# Spacetimes Admitting Concircular Curvature Tensor in f(R) Gravity

^{1}Department of Pure Mathematics, University of Calcutta, West Bengal, India^{2}Basic Science Department, Modern Academy for Engineering and Technology, Maadi, Egypt^{3}Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt^{4}Department of Mathematics, College of Science, King Saud University, Riyadh, Saudi Arabia^{5}Applied Science College, Department of Mathematical Sciences, Umm Al-Qura University, Makkah, Saudi Arabia

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 *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 *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

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)

with *κ* being the Newtonian constant and *T*_{ij} is divergence-free. This condition is satisfied whenever ∇_{l}*T*_{ij} = 0, where ∇_{l} denotes the covariant differentiation. There are many modifications of the standard relativity theory. The *R* with a generic function

where *R* and *f*(*R*) gravity represents a higher order and well-studied theory of gravity. For example, an earlier investigation of quintessence and cosmic acceleration in

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

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

where *R*, and *g*_{kl} are the Riemann curvature tensor, the scalar curvature tensor, and the metric tensor [1].

Here, we will consider

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

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

In vacuum case, we have

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

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

We can thus state following theorem:

Theorem 2. *A concircularly flat spacetime in* *gravity is vacuum if and only if* *.*The vector filed *ξ* is called Killing if

whereas *ξ* is called conformal Killing if

where *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

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].

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

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

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

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

We thus motivate to state the following theorem:

Theorem 3. *Let M be a concircularly flat spacetime satisfying* *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

Corollary 3. *A non-vacuum concircularly flat spacetime satisfying* *gravity admits the maximum number of matter collineations* *ξ* be a conformal Killing vector field, that is, Eq. 2.7 holds. Eq. 2.11 implies

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

Hence, we can state the following theorem:

Theorem 4. *Let M be a concircularly flat spacetime satisfying* *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

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

Thus, we have:

Theorem 5. *Let M be a concirculary flat spacetime satisfying* *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

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

The use of Eq. 2.3 implies that

Contracting Eq. 3.3 with *u*^{i}, we get

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

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* *and* *.*Combining Eq. 3.4 and Eq. 3.5, one easily gets

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* = 3*p*, therefor the energy-momentum tensor *T*_{ij} takes the form

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

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* *gravity, then the Radiation era in M is vacuum.*In pressureless fluid spacetime *p* = 0, the energy-momentum tensor is expressed as [25].

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

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* *gravity, then M is vacuum.*

### 3.1 Energy Conditions in Concircularly Flat Spacetime

In this subsection, some energy conditions in concircularly flat spacetimes obeying *p*^{eff} and the effective energy density *σ*^{eff} to state some of these energy conditions.

Eq. 2.4 may be rewritten as

where

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

where

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

Let us investigate certain energy conditions of a perfect fluid type effective matter in

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} + 3*p*^{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].

It is well-known that

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

Assume that the concircular curvature tensor is divergence free, that is

Contracting with *g*^{lm} and using

Utilizing (4.5) in Eq. 4.4, we have

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

Using Eq. 4.7 in Eq. 4.6, we get

Hence, we have the following corollary:

Corollary 6. *The energy-momentum tensor of a spacetime with divergence free concircular curvature tensor obeying* *gravity is of Codazzi type.*The spacetime *M* is called Ricci semi-symmetric [30] if

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

Now, Eq. 4.7 implies

Thus, we can state the following theorem:

Theorem 9. *Let M be a spacetime with divergence free concircular curvature tensor satisfying* *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

whereas *T*_{ij} is called bi-recurrent if there exists a non-zero tensor *ɛ*_{hk} such that

In view of the above definition, it is clear that every recurrent tensor field is bi-recurrent.Now assume that

Contracting with *g*^{ij}, we obtain

where

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

It follows that

Similarly, the same result holds for a bi-recurrent

Lemma 1. *A* (*bi-*)*recurrent* *symmetric tensor is semi-symmetric.*Assume that the energy-momentum tensor *M* is Ricci semi-symmetric.

Theorem 10. *Let M be a spacetime with divergence free concircular curvature tensor obeying* *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

where

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

Since the

Contracting with *u*^{i}, we obtain

But

Equivalently, it is

This equation implies the following cases:

Case 1. *If* *, 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* *, hence a contraction with g*^{hj} *implies that* *Contracting equation* (4.13) *with u*^{i} *and using* *, one gets*

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

*Using* (4.16) *in* (4.15)*, we get*

Theorem 11. *Let the energy-momentum tensor of a perfect fluid spacetime with divergence free concircular curvature tensor obeying* *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

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* *obeying* *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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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).

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

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

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

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

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

Keywords: perfect fluid, energy-momentum tensor, concircular curvature tensor, f (R) gravity theory, energy conditions in modified gravity

Citation: De UC, Shenawy S, Abu-Donia HM, Turki NB, Alsaeed S and Syied AA (2022) Spacetimes Admitting Concircular Curvature Tensor in f(R) Gravity. *Front. Phys.* 9:800060. doi: 10.3389/fphy.2021.800060

Received: 22 October 2021; Accepted: 09 December 2021;

Published: 12 January 2022.

Edited by:

Jae Won Lee, Gyeungsang National University, South KoreaReviewed by:

Peibiao Zhao, Nanjing University of Science and Technology, ChinaSalvatore Capozziello, University of Naples Federico II, Italy

İnan Ünal, Munzur University, Turkey

Oğuzhan Bahadır, Kahramanmaras Sütçü Imam University, Turkey

Copyright © 2022 De, Shenawy, Abu-Donia, Turki, Alsaeed and Syied. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Nasser Bin Turki, nassert@ksu.edu.sa; Abdallah Abdelhameed Syied, a.a_syied@yahoo.com