eq7}  
	\end{equation}
(\ref{eq7}) can be rearranged in the following form:

	\begin{equation}
	\nabla_{l}\mathfrak{f}_{ij}=-\nabla_{l}\mathfrak{R}_{ij} +\mu\left[ (\nabla_{l}\mathfrak{f}_{i})\mathfrak{f}_{j}+ \mathfrak{f}_{i}(\nabla_{l}\mathfrak{f}_{j})\right]=0.  \label{eq8}  
	\end{equation}
By changing the indices $ "j" $ and $ "l" $ in (\ref{eq8}) and then subtracting the resulting equation from (\ref{eq8}), one can obtain

	\begin{equation}
\nabla_{l}\mathfrak{f}_{ij}-\nabla_{j}\mathfrak{f}_{il}=\nabla_{j}\mathfrak{R}_{il}-\nabla_{l}\mathfrak{R}_{ij} +\mu\left[ (\nabla_{l}\mathfrak{f}_{i})\mathfrak{f}_{j}- (\nabla_{j}\mathfrak{f}_{i})\mathfrak{f}_{l}\right]. \label{eq9}
	\end{equation}
Considering Ricci identity, (\ref{eq9}) becomes
		\begin{equation}
	\mathfrak{f}_{h}\mathfrak{R}_{ijl}^{h}=\nabla_{j}\mathfrak{R}_{il}-\nabla_{l}\mathfrak{R}_{ij} +\mu\left[ (\nabla_{l}\mathfrak{f}_{i})\mathfrak{f}_{j}- (\nabla_{j}\mathfrak{f}_{i})\mathfrak{f}_{l}\right]. 
	\end{equation} \label{eq10}
Suppose that this spacetime is a PF spacetime. Then, from (\ref{eq3}), we can reach 
	\begin{equation}
	\nabla_{l}\mathfrak{R}_{ij} = \eta_{l} \mathfrak{g}_{ij} + \tau_{l} \mathfrak{u}_{i} \mathfrak{u}_{j} + \left[(\nabla_{l}\mathfrak{u}_{i})\mathfrak{u}_{j}+\mathfrak{u}_{i}(\nabla_{l}\mathfrak{u}_{j})\right] , \label{eq11}
	\end{equation}
where $\eta_{l}=\nabla_{l}\eta $ and $\tau_{l}=\nabla_{l}\tau $.

After changing the indices $ j $ and $ l $ in (\ref{eq11}) and  substituting the resulting equation and (\ref{eq11}) into (\ref{eq10}), we have
	\begin{equation*}
	\mathfrak{f}_{h}\mathfrak{R}_{ijl}^{h}=\eta_{j} \mathfrak{g}_{il}-\eta_{l} \mathfrak{g}_{ij}+\left[ \tau_{j} \mathfrak{u}_{l}-\tau_{l} \mathfrak{u}_{j}\right] \mathfrak{u}_{i}  
\end{equation*}
\begin{equation}
		+\tau\left[  (\nabla_{j}\mathfrak{u}_{i})\mathfrak{u}_{l} -(\nabla_{l}\mathfrak{u}_{i})\mathfrak{u}_{j}+\mathfrak{u}_{i}\left( (\nabla_{j}\mathfrak{u}_{l})-(\nabla_{l}\mathfrak{u}_{j})\right) \right]  \label{eq12}
\end{equation}
\begin{equation*}
	+\mu \left[ (\nabla_{l}\mathfrak{f}_{i})\mathfrak{f}_{j}- (\nabla_{j}\mathfrak{f}_{i})\mathfrak{f}_{l}\right].
\end{equation*}
From (\ref{eq2}), it follows that
\begin{equation}
	\mathfrak{f}_{ij}=-\mathfrak{R}_{ij} +  \mu \,  \mathfrak{f}_{i}  \mathfrak{f}_{j} + \lambda \mathfrak{g}_{ij}.\label{eq13}
\end{equation}
Multiplying (\ref{eq13}) by $\mathfrak{f}_{l}  $ gives
\begin{equation}
	\mathfrak{f}_{ij}\mathfrak{f}_{l}=-\mathfrak{R}_{ij}\mathfrak{f}_{l} +  \mu \,  \mathfrak{f}_{i}  \mathfrak{f}_{j}\mathfrak{f}_{l} + \lambda \mathfrak{g}_{ij}\mathfrak{f}_{l}.\label{eq14}
\end{equation}
By changing the indices $ j $ and $ l $ in (\ref{eq14}) and then subtracting the resulting equation from (\ref{eq14}),  we reveal that
\begin{equation}
	\mathfrak{f}_{ij}\mathfrak{f}_{l}-\mathfrak{f}_{il}\mathfrak{f}_{j}=\mathfrak{R}_{il}\mathfrak{f}_{j}-\mathfrak{R}_{ij}\mathfrak{f}_{l} + \lambda \left[ \mathfrak{g}_{ij}\mathfrak{f}_{l}-\mathfrak{g}_{il}\mathfrak{f}_{j}\right]. \label{eq15}
\end{equation}
	Multiplying (\ref{eq15}) by $\mathfrak{g}^{ij} $ and $\mathfrak{u}^{l} $, respectively, we can get
\begin{equation}
(\mathfrak{f}_{ij}\mathfrak{f}_{l}-\mathfrak{f}_{il}\mathfrak{f}_{j})\mathfrak{g}^{ij}\mathfrak{u}^{l}=\mathfrak{R}_{il}\mathfrak{f}^{i}\mathfrak{u}^{l} +  \left[\lambda(n-1)-\mathfrak{R}\right] \mathfrak{f}_{l}\mathfrak{u}^{l}.\label{eq16}
\end{equation}
 
On the other hand, multiplying (\ref{eq12}) by $\mathfrak{g}^{ij} $ and $\mathfrak{u}^{l} $, respectively, it can be found that
	\begin{equation}
\mathfrak{f}_{h}\mathfrak{R}_{l}^{h}\mathfrak{u}^{l}=(1-n)\eta_{l}\mathfrak{u}^{l}-\tau(\nabla_{j}\mathfrak{u}^{j})+\mu\left[\mathfrak{f}_{jl}\mathfrak{f}_{k}-\mathfrak{f}_{jk}\mathfrak{f}_{l}\right] \mathfrak{g}^{jk}\mathfrak{u}^{l}  .\label{eq17}
	\end{equation}
	Putting (\ref{eq16}) in (\ref{eq17}), we get
\begin{equation}
\mathfrak{f}_{h}\mathfrak{R}_{l}^{h}\mathfrak{u}^{l}=(1-n)\eta_{l}\mathfrak{u}^{l}-\tau(\nabla_{j}\mathfrak{u}^{j})+\mu\left[  \mathfrak{R}_{jl}\mathfrak{f}^{j}\mathfrak{u}^{l}+ \left( \lambda (n-1) -\mathfrak{R}\right) \mathfrak{f}_{l}\mathfrak{u}^{l} \right] .\label{eq18}
\end{equation}

Moreover,  multiplying (\ref{eq2}) by $ \mathfrak{f}^{j} $ and $ \mathfrak{u}^{l} $ gives
	\begin{equation}
	\mathcal{R}_{jl}\mathfrak{f}^{j}\mathfrak{u}^{l}=(\eta-\tau)\mathfrak{f}_{l}\mathfrak{u}_{l}. \label{eq19}
	\end{equation}

Let's assume that the scalar function $ \eta $ and the smooth function $\mathfrak{f}  $ satisfy $ \eta_{l}\mathfrak{u}^{l}=0  $ and $ \mathfrak{f}_{l}\mathfrak{u}^{l}=0  $. Then, considering (\ref{eq18}) and (\ref{eq19}), we achieve
	\begin{equation}
\tau(	\nabla_{j}\mathfrak{u}^{j})=0.\label{eq20}
	\end{equation}
(\ref{eq20}) means that $\tau=0  $ or $ \nabla_{j}\mathfrak{u}^{j}=0 $. Thus, if $ \nabla_{j}\mathfrak{u}^{j}=0 $, then the velocity vector is divergence free. The expansion scalar of the fluid vanishes \cite{bib25} or the spacetime represents the dark energy era for $\tau=0  $ and since the spacetime is Einstein, with the help of (\ref{eq06}), we infer that $ div \:\mathcal{C}=0$. In \cite{19}, it was demonstrated that a GRW spacetime is a PF spacetime if and only if $ div \:\mathcal{C}=0$.

	Therefore we can state the following result:
	\begin{theorem}\label{1}
	Let a  $ (GQE)_{n} $ spacetime be a PF spacetime in which the scalar function $ \eta $ and the smooth function $\mathfrak{f}  $ satisfy $ \eta_{l}\mathfrak{u}^{l}=0  $ and $ \mathfrak{f}_{l}\mathfrak{u}^{l}=0  $. Then, the spacetime could either indicate the dark energy era and become a GRW spacetime, or the expansion scalar of the fluid may vanish.
	\end{theorem}

Let's now suppose that $ \varphi^{i} $ satisfies the condition
\begin{equation}
	\theta=\mathfrak{f}_{i}\varphi^{i}.\label{eq21}
\end{equation}
Taking $ \varphi $ as a parallel vector field, that is,
\begin{equation}
\nabla_{j}\varphi_{i}=0,\label{eq22}
\end{equation}
then it follows from (\ref{eq21}) and (\ref{eq22}) that
\begin{equation}
	\theta_{j}=\mathfrak{f}_{ij}\varphi^{i}.\label{eq23}
\end{equation}
Applying the covariant derivative on (\ref{eq23}) gives
\begin{equation}
	\nabla_{k}\theta_{j}=\left( \nabla_{k}\mathfrak{f}_{ij}\right) \varphi^{i}.\label{eq24}
\end{equation}
Multiplying (\ref{eq2}) by $ \varphi^{i} $ and using (\ref{eq21})-(\ref{eq24}) and the Ricci identity, one finds
\begin{equation}
	\theta_{j}-\mu\theta\mathfrak{f}_{j}-\lambda\varphi_{j}=0.\label{eq25}
\end{equation}
Taking the covariant derivative of (\ref{eq25}) yields
\begin{equation}
	\nabla_{k}\theta_{j}=\mu\left( \theta_{k}\mathfrak{f}_{j}+\theta\mathfrak{f}_{jk}\right) .\label{eq26}
\end{equation}
Interchanging the indices $ j $ and $ k $ in (\ref{eq26}) and then, subtracting  (\ref{eq26}) from the resulting equation, one infers that
\begin{equation*}
	\mu\left( \theta_{k}\mathfrak{f}_{j}-\theta_{j}\mathfrak{f}_{k}\right) =0.
\end{equation*}
Since $ \mu\neq0 $, the above equation means that
\begin{equation}
	 \theta_{k}\mathfrak{f}_{j}=\theta_{j}\mathfrak{f}_{k}.\label{eq27}
\end{equation}
On the otherhand,  multiplying (\ref{eq25}) by $ \mathfrak{f}_{k} $ yields
\begin{equation}
	\theta_{j}\mathfrak{f}_{k}-\mu\theta\mathfrak{f}_{j}\mathfrak{f}_{k}-\lambda\varphi_{j}\mathfrak{f}_{k}=0. \label{eq28}
\end{equation}
 Interchanging the indices $ j $ and $ k $ in (\ref{eq28}) and then, subtracting the resulting equation from (\ref{eq28}), we have
 \begin{equation}
 \lambda\left( \varphi_{j}\mathfrak{f}_{k}-\varphi_{k}\mathfrak{f}_{j}\right) =0, \label{eq29}
 \end{equation}
in which (\ref{eq27}) is used. 

Since $ \lambda\neq0 $, (\ref{eq29}) implies that
\begin{equation}
	 \varphi_{j}\mathfrak{f}_{k}=\varphi_{k}\mathfrak{f}_{j}. \label{eq30}
\end{equation}
After multiplying (\ref{eq30}) by $ \varphi^{k} $ and using (\ref{eq21}) immediately give
\begin{equation}
\mathfrak{f}_{j}=\frac{\theta}{\left| \varphi\right| }\varphi_{j}, \label{eq31}
\end{equation}  
where $ \left| \varphi\right|= \varphi^{j}\varphi_{j}$.
 Taking the covariant derivative of (\ref{eq31}) and using (\ref{eq22}), we achieve
\begin{equation}
	\mathfrak{f}_{jk}=\frac{\theta_{k}}{\left| \varphi\right| }\varphi_{j}. \label{eq32}
\end{equation}   
Substuting (\ref{eq31}) and (\ref{eq32}) into (\ref{eq2}), we reach
\begin{equation}
\mathfrak{R}_{ij}=\lambda\mathfrak{g}_{ij}+\frac{1}{\left| \varphi\right| }\left( \mu \theta^{2}-\theta_{k}\varphi^{k}\right) \varphi_{i}\varphi_{j}. \label{eq33}
\end{equation}
 Assume that $ \varphi_{i} $ is also a unit time-like vector, meaning that $ \left| \varphi\right| =\varphi_{i}\varphi^{i}=-1 $. Then, (\ref{eq33}) becomes the following form:
 \begin{equation}
 	\mathfrak{R}_{ij}=\lambda\mathfrak{g}_{ij}+\gamma \varphi_{i}\varphi_{j}, \label{eq34}
 \end{equation}
where $ \gamma= \theta_{k}\varphi^{k}-\mu \theta^{2} $.

 We can thus reach the following result:
	\begin{theorem}\label{2}
 A $(GQE)_{n}$ spacetime that admits a parallel unit time-like vector field is classified as a PF spacetime.
\end{theorem}

Applying the covariant derivetive of (\ref{eq34}) with respect to $ k $, we find that
\begin{equation}
	\nabla_{k}\mathfrak{R}_{ij}=\lambda\mathfrak{g}_{ij}+\left( \nabla_{k}\gamma \right) \varphi_{i}\varphi_{j}, \label{eq35}
\end{equation}
By interchanging the indices $ k $  and $ j $  in (\ref{eq35}) and subsequently subtracting the resulting equation from (\ref{eq35}), then we can derive 
\begin{equation}
	\nabla_{k}\mathfrak{R}_{ij}=	\nabla_{j}\mathfrak{R}_{ik}, \label{eq36}
\end{equation}
in which (\ref{eq23}), (\ref{eq24}) and (\ref{eq30}) are used.

(\ref{eq36}) means that $ \mathfrak{R}_{ij} $ is of Codazzi type and $ \mathfrak{R} $ becomes constant.  Consequently, this spacetime is classified within the subspaces $\mathfrak{B}$ and $\mathfrak{B}^{\prime}$ as introduced by Gray \cite{bibG78}. 

We can hence state that
	\begin{theorem}\label{3}
 A  $ (GQE)_{n} $ spacetime admitting a parallel unit time-like vector field is included in Gray's subspaces $ \mathfrak{B} $ and $ \mathfrak{B}^{\prime} $, and it has  constant scalar curvature.
\end{theorem}

According to Theorem \ref{3}, we can conclude that this spacetime possesses the property $ \text{div} \: \mathcal{C} = 0 $. Mantica et al. \cite{19} demonstrated that if a PF spacetime has the property $  \text{div} \: \mathcal{C} = 0 $  and constant scalar curvature, then such a spacetime qualifies as a GRW spacetime.

Considering Theorem \ref{2}, Theorem \ref{3} and the foregoing discussion, we have the result:
 \begin{theorem}\label{4}
A  $ (GQE)_{n} $ spacetime that admits a parallel unit time-like vector field is classified as a GRW spacetime.
 \end{theorem}
Based on \cite{19}, a 4-dimensional GRW spacetime qualifies as a PF spacetime if and only if it is classified as a RW spacetime. 

Thus, the following theorem can be stated:

\begin{theorem}\label{5}
	A $ (GQE)_{4} $ spacetime that admits a parallel unit time-like vector field is classified as a RW spacetime.
\end{theorem}
Additionally, it is well-established \cite{bibVRL05} that a GRW spacetime qualifies as a RW spacetime if and only if it possesses two specific characteristics: it must be conformally flat and exhibit Petrov type O. 

Therefore, based on Theorem \ref{4} and Theorem \ref{5},  the following conclusion can be drawn:
\begin{theorem}\label{6}
	A $ (GQE)_{4} $ spacetime that admits a parallel unit time-like vector field becomes conformally flat and of Petrov type O.
\end{theorem}

\begin{remark}Based on Theorem \ref{6}, since this spacetime is of Petrov type O, namely, conformally flat, in this case, the curvature is referred to as pure Ricci curvature. Thus, it can be concluded that in this spacetime, any gravitational effect must be caused directly by the field energy of matter or the energy of a non-gravitational force such as an electromagnetic field.
\end{remark}
Contracting (\ref{eq34}) gives
\begin{equation}
 \mathfrak{R}=n\lambda-\gamma. \label{eq37}
\end{equation}
Since $ \mathfrak{R} $ and $ \lambda $ are constant, $ \gamma $
is constant.
On the other hand, considering (\ref{eq5}), (\ref{eq34}) and (\ref{eq37}), we have
\begin{equation}
	\mathfrak{T}_{ij}=  \frac{\gamma-(n-2)\lambda}{2\kappa} \mathfrak{g}_{ij} +\frac{\gamma}{\kappa} \varphi_{i}\varphi_{j}. \label{eq38}
\end{equation}
After substituting (\ref{eq38}) into (\ref{eq4}), multiplying the new obtained equation by $ \mathfrak{g}^{ij} $ and $ \varphi^{i}\varphi^{j} $, respectively, we infer that:
\begin{equation}
	\sigma=\frac{\gamma+(n-2)\lambda}{2\kappa} \label{eq39}
\end{equation}
and 
\begin{equation}
\mathfrak{p}=\frac{\gamma-(n-2)\lambda}{2\kappa}. \label{eq40}
\end{equation}
Summing (\ref{eq39}) and (\ref{eq40}), it can be seen that
\begin{equation}
	\mathfrak{p}+\sigma=\frac{\gamma}{\kappa}. \label{41}
\end{equation}
Consequently, according to Theorem \ref{2} and the foregoing discussion, this result can be deduced:
\begin{theorem}\label{7}
	A $ (GQE)_{n} $ spacetime that admits a parallel unit time-like vector field is consistent with the present state of the universe.
\end{theorem}
In \cite{bibST1967}, Shepley and Taub established that a $ 4 $-dimensional PF spacetime with $ div \:\mathcal{C}=0$ and an equation of state represented as $\mathfrak{p} = \mathfrak{p}(\sigma) $ is conformally flat. Additionally, this spacetime is associated with the RW metric. It is that within this framework, the flow is characterized as geodesic, irrotational, and with no shear.

Thus, based on Theorem \ref{2}, Theorem \ref{3} and Theorem \ref{7}, we can conclude the result:

\begin{theorem}
	In a $ (GQE)_{4} $ spacetime that admits a parallel unit time-like vector field, the fluid flow is characterized as geodesic, irrotational, and it has no shear. 
\end{theorem}
On the other hand, due to Theorem \ref{6}, substituting (\ref{eq34}) into (\ref{eq6}) yields
\begin{equation}
\mathfrak{R}_{hijk}= \frac{(n-2)\lambda+\gamma}{(n-1)(n-2)} \left[ \mathfrak{g}_{hk}\mathfrak{g}_{ij}-\mathfrak{g}_{hj}\mathfrak{g}_{ik}\right] \label{eq42}
\end{equation}
\begin{equation*}
	+\frac{\gamma}{n-2}\left[ \mathfrak{g}_{ij}\varphi_{k}\varphi_{h}-\mathfrak{g}_{ik}\varphi_{j}\varphi_{h}+\mathfrak{g}_{kh}\varphi_{i}\varphi_{j}-\mathfrak{g}_{hj}\varphi_{i}\varphi_{k}\right] .
\end{equation*}

Since $ \lambda $ and $ \gamma $ are constant, $ \frac{(n-2)\lambda+\gamma}{(n-1)(n-2)} $ is constant. Therefore, based on \cite{bibCY72}, the previous equation means that this spacetime  has quasi-constant curvature.

Thus, the following result can be expressed:
\begin{theorem}\label{9}
A $ (GQE)_{n} $ spacetime that admits a parallel unit time-like vector field is of quasi-constant curvature.
\end{theorem}
 Furthermore, De et al. \cite{bibDSC22} demonstrated that an n-dimensional spacetime, $ n > 3 $, is a RW spacetime if and only if it exhibits quasi-constant curvature. Therefore, based on Theorem \ref{5}, Theorem \ref{9}, and the preceding discussion, the following result can be deduced:
 \begin{theorem}
 A $ (GQE)_{4} $ spacetime that admits a parallel unit time-like vector field is a RW spacetime if and only if it is of quasi-constant curvature.
 \end{theorem}

\section{$\mathcal{F(\mathfrak{R})}$-gravity}
To address the ongoing challenge of understanding the late-time accelerated expansion of the universe, without supposing the existence of dark energy, researchers have explored modifications to EFEs. One prominent and well-established approach in this regard is the $\mathcal{F(\mathfrak{R})}$-theory of modified gravity. According to this theoretical framework, the scalar curvature $\mathfrak{R}$ within the Einstein-Hilbert action term 
\begin{equation*}
\mathcal{S}=\frac{1}{2\kappa^{2}}\int d^{4}x\sqrt{-\mathfrak{g}}\mathcal{F(\mathfrak{R})}
	 	+\int d^{4}x\sqrt{-\mathfrak{g}}\mathcal{L}_{m}
\end{equation*} 
 is replaced by an arbitrary function $\mathcal{F(\mathfrak{R})}$. In the foregoing equation, $ \kappa^{2}=8\pi G $, $ G $ represents Newton's constant. This alteration leads to the derivation of the field equations associated with $\mathcal{F(\mathfrak{R})}$-gravity
 \begin{equation}
 \mathcal{F}_{\mathfrak{R}}(\mathfrak{R})\left(\mathfrak{g}_{ij}\Box -\nabla_{i}\nabla_{j}\right) +\mathcal{F}_{\mathfrak{R}}(\mathfrak{R})\mathfrak{R}_{ij}-\frac{\mathcal{F}(\mathfrak{R})}{2}\mathfrak{g}_{ij}=\kappa^{2}\mathfrak{T}_{ij}, \label{eq43}
 \end{equation}
 in which $ \Box $ indicates D'Alembert operator, that is, $ \Box=\nabla_{k}\nabla^{k} $, and $\mathcal{F}_{\mathfrak{R}}(\mathfrak{R})  $ represents the derivative according to $ \mathfrak{R} $. The EMT is obtained from the matter Lagrangian density $ \mathcal{L}_{m}=\mathcal{L}_{m}(\mathfrak{g}^{ij}) $ by
 \begin{equation*}
 \mathfrak{T}_{ij}=-\frac{2}{\sqrt{-\mathfrak{g}}}\frac{\delta(\sqrt{-\mathfrak{g}}\mathcal{L}_{m})}{\delta \mathfrak{g}_{ij} }.
 \end{equation*}
Transvecting (\ref{eq43}) with $ \mathfrak{g}^{ij} $, one gets
\begin{equation}
-2\mathcal{F}(\mathfrak{R})+\mathfrak{R}\mathcal{F}_{\mathfrak{R}}(\mathfrak{R})+3\Box\mathcal{F}_{\mathfrak{R}}(\mathfrak{R})=\kappa^{2}\mathfrak{T}_{ij}. \label{eq44}
\end{equation}
Subtraction $ \frac{\mathfrak{R}\mathcal{F}(\mathfrak{R})}{2} $ from both sides of (\ref{eq43}) leads to the following relation:
\begin{equation}
\mathcal{F}_{\mathfrak{R}}(\mathfrak{R})\mathfrak{R}_{ij}-\frac{\mathfrak{R}\mathcal{F}_{\mathfrak{R}}(\mathfrak{R})}{2}\mathfrak{g}_{ij}=\kappa^{2}\mathfrak{T}_{ij}^{(eff)}\label{eq45}
\end{equation}
such that
\begin{equation}
	\mathfrak{T}_{ij}^{(eff)}=\mathfrak{T}_{ij}+\mathfrak{T}_{ij}^{(curve)},	\label{eq46}
\end{equation}
in which
\begin{equation}
	\mathfrak{T}_{ij}^{(curve)}=\frac{1}{\kappa^{2}}\left[ \frac{\mathcal{F}(\mathfrak{R})-\mathfrak{R}\mathcal{F}_{\mathfrak{R}}(\mathfrak{R})}{2}\mathfrak{g}_{ij}+\left( \nabla_{i}\nabla_{j}-\mathfrak{g}_{ij}\Box\right)\mathcal{F}_{\mathfrak{R}}(\mathfrak{R})\right].	\label{eq47}
\end{equation}
$\mathfrak{T}_{ij}^{(eff)}$ denotes an effective voltage-energy tensor that incorporates geometric considerations. It is essential to note that this tensor does not adhere to traditional energy conditions, resulting in an energy density that is generally not positive definite.

By (\ref{eq45}), we get
\begin{equation}
	\mathfrak{R}_{ij}-\frac{\mathfrak{R}}{2}\mathfrak{g}_{ij}=\frac{\kappa^{2}}{\mathcal{F}_{\mathfrak{R}}(\mathfrak{R})}\mathfrak{T}_{ij}^{(eff)}.\label{eq48}
\end{equation}
It is clear that (\ref{eq48}) satisfies (\ref{eq46}) and (\ref{eq47}).

For $\mathcal{F(\mathfrak{R})}$-gravity, the field equations  are as follows:

\begin{equation}
\kappa\mathfrak{T}_{ij}=\mathcal{F}^{\prime}(\mathfrak{R})\mathfrak{R}_{ij}-\mathcal{F}^{\prime\prime\prime}(\mathfrak{R})\nabla_{i}\mathfrak{R}\nabla_{j}\mathfrak{R}-\mathcal{F}^{\prime\prime}(\mathfrak{R})\nabla_{i}\nabla_{j}\mathfrak{R} \label{eq49}
\end{equation}
\begin{equation*}
	+\mathfrak{g}_{ij}\left[ \mathcal{F}^{\prime\prime\prime}(\mathfrak{R})\nabla_{h}\mathfrak{R}\nabla^{h}\mathfrak{R}+\mathcal{F}^{\prime\prime}(\mathfrak{R})\nabla^{2}\mathfrak{R}-\frac{1}{2}\mathcal{F}\right] .
\end{equation*}
In the previous equation, $\mathcal{F}^{\prime}(\mathfrak{R})$ represents the derivative according to $ \mathfrak{R} $.
Since $ \mathfrak{R} $ is constant, we infer
\begin{equation}
	\mathfrak{R}_{ij}=\frac{\mathcal{F}(\mathfrak{R})}{2\mathcal{F}^{\prime}(\mathfrak{R})}\mathfrak{g}_{ij}+\frac{\kappa}{\mathcal{F}^{\prime}(\mathfrak{R})}\mathfrak{T}_{ij}.\label{eq50}
\end{equation}
Utilizing (\ref{eq4}) in the foregoing equation, we acquire 
\begin{equation}
	\mathfrak{R}_{ij}=\left[ \frac{\mathcal{F}(\mathfrak{R})}{2\mathcal{F}^{\prime}(\mathfrak{R})}+\mathfrak{p}\frac{\kappa}{\mathcal{F}^{\prime}(\mathfrak{R})}\right] \mathfrak{g}_{ij}+\frac{\kappa}{\mathcal{F}^{\prime}(\mathfrak{R})}(\mathfrak{p}+\sigma)\varphi_{i}\varphi_{j}.\label{eq51}
\end{equation}
Comparing (\ref{eq34}) and (\ref{eq51}), with help of (\ref{eq37}), we obtain
\begin{equation}
\mathfrak{p}=\frac{1}{\kappa}\left( \mathcal{F}^{\prime}(\mathfrak{R})\lambda-\frac{\mathcal{F}(\mathfrak{R})}{2}\right), \label{eq52}
\end{equation}

\begin{equation}
	\sigma=\frac{\mathcal{F}^{\prime}(\mathfrak{R})}{\kappa}\left( 3\lambda-\mathfrak{R}\right)+ \frac{\mathcal{F}(\mathfrak{R})}{2\kappa}.\label{eq53}
\end{equation}
We can therefore state that:
\begin{theorem}\label{10}
	In a $ (GQE)_{4} $ spacetime that admits a parallel unit time-like vector field obeying $\mathcal{F}(\mathfrak{R}) $-gravity, $ \mathfrak{p} $ and $ \sigma $ are given by (\ref{eq52}) and (\ref{eq53}), respectively.
\end{theorem}
Summing the last two equation, one finds that
\begin{equation}
	\mathfrak{p}+\sigma=\frac{\mathcal{F}^{\prime}(\mathfrak{R})}{\kappa}\left( 4\lambda-\mathfrak{R}\right).\label{eq54}
\end{equation}
 We can thus deduce the result:
 \begin{corollary}
 	 Let a spacetime $ (GQE)_{4} $ admitting a parallel unit time-like vector field obey $\mathcal{F}(\mathfrak{R}) $-gravity. In this case, it cannot accept the dark energy under the condition $ \mathfrak{R}\neq 4 \lambda $ .
 \end{corollary}
 In view of (\ref{eq52}) and (\ref{eq53}), it can be obtain that
\begin{equation}
	\sigma-3\mathfrak{p}=\frac{1}{\kappa}\left( 2\mathcal{F}(\mathfrak{R})-\mathfrak{R}\right).\label{eq55}
\end{equation} 
Hence, we can reach the result:
 \begin{corollary}
A $ (GQE)_{4} $ spacetime that admits a parallel unit time-like vector field obeying $\mathcal{F}(\mathfrak{R}) $-gravity, $\mathcal{F}(\mathfrak{R})=\frac{\mathfrak{R}}{2} $,  can accept the radiation era.
\end{corollary} 
\section{Energy Conditions}
Energy conditions serve as crucial tools in the study of black holes and wormholes across a range of modified gravity theories (\cite{bibBBY2017}, \cite{bibHE2013}). In this work, it is imperative to ascertain the effective energy density, $ \sigma^{(eff)}$, and the effective isotropic pressure, $ \mathfrak{p}^{(eff)} $,  to establish the specific energy conditions.

When examining the potential of various matter sources in gravitational field equations, as well as in extended theories of gravity and general relativity, energy conditions (ECs) serve as valuable tools to restrict the EMT. This helps uphold the notion that gravity is not only attractive but also that its energy density s positive. In the case of PF-type active matter within the framework of $\mathcal{F}(\mathfrak{R})$-gravity theory, these ECs can be defined as follows: (\cite{bibDLSS2021}, \cite{bibDLS2021})
\begin{enumerate}
	\item[-]Null EC (NEC): \quad $\sigma^{(eff)}+\mathfrak{p}^{(eff)}\geq 0$,
	\item[-] Weak EC (WEC):\quad $\sigma^{(eff)} \geq 0$, \quad $\sigma^{(eff)}+\mathfrak{p}^{(eff)}\geq 0$,  
	\item[-]Strong EC (SEC):\quad $\sigma^{(eff)}+3\mathfrak{p}^{(eff)}\geq 0$, \quad $\sigma^{(eff)}+\mathfrak{p}^{(eff)}\geq 0$, 
	\item[-] Dominant EC (DEC):\quad  $\sigma^{(eff)}\pm \mathfrak{p}^{(eff)}\geq 0$,  \quad $\sigma^{(eff)} \geq 0$. 
\end{enumerate}
(\ref{eq50}) can be rewritten as follows:
\begin{equation}
	\mathfrak{R}_{ij}-\frac{\mathfrak{R}}{2}\mathfrak{g}_{ij}=\frac{\kappa}{\mathcal{F}^{\prime}(\mathfrak{R})}\mathfrak{T}^{(eff)}_{ij},\label{eq56}
\end{equation}
here 
\begin{equation}
\mathfrak{T}^{(eff)}_{ij}=\mathfrak{T}_{ij}+\frac{\mathcal{F}(\mathfrak{R})-\mathfrak{R}\mathcal{F}^{\prime}(\mathfrak{R})}{2\kappa}\mathfrak{g}_{ij}. \label{eq57}
\end{equation}
Then, (\ref{eq4}) transforms to the following equation
\begin{equation}
	\mathfrak{T}^{(eff)}_{ij} = \mathfrak{p}^{(eff)} \mathfrak{g}_{ij} + (\mathfrak{p}^{(eff)} + \sigma^{(eff)}) \varphi_{i} \varphi_{j}. \label{eq58}
\end{equation}
such that 
\begin{equation}
\mathfrak{p}^{(eff)}=\mathfrak{p}+\frac{\mathcal{F}(\mathfrak{R})-\mathfrak{R}\mathcal{F}^{\prime}(\mathfrak{R})}{2\kappa}, \label{eq59}
\end{equation}

\begin{equation}
	\sigma^{(eff)}=\sigma-\frac{\mathcal{F}(\mathfrak{R})-\mathfrak{R}\mathcal{F}^{\prime}(\mathfrak{R})}{2\kappa}.\label{eq60}
\end{equation}
Putting (\ref{eq52}) and (\ref{eq53}) in (\ref{eq59})  and (\ref{eq60}), respectively, give
\begin{equation}
	\mathfrak{p}^{(eff)}=\frac{\mathcal{F}^{\prime}(\mathfrak{R})}{\kappa}\left( \lambda-\frac{\mathfrak{R}}{2}\right),  \label{eq61}
\end{equation}

\begin{equation}
	\sigma^{(eff)}=\frac{\mathcal{F}^{\prime}(\mathfrak{R})}{\kappa}\left( 3\lambda-\frac{\mathfrak{R}}{2}\right).\label{eq62}
\end{equation}
In view of ECs, with help of (\ref{eq61}) and (\ref{eq62}), we obtain for validity of ECs in a $ (GQE)_{4} $ spacetime that admits a parallel unit time-like vector field as follows:
\begin{enumerate}
	\item[-] For NEC, the condition of validation is $ \mathfrak{R}\le4\lambda $,
	\item[-] For WEC, the conditions of validation are $ \mathfrak{R}\le3\lambda $ and $ \mathfrak{R}\le4\lambda $,
	\item[-] For SEC, the conditions of validation are $ \mathfrak{R}\le4\lambda $ and $ \mathfrak{R}\le3\lambda $,
	\item[-] For DEC, the conditions of validation are $ \mathfrak{R}\le4\lambda $,  $ \lambda\geq 0 $ and $ \mathfrak{R}\le6\lambda $.
\end{enumerate}

\subsection*{Declerations }
\begin{flushleft}
	\textbf{Author Contributions.} The authors of this manuscript contributed equally to the work.\\
	\textbf{Funding. }There is no funding.\\
	\textbf{Availability of data and materials.} There is no data set used.\\
	\textbf{Conflict of interest.} There are no included interests of a financial or personal nature.\\
	\textbf{Ethical Approval.} The paper is a theoretical study. There are no applicable neither human or animal studies.
	
\end{flushleft}


\textit{}


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