New Results on the Perfect Fluid Lorentzian Para-Sasakian Spacetimes

Year 2025, Volume: 13 Issue: 1, 1 - 13, 30.04.2025

Sunıl Yadav , Ahmet Yıldız

Abstract

In this article, we investigate $\mathcal{LP}$-Sasakian spacetimes attached with perfect fluid whose metrics are (CERY)$_{4}$-soliton admitting $% \mathcal{Z}$-tensor. Also we discuss the application of such soliton to cosmology and general relativity. Besides this, we deduce a modified Poisson equation and modified Liouville equation from the (CERY)$_{4}$-soliton on $% \mathcal{LP}$-Sasakian spacetimes . In addition, we light up the harmonic aspect of such soliton on perfect fluid $\mathcal{LP}$-Sasakian spacetimes. Moreover, we conclude a necessary and sufficient condition for a $1$-form $% \eta ^{\sharp }$, which is the $g^{\star }$-dual of the vector field $\xi $ on such a spacetime to be a solution of the Schr\"{o}dinger-Ricci equation. In conclusion, we present an instance of a $4$-dimensional $\mathcal{LP}$% -Sasakian spacetime with the (CERY)$_{4}$-soliton equipped with $\mathcal{Z}$% -tensor.

Keywords

LP-Sasakian manifold, Z-tensor, conformal eta-Ricci Yamabe soliton, Schrodinger-Ricci equation, Modified Possion equation

Ethical Statement

There are no any Ethical statement

Supporting Institution

NA

Project Number

NA

Thanks

NA

References

• [1] A. H. Alkhaldi, M. D. Siddiqi, M. A. Khan, and L. S. Alqahtani Imperfect Fluid Generalized Robertson Walker Spacetime Admitting Ricci-Yamabe Metric, Advances in Mathematical Physics Volume 2021, Article ID 2485804, 10 pages
• [2] A. Barman, and Inan Unal, Geometry of Kenmotsu manifolds admitting Z -tensor, Bulletin of the Transilvania University of Brasov. Series III: Mathematics and Computer Science (2022), 23-40.
• [3] D. E. Blair, Riemannian geometry of contact and symmetric manifolds, Birhauser, Bston. 2010.
• [4] A. M. Blaga, Harmonic aspects in an h-Ricci soliton, Int. Electron. J. Geom., 13 (2020), 41-49.
• [5] G. Catino, L. Mazzieri, Gradient Einstein solitons, Nonlinear Anal., 132 (2016), 66-94.
• [6] M. C. Chaki, S. Roy, Spacetime with covariant constant energy momentum tensor, Int. J. Theor. Physics, 35 (1996), 1027-1032.
• [7] B. Chow, S. C. Chu and D. Glickenstein et al., The Ricci flow:techniques and applications,in Part I: Geometric Aspects 135, AMS, 2007.
• [8] U. C. De, A. A. Shaikh, and A. Sengupta, On LP-Sasakian manifolds with a coefficient a., Kyungpook Math. J., 42 (2002), 177-186.
• [9] K. De, U. C. De, A. A. Syied, N. B. Turki and S. Alsaeed, Perfect fluid spacetimes and Gradient Solitons, J. Nonlin. Math. Phys., 29, 4 (2022), 843-858.
• [10] A. E. Fischer, An introduction to conformal Ricci flow, Classical and Quantum Gravity., 21, (2004) 171-218.
• [11] S. Güler, M. Crasmareanu, Ricci-Yamabe maps for Riemannian flows and their volume variation and volume entropy, Turk. J. Math., 43 (2019), 261-2641.
• [12] R. S. Hamilton, Three Manifold with positive Ricci curvature, J. Differ. Geom. 17 (1982), 255-306.
• [13] R. S. Hamilton, The Ricci flow on surfaces, Contemp. Math., 71 (1988), 237-261.
• [14] Y. Li, A. Haseeb and M. Ali, LP-Kenmotsu Manifolds Admitting jeta-Ricci Solitons and Spacetime, Journal of Mathematics, (2022), Article ID 6605127, 10 pages.
• [15] V. R. Kaigorodov, The curvature structure of spacetime, Prob. Geom. 14 (1983), 177-202.
• [16] H. Kachar, Infinitesimal characterization of Friedmann Universes,Arch. Math. Basel., 38 (1982), 58-64.
• [17] C. A. Mantica, U. C. De, Y. J. Suh, and L.G. Molinari,Perfect fluid spacetimes with harmonic generalized curvature tensor, Osaka J. Math. 56, (2019), 173-182.
• [18] C. A. Mantica L. G. and Molinari, Weakly Z symmetric manifolds,Acta Math. Hungar., 135 (2012), 80-96.
• [19] C. A. Mantica and Y. J. Suh, Pseudo Z symmetric Riemannian manifolds with harmonic curvature tensors, Int. J. Geom. Methods Mod. Phys., 9 1 (2012), 1250004.
• [20] C. A. Mantica and Y. J. Suh,, Pseudo Z symmetric spacetimes, J. Math. Phys, 55 (2014), 042502.
• [21] C. A. Mantica and Y. J. Suh, Recurrent Z forms on Riemannian and Kaehler manifolds, Int. J. Geom. Methods Mod. Phys., 9 (2012), 1250059.
• [22] K. Matsumoto, On Lorentzian paracontact manifolds, Bull. of Yamagata Univ. Nat. Sci., 12 (1989), 151-156.
• [23] B. O’Neill, Semi-Riemannian geometry with application to relativity, Pure and applied mathematics, Academic press, New York. 1983
• [24] A. G. Popov, Exact formula for constructing solutions of the Liouville equation 42u=ev from solutions of the Laplace equation 42v=0. (Russian).
• Dokl. Akad. Nauk. 1993, 333, 440-441. Translation in Russian Acad. Sci. Dokl. Math. 1994, 2, 55-71.
• [25] A. Z. Petro, Einstein space, Pergamon Press, Oxford, (1967).
• [26] A. K. Raychaudhary, S. Banerji and A. Banerjee , General relatively, Astrophysics and cosmology, Springer-Verlag. 1942.
• [27] R. N. Singh, S. K. Pandey and G. Pandey, Second Order Parallel Tensors on LP-Sasakian Manifolds, Journal of International Academy of Physical Sciences, 13, (2009), 383-388.
• [28] S. Sarkara, S. Dey and X. Chen, Certain Results of Conformal and -Conformal Ricci Soliton on Para-Cosymplectic and Para-Kenmotsu Manifolds, Filomat, 35, 15 (2021), 5001-5015.
• [29] Y. J. Suh and S. K. Chaubey, Ricci solitons on general relativistic spacetime, Physica Scripta, 98 6,(2023).
• [30] M. D. Siddiqi, F. Mofarreh, M. A. Akyol and A. H. Hakami, h-Ricci-Yamabe Solitons along Riemannian Submersions, Axioms, 12 (2023), 796.
• [31] A. Y. Tasci and F. O. Zengin, On Z -symmetric manifold with conharmonic curvature tensor in special conditions, Kragujevac Journal of Mathematics, 49, 1 (2025), 65-80.
• [32] K. Yano, M. Kon, Structures on manifolds, Series in Pure Math., World Scientific. 1984.
• [33] K. Yano, Integral formulas in Riemannian geometry, Pure and Applied Mathematical, Marcel Dekkr, Inc., New York. 1970.
• [34] S. K. Yadav, A note on space time in f (R)-gravity, Annals of communications in Mathematics, 6, 2 (2023), 99-108.
• [35] S. K. Yadav, D. L.Suthar and S. K. Chaubey, Almost conformal Ricci soliton on generalized Robertson walker-space time, Research in Mathematics, 10, 1 (2023), 1-10.
• [36] S. K. Yadav, P. K. Dwivedi and D. L. Suthar, On (LCS)n-manifolds satisfying certain conditions on the concircular curvature tensor,Thiland Journal of Mathematics, l, 9 (2011), 597-603.
• [37] S. K. Yadav, A. Haseeb and A. Yildiz, Conformal h-Ricci-Yamabe solitons on submanifolds of an (LCS)n-manifold admitting a quarter-symmetric metric connection, Commun. Fac. Sci. Univ. Ank.Ser. A1 Math. Stat, 73, 3, (2024),1-19
• [38] S. K. Yadav and D. L. Suthar, Kaehlerian Norden spacetime admitting conformal h-Ricci-Yamabe Metric, Int. J. of Geometric Methods in Modern Physics, https://doi.org/10.1142/S0219887824502347.
• [39] S. K. Yadav and X. Chen, Z -tensor in mixed generalized quasi-Einstein GRW space-time, Sohag Journal of mathematics, 11, 1, (2024), 1-9.
• [40] S. K. Yadav, Sumesh Senway, N. B. Turki and Rajendra Prasad, The Z -Tensor on Almost Co-K¨ahlerian Manifolds Admitting Riemann Soliton Structure, Advance in Mathematical Physics, Volume 2024, Article ID 7445240, 14 pages, https://doi.org/10.1155/2024/7445240
• [41] P. Zhang, Y. Li,S. Roy, S. Dey and A. Bhattacharyya , Geometrical Structure in a Perfect Fluid Spacetime with Conformal Ricci-Yamabe Soliton, Symmetry, 14, 3 (2022), 594.