Spacetimes Admitting Concircular Curvature Tensor in f(R) Gravity

The main object of this paper is to investigate spacetimes admitting concircular curvature tensor in f(R) gravity theory. At first, concircularly flat and concircularly flat perfect fluid spacetimes in $f\left(R\right)$ gravity are studied. In this case, the forms of the isotropic pressure p and the energy density σ are obtained. Next, some energy conditions are considered. Finally, perfect fluid spacetimes with divergence free concircular curvature tensor in f(R) gravity are studied; amongst many results, it is proved that if the energy-momentum tensor of such spacetimes is recurrent or bi-recurrent, then the Ricci tensor is semi-symmetric and hence these spacetimes either represent inflation or their isotropic pressure and energy density are constants.

1 Introduction

A concircular transformation was first coined by Yano in 1940 [1]. Such a transformation preserves geodesic circles. The geometry that deals with a concircular transformation is called concircular geometry. Under concircular transformation the concircular curvature tensor $\mathcal{M}$ remains invariant. Every spacetime M has vanishing concircular curvature tensor is called concirculary flat. A concircularly flat spacetime is of constant curvature. As a result, the deviation of a spacetime from constant curvature is measured by the concircular curvature tensor $\mathcal{M}$. Researchers have shown the curial role of the concircular curvature tensor in mathematics and physics (for example, see [2–6] and references therein).

In Einstein’s theory of gravity, the relation between the matter of spacetimes and the geometry of the spacetimes is given by Einstein’s field equations (EFE)

${\mathcal{R}}_{ij}-\frac{R}{2}{g}_{ij}=\kappa {\mathcal{T}}_{ij,}$

with κ being the Newtonian constant and ${\mathcal{T}}_{ij}$ is the energy-momentum tensor [7]. These equations imply that the energy-momentum tensor T_ij is divergence-free. This condition is satisfied whenever ∇_lT_ij = 0, where ∇_l denotes the covariant differentiation. There are many modifications of the standard relativity theory. The $f\left(R\right)$ gravity theory is the most popular of such modification of the standard theory of gravity. This important modification was first introduced in 1970 [8]. This modified theory can be obtained by replacing the scalar curvature R with a generic function $f\left(R\right)$ in the Einstein-Hilbert action. The field equations of $f\left(R\right)$ gravity are given as

$\begin{array}{ccc}\hfill \kappa {\mathcal{T}}_{ij}& =\hfill & f^{\prime }\left(R\right){\mathcal{R}}_{ij}-f^{\prime \prime }\left(R\right)f{\nabla }_{i}R{\nabla }_{j}R-f^{\prime \prime }\left(R\right){\nabla }_i{\nabla }_{j}R\hfill \\ \hfill & \hfill & +g_{ij}\left[f^{\prime \prime \prime }\left(R\right){\nabla }_{k}R{\nabla }^{k}R+f^{\prime \prime }\left(R\right){\nabla }^{2}R-\frac{1}{2}f\left(R\right)\right],\hfill \end{array}$(1.1)

where $f\left(R\right)$ is an arbitrary function of the scalar curvature R and $f^{\prime }\left(R\right)=\frac{df}{dR}$ which must be positive to ensure attractive gravity [9]. The f(R) gravity represents a higher order and well-studied theory of gravity. For example, an earlier investigation of quintessence and cosmic acceleration in $f\left(R\right)$ gravity theory as a higher order gravity theory are considered in [10]. Also, Capoziello et al. proved that, in a generalized Robertson-Walker spacetime with divergence free conformal curvature tensor, the higher order gravity tensor has the form of perfect fluid [11].

In a series of recent studies, weakly Ricci symmetric spacetimes (WRS)₄, almost pseudo-Ricci symmetric spacetimes(APRS)₄, and conformally flat generalized Ricci recurrent spacetimes are investigated in $f\left(R\right)$ gravity theory [12–14]. Motivated by these studies and many others, the main aim of this paper is to study concircularly flat and concircularly flat perfect fluid spacetimes in $f\left(R\right)$ gravity. Also, spacetimes with divergence free concircular curvature tensor in $f\left(R\right)$ gravity are considered.

This article is organized as follows. In Section 2, concircularly flat spacetimes in f(R) gravity are considered. In Section 3, we study concircularly flat perfect fluid spacetimes in f(R) gravity as well as we consider some energy conditions. Finally, spacetimes with divergence free concircular curvature tensor in f(R) gravity are investigated.

2 Concircularly Flat Spacetimes in f(R) Gravity

The concircular curvature tensor of type $\left(\mathrm{0,4}\right)$ is defined locally as

${\mathcal{M}}_{jklm}={\mathcal{R}}_{jklm}-\frac{R}{n\left(n-1\right)}\left[g_{kl}{g}_{jm}-g_{km}{g}_{jl}\right],$(2.1)

where ${\mathcal{R}}_{jklm}$, R, and g_kl are the Riemann curvature tensor, the scalar curvature tensor, and the metric tensor [1].

Here, we will consider ${\mathcal{M}}_{jklm}=0$, thus it follows form Eq. 2.1 that

${\mathcal{R}}_{jklm}=\frac{R}{n\left(n-1\right)}\left[g_{kl}{g}_{jm}-g_{km}{g}_{jl}\right].$(2.2)

This equation leads us to state the following theorem:

Theorem 1. A concircularly flat spacetime is of constant curvature.

Corollary 1. A concircularly flat spacetime is of constant scalar curvature.Contracting Eq. 2.2 with g^jm, we get

${\mathcal{R}}_{kl}=\frac{R}{n}{g}_{lk}.$(2.3)

In view of Eq. 2.3 we can state the following corollary:

Corollary 2. A concircularly flat spacetime is Einstein.In view of corollary 1, the field Eq. 1.1 in $f\left(R\right)$ gravity become

${\mathcal{R}}_{ij}-\frac{f}{2f^{\prime }}{g}_{ij}=\frac{\kappa }{f^{\prime }}{\mathcal{T}}_{ij}.$(2.4)

In vacuum case, we have

${\mathcal{R}}_{ij}-\frac{f}{2f^{\prime }}{g}_{ij}=0.$

Contracting with g^ij and integrating the result, one gets

$f=\lambda R^{\frac{n}{2}},$(2.5)

where λ is a constant.Conversely, if Eq. 2.5 holds, then

${\mathcal{T}}_{ij}=0.$

We can thus state following theorem:

Theorem 2. A concircularly flat spacetime in $f\left(R\right)$ gravity is vacuum if and only if $f=\lambda R^{\frac{n}{2}}$.The vector filed ξ is called Killing if

${\mathcal{L}}_{\xi }{g}_{ij}=0,$(2.6)

whereas ξ is called conformal Killing if

${\mathcal{L}}_{\xi }{g}_{ij}=2\phi g_{ij},$(2.7)

where ${\mathcal{L}}_{\xi }$ is the Lie derivative with respect to the vector filed ξ and φ is a scalar function [15–17]. The symmetry of a spacetime is measured by the number of independent Killing vector fields the spacetime admits. A spacetime of maximum symmetry has a constant curvature.A spacetime M is said to admit a matter collineation with respect to a vector field ξ if the Lie derivative of the energy-momentum tensor T with respect to ξ satisfies

${\mathcal{L}}_{\xi }{\mathcal{T}}_{ij}=0.$(2.8)

It is clear that every Killing vector field is a matter collineation, but the converse is not generally true. The energy-momentum tensor T_ij has the Lie inheritance property along the flow lines of the vector field ξ if the Lie derivative of T_ij with respect to ξ satisfies [15–17].

${\mathcal{L}}_{\xi }{\mathcal{T}}_{ij}=2\phi {\mathcal{T}}_{ij}.$(2.9)

Now using Eq. 2.3 in Eq. 2.4, one gets

$\left(\frac{R}{n}-\frac{f}{2f^{\prime }}\right)g_{ij}=\frac{\kappa }{f^{\prime }}{\mathcal{T}}_{ij}.$(2.10)

In a concircularly flat spacetime the scalar curvature R is constant, and hence f and f′ are also constants. Now, we consider a non-vacuum concircularly flat spacetime M. Therefore the Lie derivative ${\mathcal{L}}_{\xi }$ of Eq. 2.10 implies that

$\left(\frac{R}{n}-\frac{f}{2f^{\prime }}\right){\mathcal{L}}_{\xi }{g}_{ij}=\frac{\kappa }{f^{\prime }}{\mathcal{L}}_{\xi }{\mathcal{T}}_{ij}.$(2.11)

Assume that the vector field ξ is Killing on M, that is, Eq. 2.6 holds, thus we have

${\mathcal{L}}_{\xi }{\mathcal{T}}_{ij}=0.$

Conversely, if Eq. 2.8 holds, then form Eq. 2.11 it follows that

${\mathcal{L}}_{\xi }{g}_{ij}=0.$

We thus motivate to state the following theorem:

Theorem 3. Let M be a concircularly flat spacetime satisfying $f\left(R\right)$ gravity, then the vector field ξ is Killing if and only if M admits matter collineation with respect to ξ.The isometry of spacetimes prescriped by Killing vector fields represents a very important type of spacetime symmetry. Spacetimes of constant curvature are known to have maximum such symmetry, that is, they admit the maximum number of linearly independent Killing vector fields. The maximum numer of linearly independent Killing vector fields in an n − dimensional spacetime is $\frac{n\left(n+1\right)}{2}$ (The reader is referred to [18–22] and references therein for a more discussion on this topic). This fact with the above theorem leads to the following corollary.

Corollary 3. A non-vacuum concircularly flat spacetime satisfying $f\left(R\right)$ gravity admits the maximum number of matter collineations $\frac{n\left(n+1\right)}{2}$.Let ξ be a conformal Killing vector field, that is, Eq. 2.7 holds. Eq. 2.11 implies

${\mathcal{L}}_{\xi }{\mathcal{T}}_{ij}=2\phi {\mathcal{T}}_{ij}.$

Conversely, assume that Eq. 2.9 holds, then from Eq. 2.11 we obtain

${\mathcal{L}}_{\xi }{g}_{ij}=2\phi g_{ij}.$

Hence, we can state the following theorem:

Theorem 4. Let M be a concircularly flat spacetime satisfying $f\left(R\right)$ gravity, then M has a conformal Killing vector filed ξ if only if the energy-momentum tensor T_ij has the Lie inheritance property along ξ .The covariant derivative of both sides of Eq. 2.10 implies that

${\nabla }_k{\mathcal{T}}_{ij}=0.$(2.12)

Since in a concircularly flat spacetime R is constant, then f and f′ are constant. Inserting Eq. 2.4 in Eq. 2.12, we get

${\nabla }_k{\mathcal{R}}_{ij}=0.$

Thus, we have:

Theorem 5. Let M be a concirculary flat spacetime satisfying $f\left(R\right)$ gravity, then M is Ricci symmetric.

3 Concircularly Flat Perfect Fluid Spacetimes in f(R) Gravity

In a perfect fluid 4 − dimensional spacetime, the energy-momentum tensor ${\mathcal{T}}_{ij}$ obeys

${\mathcal{T}}_{ij}=\left(p+\sigma \right)u_i{u}_j+p{g}_{ij},$(3.1)

where p is the isotropic pressure, σ is the energy density, and u_i is a unit timelike vector field [7, 23].

Making use of Eq. 3.1 in Eq. 2.4, we get

${\mathcal{R}}_{ij}=\frac{\kappa }{f^{\prime }}\left[\left(p+\sigma \right)u_i{u}_j+p{g}_{ij}\right]+\frac{f}{2f^{\prime }}{g}_{ij}.$(3.2)

The use of Eq. 2.3 implies that

$\frac{R}{4}{g}_{ij}=\frac{\kappa }{f^{\prime }}\left[\left(p+\sigma \right)u_i{u}_j+p{g}_{ij}\right]+\frac{f}{2f^{\prime }}{g}_{ij}.$(3.3)

Contracting Eq. 3.3 with uⁱ, we get

$\sigma =\frac{2f-R{f}^{\prime }}{4\kappa }.$(3.4)

Transvecting Eq. 3.3 with g^ij and using Eq. 3.4, one obtains

$p=-\frac{2f-R{f}^{\prime }}{4\kappa }.$(3.5)

In consequence of the above we can state the following theorem:

Theorem 6. In a concircularly flat perfect fluid spacetime obeying f(R) gravity, the isotropic pressure p and the energy density σ are constants and $p=-\frac{2f-R{f}^{\prime }}{4\kappa }$ and $\sigma =\frac{2f-R{f}^{\prime }}{4\kappa }$.Combining Eq. 3.4 and Eq. 3.5, one easily gets

$p+\sigma =0,$(3.6)

which means that the spacetime represents dark matter era or alternatively the perfect fluid behaves as a cosmological constant [ [24]]. Thus we can state the following theorem:

Theorem 7. A concircularly flat perfect fluid spacetime obeying f(R) gravity represents dark matter era.In radiation era σ = 3p, therefor the energy-momentum tensor T_ij takes the form

${\mathcal{T}}_{ij}=4p{u}_j{u}_k+p{g}_{jk}.$(3.7)

Eq. 3.6 implies that p = 0. It follows that

${\mathcal{T}}_{ij}=0.$

which means that the spacetime is devoid of matter.Thus we motivate to state the following corollary:

Corollary 4. Let M be a concircularly flat spacetime obeying $f\left(R\right)$ gravity, then the Radiation era in M is vacuum.In pressureless fluid spacetime p = 0, the energy-momentum tensor is expressed as [25].

${\mathcal{T}}_{ij}=\sigma u_i{u}_j.$(3.8)

From Eq. 3.6 it follows that σ = 0. And consequently from Eq. 3.8 we infer

${\mathcal{T}}_{ij}=0,$

which means that the spacetime is vacuum.We thus can state the following:

Corollary 5. Let M be a concircularly flat dust fluid spacetime obeying $f\left(R\right)$ gravity, then M is vacuum.

3.1 Energy Conditions in Concircularly Flat Spacetime

In this subsection, some energy conditions in concircularly flat spacetimes obeying $f\left(R\right)$ gravity are considered. Indeed, energy conditions serve as a filtration system of the energy-momentum tensor in standard theory of gravity and the modified theories of gravity.[12–14]. In [26], the authors studied weak energy condition (WEC), dominant energy conditions (DEC), null energy conditions (NEC), and strong energy conditions in two extended theories of gravity. As a starting point, we need to determine the effective isotropic pressure p^eff and the effective energy density σ^eff to state some of these energy conditions.

Eq. 2.4 may be rewritten as

${\mathcal{R}}_{ij}-\frac{1}{2}R{g}_{ij}=\frac{\kappa }{f^{\prime }}{T}_{ij}^{e\mathit{ff}},$(3.9)

where

${\mathcal{T}}_{ij}^{e\mathit{ff}}={\mathcal{T}}_{ij}+\frac{f-R{f}^{\prime }}{2\kappa }{g}_{ij}.$

This leads us to rewrite Eq. 3.1 in the following form

${\mathcal{T}}_{ij}^{e\mathit{ff}}=\left(p^{e\mathit{ff}}+{\sigma }^{e\mathit{ff}}\right)u_i{u}_j+p^{e\mathit{ff}}{g}_{ij},$

where

$p^{e\mathit{ff}}=p+\frac{f-R{f}^{\prime }}{2\kappa }\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\sigma }^{e\mathit{ff}}=\sigma -\frac{f-R{f}^{\prime }}{2\kappa }.$

The use of Eq. 3.4 and Eq. 3.5 entails that

$\begin{array}{ccc}\hfill p^{e\mathit{ff}}& =\hfill & -\frac{R{f}^{\prime }}{4\kappa },\hfill \\ \hfill {\sigma }^{e\mathit{ff}}& =\hfill & \frac{R{f}^{\prime }}{4\kappa }.\hfill \end{array}$

Let us investigate certain energy conditions of a perfect fluid type effective matter in $f\left(R\right)$ gravity theory [12, 26, 27]:

1) Null energy condition (NEC): it says that p^eff + σ^eff ≥ 0.

2) Weak energy condition (WEC): it states that σ^eff ≥ 0 and p^eff + σ^eff ≥ 0.

3) Dominant energy condition (DEC): it states that σ^eff ≥ 0 and p^eff ± σ^eff ≥ 0.

4) Strong energy condition (SEC): it states that σ^eff + 3p^eff ≥ 0 and p^eff + σ^eff ≥ 0.

In this context, all mentioned energy conditions are consistently satisfied if Rf’ ≥ 0. As mentioned earlier, f′ must be positive to ensure attractive gravity. Therefore, the previous energy conditions are always satisfied if R ≥ 0.

4 Spacetimes With Divergence Free Concircular Curvature Tensor in f(R) Gravity

The divergence of the concircular curvature tensor, for n = 4, is given by [28].

${\nabla }_h{\mathcal{M}}_{klm}^h={\nabla }_h{\mathcal{R}}_{klm}^h-\frac{1}{12}\left[g_{lm}{\nabla }_{k}R-g_{km}{\nabla }_{l}R\right].$(4.1)

It is well-known that

${\nabla }_h{\mathcal{R}}_{klm}^h={\nabla }_k{\mathcal{R}}_{lm}-{\nabla }_l{\mathcal{R}}_{km}.$(4.2)

The use of Eq. 4.2 in Eq. 4.1 implies that

${\nabla }_h{\mathcal{M}}_{klm}^h={\nabla }_k{\mathcal{R}}_{lm}-{\nabla }_l{\mathcal{R}}_{km}-\frac{1}{12}\left[g_{lm}{\nabla }_{k}R-g_{km}{\nabla }_{l}R\right].$(4.3)

Assume that the concircular curvature tensor is divergence free, that is ${\nabla }_h{\mathcal{M}}_{klm}^h=0$, then

${\nabla }_k{\mathcal{R}}_{lm}-{\nabla }_l{\mathcal{R}}_{km}-\frac{1}{12}\left[g_{lm}{\nabla }_{k}R-g_{km}{\nabla }_{l}R\right]=0.$(4.4)

Contracting with g^lm and using ${\nabla }_j{\mathcal{R}}_i^j=\frac{1}{2}{\nabla }_{i}R$, we obtain

${\nabla }_{k}R=0.$(4.5)

Utilizing (4.5) in Eq. 4.4, we have

${\nabla }_k{\mathcal{R}}_{lm}={\nabla }_l{\mathcal{R}}_{km},$(4.6)

which means that the Ricci tensor is of Codazzi type [29]. The converse is trivial. Thus we can state the following theorem:

Theorem 8. Let M be a spacetime with concircular curvature tensor, then M has Codazzi type of Ricci tensor if and only if the concircular curvature tensor is divergence free.In view of Eq. 4.5, the field Eq. 1.1 in $f\left(R\right)$ gravity are

${\mathcal{R}}_{ij}-\frac{f}{2f^{\prime }}{g}_{ij}=\frac{\kappa }{f^{\prime }}{\mathcal{T}}_{ij}.$(4.7)

Using Eq. 4.7 in Eq. 4.6, we get

${\nabla }_k{\mathcal{T}}_{lm}={\nabla }_l{\mathcal{T}}_{km}.$

Hence, we have the following corollary:

Corollary 6. The energy-momentum tensor of a spacetime with divergence free concircular curvature tensor obeying $f\left(R\right)$ gravity is of Codazzi type.The spacetime M is called Ricci semi-symmetric [30] if

$\left({\nabla }_h{\nabla }_k-{\nabla }_k{\nabla }_h\right){\mathcal{R}}_{ij}=0,$(4.8)

whileas the energy-momentum tensor is called semi-symmetric if

$\left({\nabla }_h{\nabla }_k-{\nabla }_k{\nabla }_h\right){\mathcal{T}}_{ij}=0.$(4.9)

Now, Eq. 4.7 implies

$\left({\nabla }_h{\nabla }_k-{\nabla }_k{\nabla }_h\right){\mathcal{R}}_{ij}=\frac{\kappa }{f^{\prime }}\left({\nabla }_h{\nabla }_k-{\nabla }_k{\nabla }_h\right){\mathcal{T}}_{ij}.$

Thus, we can state the following theorem:

Theorem 9. Let M be a spacetime with divergence free concircular curvature tensor satisfying $f\left(R\right)$ gravity, then M is Ricci semi-symmetric if and only if the energy-momentum tensor of M is semi-symmetric.The energy-momentum tensor T_ij is called recurrent if there exists a non-zero 1 − form λ_k such that

${\nabla }_k{\mathcal{T}}_{ij}={\lambda }_k{\mathcal{T}}_{ij},$

whereas T_ij is called bi-recurrent if there exists a non-zero tensor ɛ_hk such that

${\nabla }_h{\nabla }_k{\mathcal{T}}_{ij}={\epsilon }_{hk}{\mathcal{T}}_{ij}.$

In view of the above definition, it is clear that every recurrent tensor field is bi-recurrent.Now assume that ${\mathcal{T}}_{ij}$ is any $\left(\mathrm{0,2}\right)$ symmetric recurrent tensor, that is,

${\nabla }_k{\mathcal{T}}_{ij}={\lambda }_k{\mathcal{T}}_{ij}.$(4.10)

Contracting with g^ij, we obtain

${\lambda }_k=\frac{1}{T}{\nabla }_{k}T,$(4.11)

where $T=g^{ij}{\mathcal{T}}_{ij}$.Applying the covariant derivative on both sides and using Eq. 4.11, we find

${\nabla }_l{\lambda }_k+{\lambda }_k{\lambda }_l=\frac{1}{T}{\nabla }_l{\nabla }_{k}T.$(4.12)

Taking the covariant derivative of Eq. 4.10 and utilizing Eq. 4.12, we get

${\nabla }_l{\nabla }_k{\mathcal{T}}_{ij}=\left(\frac{1}{T}{\nabla }_l{\nabla }_{k}T\right)T_{ij}.$

It follows that

$\left({\nabla }_l{\nabla }_k-{\nabla }_k{\nabla }_l\right){\mathcal{T}}_{ij}=0.$

Similarly, the same result holds for a bi-recurrent $\left(\mathrm{0,2}\right)$ symmetric tensor. In view of the above discussion, we have the following:

Lemma 1. A (bi-)recurrent $\left(\mathrm{0,2}\right)$ symmetric tensor is semi-symmetric.Assume that the energy-momentum tensor ${\mathcal{T}}_{ij}$ is recurrent or bi-recurrent, it follows form Lemma one that ${\mathcal{T}}_{ij}$ is semi-symmetric. Consequently, M is Ricci semi-symmetric.

Theorem 10. Let M be a spacetime with divergence free concircular curvature tensor obeying $f\left(R\right)$ gravity. If the energy-momentum (Ricci) tensor is recurrent or bi-recurrent, then the Ricci (energy-momentum) tensor is semi-symmetric.Let us now consider that M be a perfect fluid spacetime with divergence free concircular curvature tensor and whose energy-momentum tensor is recurrent or bi-recurrent. Thus the use of Eq. 3.1 in Eq. 4.7 implies that

${\mathcal{R}}_{ij}=a{g}_{ij}+b{u}_i{u}_j,$(4.13)

where

$a=\frac{1}{2f^{\prime }}\left(2\kappa p+f\right)\text{\hspace{0.17em}and\hspace{0.17em}}b=\frac{\kappa }{f^{\prime }}\left(p+\sigma \right).$(4.14)

Making a contraction of Eq. 4.13 with g^ij, we get

$R=\frac{1}{f^{\prime }}\left(3\kappa p-\kappa \sigma +2f\right).$(4.15)

Since the ${\mathcal{T}}_{ij}$ is recurrent or bi-recurrent, then $\left({\nabla }_h{\nabla }_k-{\nabla }_k{\nabla }_h\right){\mathcal{T}}_{ij}=0$ it follows that $\left({\nabla }_h{\nabla }_k-{\nabla }_k{\nabla }_h\right){\mathcal{R}}_{ij}=0$ Thus Eq. 4.13 implies that

$b{u}_i\left({\nabla }_h{\nabla }_k-{\nabla }_k{\nabla }_h\right)u_j+b{u}_j\left({\nabla }_h{\nabla }_k-{\nabla }_k{\nabla }_h\right)u_i=0.$

Contracting with uⁱ, we obtain

$-b\left({\nabla }_h{\nabla }_k-{\nabla }_k{\nabla }_h\right)u_j+b{u}_j{u}^i\left({\nabla }_h{\nabla }_k-{\nabla }_k{\nabla }_h\right)u_i=0.$

But $u^i\left({\nabla }_h{\nabla }_k-{\nabla }_k{\nabla }_h\right)u_i=0$, thus we have

$b\left({\nabla }_h{\nabla }_k-{\nabla }_k{\nabla }_h\right)u_j=0.$

Equivalently, it is

$b{\mathcal{R}}_{hkj}^m{u}_m=0.$

This equation implies the following cases:

Case 1. If ${\mathcal{R}}_{hkj}^m{u}_m=0\ne 0$, then b = 0 and hence we get p + σ = 0 which means that the spacetime represents inflation and the fluid behaves as a cosmological constant.

Case 2. If b ≠ 0, then ${\mathcal{R}}_{hkj}^m{u}_m=0$, hence a contraction with g^hj implies that ${\mathcal{R}}_{mk}{u}^m=0$. Contracting equation (4.13) with uⁱ and using ${\mathcal{R}}_{mk}{u}^m=0$, one gets

$\left(a-b\right)u_i=0.$

Then a − b = 0. With the help of equation (4.14), we have

$\sigma =\frac{f}{2\kappa }.$(4.16)

Using (4.16) in (4.15), we get

$p=\frac{2f^{\prime }R-3f}{6\kappa }.$(4.17)

Theorem 11. Let the energy-momentum tensor of a perfect fluid spacetime with divergence free concircular curvature tensor obeying $f\left(R\right)$ gravity be recurrent or bi-recurrent. Then,

1) The spacetime represents inflation, or

2) The isotropic pressure p and the energy density σ are constants. Moreover, they are given by Eq. 4.16 and Eq. 4.17.

In virtue of Eq. 4.16 and Eq. 4.17 we have

$\frac{p}{\sigma }=\frac{2f^{\prime }R-3f}{3f}.$

In view of the previous theorem we can state the following:

Remark 1. According to the different states of cosmic evolution of the Universe we can conclude that the spacetime with ${\nabla }_h{\mathcal{M}}_{klm}^h=0$ obeying $f\left(R\right)$ gravity either represents inflation or

Data Availability Statement

The original contributions presented in the study are included in the article/Supplementary material, further inquiries can be directed to the corresponding authors.

Author Contributions

Conceptualization and methodology, SS, UD, AS, NT, HA-D, and SA; formal analysis, SS, UD and AS; writing original draft preparation, SS, AS and NT; writing-review and editing, SS, UD, HA-D, and SA; supervision, SS and UD; project administration, NT and AS; and funding acquisition, NT and SA. All authors have read and agreed to the published version of the manuscript.

Funding

This project was supported by the Researchers Supporting Project number (RSP2022R413), King Saud University, Riyadh, Saudi Arabia.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

References

1. Yano K. Concircular Geometry I. Concircular Transformations. Proc Imperial Acad (1940) 16:195–200. doi:10.3792/pia/1195579139

CrossRef Full Text | Google Scholar

2. De UC, Shenawy S, Ünal B. Concircular Curvature on Warped Product Manifolds and Applications. Bull Malays Math Sci Soc (2020) 43:3395–409. doi:10.1007/s40840-019-00874-x

CrossRef Full Text | Google Scholar

3. Blair DE, Kim J-S, Tripathi MM. On the Concircular Curvature Tensor of a Contact Metric Manifold. J Korean Math Soc (2005) 42:883–92. doi:10.4134/jkms.2005.42.5.883

CrossRef Full Text | Google Scholar

4. Majhi P, De UC. Concircular Curvature Tensor on K-Contact Manifolds. Acta Mathematica Academiae Paedagogicae Nyregyhaziensis (2013) 29:89–99.

Google Scholar

5. Olszak K, Olszak Z. On Pseudo-riemannian Manifolds with Recurrent Concircular Curvature Tensor. Acta Math Hung (2012) 137:64–71. doi:10.1007/s10474-012-0216-5

CrossRef Full Text | Google Scholar

6. Vanli AT, Unal I. Conformal, Concircular, Quasi-Conformal and Conharmonic Flatness on normal Complex Contact Metric Manifolds. Int J Geom Methods Mod Phys (2017) 14:1750067. doi:10.1142/s0219887817500670

CrossRef Full Text | Google Scholar

7. O’Neill B. Semi-semi-Riemannian Geometry with Applications to Relativity. New York-London): Academic Press (1983).

Google Scholar

8. Buchdahl HA. Non-linear Lagrangians and Cosmological Theory. Monthly Notices R Astronomical Soc (1970) 150:1–8. doi:10.1093/mnras/150.1.1

CrossRef Full Text | Google Scholar

9. Capozziello S, Mantica CA, Molinari LG. General Properties of F(r) Gravity Vacuum Solutions. Int J Mod Phys D (2020) 29:2050089. doi:10.1142/s0218271820500893

CrossRef Full Text | Google Scholar

10. Capozziello S. Curvature Quintessence. Int J Mod Phys D (2002) 11:483–91. doi:10.1142/s0218271802002025

CrossRef Full Text | Google Scholar

11. Capozziello S, Mantica CA, Molinari LG. Cosmological Perfect Fluids in Higher-Order Gravity. Gen Relativity Gravitation (2020) 52:1–9. doi:10.1007/s10714-020-02690-2

CrossRef Full Text | Google Scholar

12. De A, Loo T-H, Arora S, Sahoo P. Energy Conditions for a (Wrs)_4(wrs) 4 Spacetime in F(r)-Gravity. The Eur Phys J Plus (2021) 136:1–10. doi:10.1140/epjp/s13360-021-01216-2

CrossRef Full Text | Google Scholar

13. De A, Loo T-H. Almost Pseudo-ricci Symmetric Spacetime Solutions in F(r)-Gravity. Gen Relativity Gravitation (2021) 53:1–14. doi:10.1007/s10714-020-02775-y

CrossRef Full Text | Google Scholar

14. De A, Loo T-H, Solanki R, Sahoo PK. A Conformally Flat Generalized Ricci Recurrent Spacetime in F(r)-Gravity. Phys Scr (2021) 96:085001. doi:10.1088/1402-4896/abf9d2

CrossRef Full Text | Google Scholar

15. Abu-Donia H, Shenawy S, Syied AA. The W* − Curvature Tensor on Relativistic Space-Times. Kyungpook Math J (2020) 60:185–95. doi:10.5666/KMJ.2020.60.1.185

CrossRef Full Text | Google Scholar

16. Mallick S, De UC. Spacetimes Admitting W2-Curvature Tensor. Int J Geom Methods Mod Phys (2014) 11:1450030. doi:10.1142/s0219887814500303

CrossRef Full Text | Google Scholar

17. Zengin FO. M-projectively Flat Spacetimes. Math Rep (2012) 14:363–70.

Google Scholar

18. De U, Shenawy S, Ünal B. Sequential Warped Products: Curvature and Conformal Vector fields. Filomat (2019) 33:4071–83. doi:10.2298/fil1913071d

CrossRef Full Text | Google Scholar

19. El-Sayied H, Shenawy S, Syied N. Conformal Vector fields on Doubly Warped Product Manifolds and Applications. Adv Math Phys (2016) 2016:6508309. doi:10.1155/2016/6508309

CrossRef Full Text | Google Scholar

20. El-Sayied HK, Shenawy S, Syied N. Symmetries Off−associated Standard Static Spacetimes and Applications. J Egypt Math Soc (2017) 25:414–8. doi:10.1016/j.joems.2017.07.002

CrossRef Full Text | Google Scholar

21. El-Sayied HK, Shenawy S, Syied N. On Symmetries of Generalized Robertson-walker Spacetimes and Applications. J Dynamical Syst Geometric Theories (2017) 15:51–69. doi:10.1080/1726037x.2017.1323418

CrossRef Full Text | Google Scholar

22. Shenawy S, Ünal B. 2-killing Vector fields on Warped Product Manifolds. Int J Math (2015) 26:1550065. doi:10.1142/s0129167x15500652

CrossRef Full Text | Google Scholar

23. Blaga AM. Solitons and Geometrical Structures in a Perfect Fluid Spacetime. Rocky Mountain J Math (2020) 50:41–53. doi:10.1216/rmj.2020.50.41

CrossRef Full Text | Google Scholar

24. Stephani H, Kramer D, MacCallum M, Hoenselaers C, Herlt E. Exact Solutions of Einstein’s Field Equations. Cambridge, United Kingdom: Cambridge University Press (2009).

Google Scholar

25. Srivastava SK. General Relativity and Cosmology. New Delhi, Delhi: PHI Learning Pvt. Ltd. (2008).

Google Scholar

26. Capozziello S, Lobo FSN, Mimoso JP. Generalized Energy Conditions in Extended Theories of Gravity. Phys Rev D (2015) 91:124019. doi:10.1103/physrevd.91.124019

CrossRef Full Text | Google Scholar

27. Santos J, Alcaniz J, Reboucas M, Carvalho F. Energy Conditions in F(r) Gravity. Phys Rev D (2007) 76:083513. doi:10.1103/physrevd.76.083513

CrossRef Full Text | Google Scholar

28. Ahsan Z, Siddiqui SA. Concircular Curvature Tensor and Fluid Spacetimes. Int J Theor Phys (2009) 48:3202–12. doi:10.1007/s10773-009-0121-z

CrossRef Full Text | Google Scholar

29. Gray A. Einstein-like Manifolds Which Are Not Einstein. Geometriae dedicata (1978) 7:259–80. doi:10.1007/bf00151525

CrossRef Full Text | Google Scholar

30. İnan Ü. On Metric Contact Pairs with Certain Semi-symmetry Conditions. Politeknik Dergisi (2021) 24:333–8. doi:10.2339/politeknik.769662

CrossRef Full Text | Google Scholar