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On $m$ $m$ -quasi-Einstein spacetimes

Absos Ali Shaikh

Department of Mathematics, The University of Burdwan, Golapbag, Burdwan 713104, West Bengal, India

E-mail Address: aask2003@yahoo.co.in

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Shyamal Kumar Hui

Department of Mathematics, The University of Burdwan, Golapbag, Burdwan 713104, West Bengal, India

E-mail Address: skhui@math.buruniv.ac.in

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, and

Akshoy Patra

Department of Mathematics, The University of Burdwan, Golapbag, Burdwan 713104, West Bengal, India

E-mail Address: akshoyp@gmail.com

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https://doi.org/10.1142/S0219887821500663Cited by:1 (Source: Crossref)

Abstract

In this paper, we have studied $m$ $m$ -quasi-Einstein spacetimes. Some basic results of such spacetimes are derived. Perfect and viscous fluid $m$ $m$ -quasi-Einstein spacetimes are also studied and the expressions of pressure, cosmological constant and energy density are obtained. We have proved that if the generator $ρ$ $ρ$ of an $m$ $m$ -quasi-Einstein spacetime is a Killing vector field, then the spacetime is either conformally flat or of Petrov-type $N$ $N$ . It is also shown that if the function $f$ $f$ of an $m$ $m$ -quasi-Einstein spacetime satisfying Einstein’s equation is harmonic and the matter distribution is perfect fluid, then Segre’ characteristics of the Ricci tensor is [(1,1), 1]. Finally, an example is constructed for the proper existence of such a spacetime.

Keywords:

AMSC: 53C15, 53C50

References

1. A. Barnes, On shear free normal flows of a perfect fluid, Gen Relativ. Gravit. 4 (1973) 105–129. Google Scholar
2. A. Barros and J. N. Gomes, Triviality of compact $m$ $m$ -quasi-Einstein manifolds, Res. Math. 71(1) (2017) 241–250. Web of Science, Google Scholar
3. A. Barros and E. Gomes and Ribeiro, Jr., Integral formulae on quasi Einstein manifolds and applications, Glasgow Math. J. 54(1) (2012) 212–223. Web of Science, Google Scholar
4. J. K. Beem and P. E. Ehrlich, Global Lorentzian Geometry (Marcel Dekker, 1981). Google Scholar
5. A. L. Besse, Einstein Manifolds (Springer, Berlin, 1987). Google Scholar
6. R. Deszcz, P. Verheyen and L. Verstraelen, On some generalized Einstein metric conditions, Publi. Inst. Math. 60 :74 (1996) 108–120. Google Scholar
7. D. Ferus, A remark on Codazzi tensors on constant curvature space, in Global Differential Geometry and Global Analysis, Lecture Notes in Mathematics, Vol. 838 (Springer-Verlag, New York, 1981), pp. 257–257. Google Scholar
8. A. Ghosh, $m$ $m$ -quasi Einstein metric and contact geometry, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. 113 (2019) 2587–2600. Google Scholar
9. A. Gray, Einstein-like manifolds which are not Einstein, Geom. Dedicata 7 (1978) 259–280. Google Scholar
10. S. Guha, On quasi Einstein and generalized quasi-Einstein manifolds, Facta Univ. Ser. Mech. Automat. Control Robot. 3(14) (2003) 821–842. Google Scholar
11. J. Kim and J. Shin, Three dimensional $m$ $m$ -quasi Einstein manifolds with degenerate Ricci tensor, Math. Nachr. 292 (2018) 1727–1750. Web of Science, Google Scholar
12. H. Karehar, Infinitesimal characterization of Friedman Universe, Arch. Math. Basel. 38 (1992) 58–64. Google Scholar
13. H. K. Lee, The fundamental groups of $m$ $m$ -quasi Einstein manifolds, ISRN Geom. 2011 (2011) 812541. Google Scholar
14. C. A. Mantica and G. L. Molinary, On the Weyl and Ricci tensors of generalized Robertson–Walker spacetimes, J. Math. Phys. 57(10) (2016) 102502. Web of Science, Google Scholar
15. C. A. Mantica and G. L. Molinary, A note on concircular structures spacetimes, Commun. Korean. Math. Soc. 34(2) (2019) 633–635. Web of Science, Google Scholar
16. K. Matsumoto and I. Mihai, On a certain transformation in a Lorentzian para-Sasskian manifold, Tensor (N.S.) 47 (1988) 189–197. Google Scholar
17. I. Mihai and R. Rosca, On Lorentzian P-Sassakian Manifolds, Classical Analysis, (World Scientific Publishing, Singapore, 1992), pp. 155–169. Google Scholar
18. I. Mihai, A. A. Shaikh and U. C. De, On Lorentzian para-Sassakian manifolds, J. Korean Math. Sci. 6 (1999) 1–13. Google Scholar
19. B. O’Neill, Semi Riemannian Geometry with Applications to Relativity (Academy Press, 1983). Google Scholar
20. A. Z. Petrov, Einstein Spaces (Pergamon Press, 1969). Google Scholar
21. A. K. Roychaudhury, S. Banerjee and A. Banerjee, General Relativity, Astrophysics and Cosmology (Springer-Verlag, 1992). Google Scholar
22. A. A. Shaikh, On Lorentzian almost paracompact manifolds with a structure of the concircular type, Kyungpook. Math. J. 43 (2001) 305–324. Google Scholar
23. A. A. Shaikh and K. K. Baishya, On concircular structure spacetimes, J. Math. Stat. 1(2) (2005) 129–132, https://doi.org/10.3844/jmssp.2005.129.132. Google Scholar
24. A. A. Shaikh and K. K. Baishya, On concircular structure spacetimes II, Amer. J. Appl. Sci. 3(3) (2006) 1790–1794. Google Scholar
25. A. A. Shaikh, Y. H. Kim and S. K. Hui, On Lorentzian quasi-Einstein manifolds, J. Korean Math. Soc. 48(4) (2011) 669–689. Web of Science, Google Scholar
26. A. A. Shaikh and S. K. Hui, On decomposable quasi-Einstein spaces, Math. Rep. 13(63) (2011) 89–94. Google Scholar
27. A. A. Shaikh, D. W. Yoon and S. K. Hui, On quasi-Einstein spacetimes, Tsukuba J. Math. 33(2) (2009) 305–326. Google Scholar
28. R. Sharma, Proper conformal symmetries of spacetimes with divergence Weyl conformal tensor, J. Math. Phys. 38 (1993) 3582–3587. Web of Science, Google Scholar