Ricci-Yamabe Solitons in f(R)-gravity

Year 2023, , 334 - 342, 30.04.2023

Krishnendu De , U.c. De

https://doi.org/10.36890/iejg.1234057

Cited By: 3

Abstract

The main objective of this paper is to describe the perfect fluid spacetimes fulfilling $f(R)$-gravity, when Ricci-Yamabe, gradient Ricci-Yamabe and $\eta$-Ricci-Yamabe solitons are its metrics. We acquire conditions for which the Ricci-Yamabe and the gradient Ricci-Yamabe solitons are expanding, steady or shrinking. Furthermore, we investigate $\eta$-Ricci-Yamabe solitons and deduce a Poisson equation and with the help of this equation, we acquire some significant results.

Keywords

Perfect fluids, f(R)-gravity, Ricci-Yamabe solitons, η-Ricci-Yamabe solitons

References

• [1] Barbosa, E., Pina, R., Tenenblat, K.: On gradient Ricci solitons conformal to a pseudo-Euclidean space. Israel J. Math. 200 , 213-224 (2014).
• [2] Batat, W., Brozos-Vazquez, M., Garcia-Rio, E., Gavino-Fernandez, S.: Ricci soliton on Lorentzian manifolds with large isometry groups. Bull. Lond. Math. Soc. 43, 1219-1227 (2011).
• [3] Blaga, A. M.: Solitons and geometrical structures in a perfect fluid spacetime. Rocky Mountain J. Math. 50, 41-53 (2020).
• [4] Blaga, A. M.: $\eta$-Ricci solitons on Lorentzian para-Sasakian manifolds. Filomat \textbf 30 (2) , 489-496 (2016).
• [5] Brozos-Vazqnez, M., Calvaruso, G., Garcia-Rio, E., Gavino-Fernandez, S.: Three-dimensional Lorentzian homogeneous Ricci solitons. Israel J. Math. 188, 385-403 (2012).
• [6] Buchdahl, H. A.: Non-linear Lagrangians and cosmological theory. Mon. Not. Roy. Astron. Soc., 150 1, (1970).
• [7] Capozziello, S., Cardone, V.F., Salzano, V.: Cosmography of f(R) gravity. Phys. Rev. D, 78, 063504 (2008).
• [8] Capozziello, S., Mantica, C. A., Molinari, L. G.: Cosmological perfect fluids in higher-order gravity. Gen. Relativ. Gravit. 52, 36 (2020). https://doi.org/10.1007/s10714-020-02690-2.
• [9] Capozziello, S., D’Agostino, R., Luongo, O.: Extended Gravity Cosmography. Int. J. Mod. Phys. D (2019) doi:10.1142/S0218271819300167
• [10] Chavanis, P. H.: Cosmology with a stiff matter era, Phys. Rev. D 92, 103004 (2015).
• [11] Catino, G., Mazzieri, L.: Gradient Einstein solitons. Nonlinear Anal., 132, 66-94 (2016).
• [12] Chen, B. Y., Deshmukh, S.: A note on Yamabe solitons. Balkan J. Geom. Appl., 23, 37-43 (2018).
• [13] Cho, J. T., Kimura, M., Ricci solitons and real hypersurfaces in a complex space form. Tohoku Math. J., 61, 205-212 (2009).
• [14] De, A., Loo, T. H., Arora, S., Sahoo, P. K.: Energy condition for a $(W$RS$)_4$ spacetime in $f(R)$-gravity, Eur. Phys. J. Plus, https://doi.org/10.1140/epjp/s13360-021-01216-2
• [15] De, A., Loo, T.H.: Almost pseudo-Ricci symmetric spacetime solutions in $f(R)$-gravity. Gen. Relativ. Gravit. 53, 5 (2021).
• [16] De, K., De, U.C.: A note on gradient Solitons on para-Kenmotsu manifolds. Int.J.Geom. Methods Mod. Phys. 18 01, 2150007 (11 pages) (2021).
• [17] De, K., De, U.C.: $\delta$-almost Yamabe solitons in paracontact metric manifolds. Mediterr. J. Math. 18, 218 (2021).
• [18] De, K., De, U.C.: Investigations on solitons in $f(\mathcal{R})$-gravity. Eur. Phys. J. Plus (2022) 137:180. https://doi.org/10.1140/epjp/s13360-022-02399-y
• [19] De, U.C., Chaubey, S. K., Shenawy, S.: Perfect fluid spacetimes and Yamabe solitons. J. Math. Phys. 62, 032501 (2021).
• [20] Duggal, Krishan L.: Almost Ricci Solitons and physical applications. Int. Elect. J. Geo., 10, 1-10 (2017).
• [21] Duggal, Krishan L.: A new class of Almost Ricci Solitons and their physical interpretion. Int. scholarly research Notices, 4903520, 6 pages (2016).
• [22] Guler, S., Crasmareanu, M.: Ricci-Yamabe maps for Riemannian flow and their volume variation and volume entropy. Turk. J. Math., 43, 2631-2641 (2019).
• [23] Hamilton, R. S.: Lectures on geometric flows. 1989 (unpublished).
• [24] Hamilton, R.S.: The Ricci flow on surfaces. Contemp. Math. 71, 237-261 (1988).
• [25] O’Neill, B.: Semi-Riemannian geometry with applications to relativity. Academic Press. London (1983).
• [26] Akyol, M. A., Siddiqi, M. D.: $\eta$-Ricci-Yamabe solitons on Riemannian submersions from Riemannian manifolds. arXiv:2004.14124
• [27] Starobinsky, A. A.: A new type of isotropic cosmological models without singularity. Phys. Lett. B 91 1, 99-102 (1980).
• [28] Naik, D. M., Venkatesha, V., Kumara, H. A.: Ricci solitons and certain related metrics on almost co-Kaehler manifolds. J. Math. Phys. Anal. Geom., 16, 402-417 (2020).
• [29] Wang, Y.: Yamabe solitons on three-dimensional Kenmotsu manifolds. Bull. Belg. Math. Soc. Simon Stevin, 23, 345-355 (2016).