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Nearly vacuum static equations on K-contact manifolds and its applications in spacetimes

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Abstract

In the present paper, we study nearly vacuum static equations on K-contact manifolds. Also, we investigate nearly vacuum static equations on \(\eta \)-Einstein K-contact manifolds and construct an example to verify the deduced results. Moreover, we apply nearly vacuum static equations on super quasi-Einstein spacetimes.

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References

  1. M. Anderson, Scalar curvature, metric degenerations and the static vacuum Einstein equations on 3-manifolds I. Geom. Funct. Anal. 9(5), 855–967 (1999)

    Article  MathSciNet  Google Scholar 

  2. M. Anderson, On stationary solutions to the vacuum Einstein equations, to appear in annales Henri poincare. http://www.math.sunysb.edu/-anderson

  3. M. Anderson, Extrema of curvature functionals on the space of metrics on 3-manifolds. Calc. Var. P.D.E. 5, 199–269 (1997)

    Article  MathSciNet  Google Scholar 

  4. G. Bunting, A. Massoud-ul-Alam, Non-existence of multiple black holes in asymptotically Euclidean static vacuum space-times. Gen. Rel. Grav. 19, 147–154 (1987)

    Article  ADS  Google Scholar 

  5. W.B. Bonnor, Physical interpretation of vacuum solutions of Einstein’s equations I. Gen. Rel. Grav. 24(5), 551–574 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  6. G.F.R. Ellis, B.G. Schmidt, Singular space-times. Gen. Rel. Grav. 8(11), 915–953 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  7. A.E. Fischer, J.E. Marsden, Manifolds of Riemannian metrics with prescribed scalar curvature. Bull. Am. Math. Soc. 80, 479–484 (1974)

    Article  MathSciNet  Google Scholar 

  8. M. Anderson, On the structure of solutions to the static vacuum Einstein equations. Ann. Henri Poincare 1, 995–1042 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  9. A. Bhattacharyya, D.S. Patra, M. Tarafder, Certain almost Kenmotsu metrics satisfying the vacuum static equation. Publ. Inst. Math. 113(127), 109–119 (2023)

    Article  MathSciNet  Google Scholar 

  10. O. Kobayashi, M. Obata, Conformally-flatness and static space-times, in Manifolds and Lie Groups, Progress in Mathematics, vol. 14, ed. by J.-I. Hano, A. Morimoto, S. Murakami, K. Okamoto, H. Ozeki (Birkhäuser, Boston, MA, 1981), pp.197–206

    Chapter  Google Scholar 

  11. T. Mandal, A. Sarkar, U.C. De, On nearly vacuum static equations in almost cokähler manifolds with applications to spacetimes (communicated)

  12. J. Qing, W. Yuan, A note on static spaces and related problems. J. Geom. Phys. 74, 18–27 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  13. J. Qing, W. Yuan, On scalar curvature rigidity of vacuum static spaces. Math. Ann. 365, 1257–1277 (2016)

    Article  MathSciNet  Google Scholar 

  14. R.S. Hamilton, The Ricci flow on surfaces, in Mathematics and General Relativity, Contemporary Mathematics, vol. 71, (American Mathematical Society, Providence, 1988), pp.237–262

    Chapter  Google Scholar 

  15. P. Nurowski, M. Randall, Generalized Ricci solitons. J. Geom. Anal. 26, 1280–1345 (2016)

    Article  MathSciNet  Google Scholar 

  16. M.D. Siddiqi, Ricci \(\rho \)-soliton and geometrical structures in a dust fluid and viscous fluid spacetime. Bulg. J. Phys. 46, 163–173 (2019)

    Google Scholar 

  17. W. Wang, Almost cosymplectic (\(\kappa \), \(\mu \))-metrics as \(\eta \)-Ricci solitons. J. Nonlinear Math. Phys. (2021). https://doi.org/10.1007/s44198-021-00019-4

    Article  ADS  Google Scholar 

  18. Y. Wang, Ricci solitons on 3-dimensional cosymplectic manifolds. Math. Slovaca 67, 979–984 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  19. D.E. Blair, Riemannian geometry of contact and symplectic manifolds. Progress in Mathematics, vol 203 (Birkhäuser, New York, 2010)

  20. D.E. Blair, T. Koufogiorgos, B.J. Papantoniou, Contact metric manifolds satisfying a nullity condition. Israel J. Math. 91, 189–214 (1995)

    Article  MathSciNet  Google Scholar 

  21. D.E. Blair, T. Koufogiorgos, R. Sharma, A classification of 3-dimensional contact metric manifolds with \({\cal{Q} }\psi =\psi {\cal{Q} }\). Kodai Math. J. 13, 391–401 (1990)

    Article  MathSciNet  Google Scholar 

  22. C.P. Boyer, K. Galicki, Einstein manifolds and contact geometry. Proc. Am. Math. Soc. 129, 2419–2430 (2001)

    Article  MathSciNet  Google Scholar 

  23. R. Sharma, Certain results on \(K\)-contact and \((\kappa,\mu )\)-contact manifolds. J. Geom. 89, 138–147 (2008)

    Article  MathSciNet  Google Scholar 

  24. M.C. Chaki, On super quasi-Einstein manifolds. Publ. Math. Debr. 64, 481–488 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  25. R.L. Bishop, S.L. Goldberg, On conformally flat spaces with commuting curvature and Ricci transformations. Can. J. Math. 14, 799–804 (1972)

    Article  MathSciNet  Google Scholar 

  26. C.A. Mantica, U.C. De, Y.J. Suh, L.G. Molinari, Perfect fluid spacetimes with harmonic generalized curvature tensor. Osaka J. Math. 56, 173–182 (2019)

    MathSciNet  Google Scholar 

  27. U.C. De, D. Hazra, Characterizations of super quasi-Einstein spacetimes (communicated)

  28. S. Mallick, Super quasi-Einstein manifolds with applications to general relativity. Kyungpook Math. J. 58, 361–375 (2018)

    MathSciNet  Google Scholar 

  29. A. Sarkar, U. Biswas, \((m,\rho )\)-quasi-Einstein solitons on 3-dimensional trans-Sasakian manifolds and its applications in spacetimes. Int. J. Geom. Methods Modern Phys. 2450002, 12 (2024)

    Google Scholar 

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Acknowledgements

We would like to thank the referee and the editor for reviewing the paper carefully and their valuable comments to improve the quality of the paper.

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Correspondence to Tarak Mandal.

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Mitra, G., Mandal, T. & Sarkar, A. Nearly vacuum static equations on K-contact manifolds and its applications in spacetimes. Eur. Phys. J. Plus 139, 182 (2024). https://doi.org/10.1140/epjp/s13360-024-04964-z

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