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On nearly vacuum static equations in almost coKähler manifolds with applications to spacetimes

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Abstract

In the present article, we extend the notion of vacuum static equations on almost coKähler manifolds and rename them as nearly vacuum static equations. It is shown that if an \(\eta \)-Einstein almost coKähler manifold admits a non-trivial solution of a nearly vacuum static equation, then the solution must be a constant. In \((\kappa ,\mu )\)-almost coKähler manifolds, the non-trivial solutions of nearly vacuum static equations do not exist. We also apply nearly vacuum static equations on perfect fluid spacetimes as well as generalized Robertson–Walker spacetimes. Among others, it is shown that a perfect fluid spacetime admitting nearly vacuum static equations is of constant scalar curvature and a generalized Robertson–Walker spacetime obeying nearly vacuum static equations represents a dark matter era.

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References

  1. Alías, L.J., Romero, A., Sánchez, M.: Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson–Walker spacetimes. Gen. Relativ. Gravit. 27, 71–84 (1995)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Balkan, Y.S., Uddin, S., Alkhaldi, A.H.: A class of \(\phi \)-recurrent almost cosymplectic space. Honam Math. J. 40(2), 293–304 (2018)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. Blaga, A.M.: Solitons and geometrical structures in a perfect fluid spacetime. Rocky Mt. J. Math. 50, 41–53 (2020)

    Article  MathSciNet  Google Scholar 

  7. Blair, D.E.: Riemannian Geometry of Contact and Symplectic Manifolds. Progress in Mathematics, vol. 203. Birkhäuser, New York (2010)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. Chavanis, P.H.: Cosmology with a stiff matter era. Phys. Rev. D 92, 103004 (2015)

    Article  MathSciNet  Google Scholar 

  10. Chen, B.-Y.: A simple characterization of generalized Robertson–Walker spacetimes. Gen. Relativ. Gravit. 46, 1833 (2014)

    Article  MathSciNet  Google Scholar 

  11. Chen, B.-Y.: Differential Geometry of Warped Product Manifolds and Submanifolds. World Scientific Publishing, Hackensack (2017)

    Book  Google Scholar 

  12. Chen, X.: Einstein–Weyl structures on almost cosymplectic manifolds. Period. Math. Hung. 79, 191–203 (2019)

    Article  MathSciNet  Google Scholar 

  13. Chen, X.: Quasi-Einstein structure and almost co-symplectic manifolds. RACSAM 114(2), 1–14 (2020)

    Article  Google Scholar 

  14. Chen, X.: Almost quasi-Yamabe solitons on almost cosymplectic manifolds. Int. J. Geom. Methods Mod. Phys. 17, 2050070 (2020)

    Article  MathSciNet  Google Scholar 

  15. Chen, X., Yang, Y.: Static perfect fluid spacetimes on contact metric manifolds. Period. Math. Hung. 86, 160–171 (2023)

    Article  Google Scholar 

  16. De, U.C., Chaubey, S.K., Suh, Y.J.: A note on almost co-Kählar manifolds. Int. J. Geom. Methods Mod. Phys. 1710, 2050153 (2020)

    Article  Google Scholar 

  17. De, K., De, U.C., Syied, A.A., Turki, N.B., Alsaeed, S.: Perfect fluid spacetimes and gradient solitons. J. Nonlinear Math. Phys. 29, 843–858 (2022)

    Article  MathSciNet  Google Scholar 

  18. De, U.C., Mantica, C.A., Suh, Y.J.: Perfect fluid spacetimes and gradient solitons. Filomat 36, 829–842 (2022)

    Article  MathSciNet  Google Scholar 

  19. Deshmukh, S., Turki, N.B., Vilcu, G.E.: A note on static spaces. Results Phys. 27, 104519 (2021)

    Article  Google Scholar 

  20. Ferus, D.: Global Differential Geometry and Global Analysis. Springer, New York (1981)

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  22. Gray, A.: Einstein-like manifolds which are not Einstein. Geom. Dedicata. 7, 259–280 (1978)

    Article  MathSciNet  Google Scholar 

  23. Guilfoyle, B.S., Nolan, B.C.: Yang’s Gravitational theory. Gen. Relativ. Gravit. 30, 473–495 (1998)

    Article  MathSciNet  Google Scholar 

  24. Hawking, S., Ellis, G.: The Large Scale Structure of Space-Times. Cambridge University Press, Cambridge (1975)

    Google Scholar 

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

    Chapter  Google Scholar 

  26. Mantica, C.A., Molinari, L.G.: Generalized Robertson–Walker spacetimes—a survey. Int. J. Geom. Methods Mod. Phys. 14, 1730001 (27 pages) (2017)

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

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  30. Sharma, R.: Proper conformal symmetries of conformal symmetric space-times. J. Math. Phys. 29(11), 2421–2422 (1988)

    Article  MathSciNet  Google Scholar 

  31. Sharma, R.: Proper conformal symmetries of space-times with divergence-free Weyl conformal tensor. J. Math. Phys. 34(8), 3582–3587 (1993)

    Article  MathSciNet  Google Scholar 

  32. Sharma, R., Ghosh, A.: Perfect fluid space-times whose energy-momentum tensor is conformal Killing. J. Math. Phys. 51(2), 022504 (2010)

    Article  MathSciNet  Google Scholar 

  33. Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C., Herlt, E.: Exact solutions of Einstein’s Field Equations. Cambridge University Press, Cambridge (2003)

    Book  Google Scholar 

  34. Venkatesha, V., De, U.C., Aruna Kumara, H., Naik, D.M.: \(\star \)-Ricci tensor on three dimensional almost coKähler manifolds. Filomat 37 (2023)

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

    Article  MathSciNet  Google Scholar 

  36. Wang, Y.: Ricci solitons on almost coKähler manofolds. Can. Math. Bull. 62, 912–922 (2019)

    Article  Google Scholar 

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

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Acknowledgements

The authors are very much thankful to the reviewers and the editor for their constructive and valuable suggestions towards the improvement of the paper.

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  1. Avijit Sarkar and Uday Chand De contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics, University of Kalyani, Kalyani, West Bengal, 741235, India

    Tarak Mandal & Avijit Sarkar

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

    Uday Chand De

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  1. Tarak Mandal
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  2. Avijit Sarkar
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Correspondence to Tarak Mandal.

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Mandal, T., Sarkar, A. & De, U.C. On nearly vacuum static equations in almost coKähler manifolds with applications to spacetimes. Anal.Math.Phys. 14, 66 (2024). https://doi.org/10.1007/s13324-024-00928-9

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  • DOI: https://doi.org/10.1007/s13324-024-00928-9

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