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Relativistic spacetimes admitting almost Schouten solitons

Department of Pure Mathematics, University of Calcutta, 35, Ballygunge Circular Road, Kolkata 700019, West Bengal, India

E-mail Address: uc_de@yahoo.com

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Arpan Sardar

https://orcid.org/0000-0001-6602-9357

Department of Mathematics, University of Kalyani, Kalyani 741235, Nadia, West Bengal, India

E-mail Address: arpansardar51@gmail.com

Corresponding author.

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, and

Fatemah Mofarreh

Department of Mathematical Science, Faculty of Science, Princess Nourah Bint Abdulrahman University, Riyadh 11546, Saudi Arabia

E-mail Address: fyalmofarrah@pnu.edu.sa

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https://doi.org/10.1142/S0219887823501475Cited by:4 (Source: Crossref)

Abstract

In this paper, we investigate almost Schouten solitons and almost gradient Schouten solitons in spacetimes of general relativity. At first, it is proven that if a generalized Robertson–Walker spacetime permits an almost Schouten soliton, then it becomes a perfect fluid spacetime as well as the spacetime represents a dark matter era. Besides this, we investigate almost gradient Schouten solitons in generalized Robertson–Walker spacetimes. Moreover, a spacetime obeying almost Schouten solitons whose potential vector field is a non-homothetic conformal vector field is of Petrov type $O$ $O$ or $N$ $N$ .

Keywords:

AMSC: 53C25, 53E20, 53Z05

References

1. L. Alias, A. Romero and M. Sanchez , Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson–Walker spacetimes, Gen. Relat. Gravit. 27 (1995) 71–84. Crossref, Web of Science, Google Scholar
2. L. Alias, A. Romero and M. Sanchez , Compact spacelike hypersurfaces of constant mean curvature in generalized Robertson–Walker spacetimes, in Geometry and Topology of Submanifolds VII (World Scientific, River Edge New Jersey, USA, 1995), pp. 67–70. Google Scholar
3. A. M. Blaga , Solitons and geometrical structures in a perfect fluid spacetime, Rocky Mountain J. Math. 50 (2020) 41–53. Google Scholar
4. A. M. Blaga and B.-Y. Chen , Gradient solitons on statistical manifolds, J. Geom. Phys. 164 (2021) 104195. Crossref, Web of Science, Google Scholar
5. V. Borges , On complete gradient Schouten solitons, Nonlinear Anal. 221 (2022) 112883, https://doi.org/10.2016/j.na.2022.112883. Crossref, Web of Science, Google Scholar
6. G. Catino and L. Mazzieri , Gradient Einstein solitons, Nonlinear Anal. 132 (2016) 66–94. Crossref, Web of Science, Google Scholar
7. P. H. Chavanis , Cosmology with a stiff matter era, Phys. Rev. D 92 (2015) 103004. Crossref, Web of Science, Google Scholar
8. B.-Y. Chen , Classification of torqued vector fields and its applications to Ricci solitons, Kragujevac J. Math. 41 (2017) 239–250. Crossref, Web of Science, Google Scholar
9. B.-Y. Chen , A simple characterization of generalized Robertson-Walker spacetimes, Gen. Relativ. Gravit. 46 (2014) 1833. Crossref, Web of Science, Google Scholar
10. B.-Y. Chen and S. Deshmukh , A note on Yamabe solitons, Balkan J. Geom. Appl. 23 (2018) 37–43. Google Scholar
11. B.-Y. Chen and S. Deshmukh , Yamabe and Quasi-Yamabe Solitons on Euclidean Submanifolds, Mediterranean J. Math. 15 (2018) 1–9. Google Scholar
12. K. De, U. C. De, A. A. Syied, N. B. Turki and S. Alsaeed , Perfect fluid spacetimes and gradient solitons, J. Nonlinear Math. Phys. 29 (2022) 843–858, https://doi.org/10.1007/s44198-022-00066-5. Crossref, Web of Science, Google Scholar
13. U. C. De, S. K. Chaubey and S. Shenawy , Perfect fluid spacetimes and Yamabe solitons, J. Math. Phys. 62 (2021) 032501. Crossref, Web of Science, Google Scholar
14. U. C. De, C. A. Mantica and Y. J. Suh , Perfect fluid spacetimes and gradient solitons, Filomat 36 (2022) 829–842. Crossref, Web of Science, Google Scholar
15. A. Fialkow , Conformals geodesics, Trans. Amer. Math. Soc. 45 (1939) 443–473. Crossref, Google Scholar
16. M. Gutierrez and B. Olea , Global decomposition of a Lorentzian manifold as a generalized Robertson-Walker space, Differ. Geom. Appl. 27 (2009) 146–156. Crossref, Web of Science, Google Scholar
17. S. W. Hawking and G. F. R. Ellis , The Large Scale Structure of Spacetime (Cambridge University Press, Cambridge, 1973). Crossref, Google Scholar
18. C. A. Mantica, L. G. Molinari and U. C. De , A condition for a perfect-fluid space-time to be a generalized Robertson–Walker space-time, J. Math. Phys. 57 (2016) 022508. Crossref, Web of Science, Google Scholar
19. C. A. Mantica, Y. J. Suh and U. C. De , A note on generalized Robertson–Walker space-times, Int. J. Geom. Methods Mod. Phys. 13 (2016) 1650079. Link, Web of Science, Google Scholar
20. C. A. Mantica and L. G. Molinari , Generalized Robertson–Walker spacetimes-A survey, Int. J. Geom. Methods Mod. Phys. 14 (2017) 1730001. Link, Web of Science, Google Scholar
21. B. O’Neill , Semi-Riemannian Geometry with Applications to Relativity (Academic Press, New York, 1983). Google Scholar
22. R. Pina and I. Menezes, Rigidity results on gradient Schouten solitons, preprint (2020), arXiv: 2010.06729V1 [math.DG]. Google Scholar
23. A. Romero, R. N. Rubio and J. J. Salamanca , Uniqueness of complete maximal hypersurfaces in spatially parabolic generalized Robertson–Walker spacetimes, Class. Quantum Grav. 30 (2013) 115007. Crossref, Web of Science, Google Scholar
24. R. Sharma , Conformal vector fields in symmetric and conformal symmetric spaces, Int. J. Math. Sci. 12 (1989) 85–88. Crossref, Google Scholar
25. R. Sharma and A. Ghosh , Perfect fluid space-times whose energy-momentum tensor is conformal Killing, J. Math. Phys. 51 (2010) 022504. Crossref, Web of Science, Google Scholar
26. M. D. Siddiqi and U. C. De , Relativistic perfect fluid spacetimes and Ricci–Yamabe solitons, Lett. Math. Phys. 112 (2022) 1. Crossref, Web of Science, Google Scholar
27. K. Yano , On torse forming direction in a Riemannian space, Proc. Imp. Acad. Tokyo 20(6) (1944) 340–345. Google Scholar