Lorentzian Para-Sasakian Manifolds and *-Ricci Solitons

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DOI: 10.46793/KgJMat2402.167H


We study the properties of Lorentzian para-Sasakian manifolds endowed with -Ricci solitons and gradient -Ricci solitons. Finally, the existence of -Ricci soliton on a 4-dimensional Lorentzian para-Sasakian manifold is proved by constructing a non-trivial example.


Lorentzian para-Sasakian manifolds, -Ricci solitons, gradient -Ricci solitons, generalized η-Einstein manifolds.


[1]   A. A. Shaikh and K. K. Baishya, Some results on LP-Sasakian manifolds, Bull. Math. Soc. Sci. Math. Roumanie 49(97)(2) (2006), 193–205.

[2]   A. A. Aqeel, U. C. De and G. C. Ghosh, On Lorentzian para-Sasakian manifolds, Kuwait J. Sci. Eng. 31(2) (2004), 1–13.

[3]   A. Yildiz, U. C. De and E. Ata, On a type of Lorentzian para-Sasakian manifolds, Math. Reports 16(66)(1) (2014), 61–67.

[4]   B. Y. Chen, Some results on concircular vector fields and their applications to Ricci solitons, Bull. Korean Math. Soc. 52(5) (2015), 1535–1547. https://doi.org/10.4134/BKMS.2015.52.5.1535

[5]   D. E. Blair, Contact Manifolds in Riemannian Geometry, Lecture Notes in Math. 509, Springer-Verlag, Berlin, 1976.

[6]   D. E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds, Second Edition, Progress in Mathematics 203, Birkhauser Boston, Inc., Boston, MA, 2010.

[7]   U. C. De and A. Sardar, Some results on LP-Sasakian manifolds, Bulletin of the Transilvania University of Brasov-Series III: Mathematics, Informatics, Physics, 13(62)(1) (2020), 89–100. https://doi.org/10.31926/but.mif.2020.

[8]   U. C. De, J. B. Jun and A. A. Shaikh, On conformally flat LP-Sasakian manifolds with a coefficient α, Nihonkai Math. J. 13(2) (2002), 121–131.

[9]   U. C. De, K. Matsumoto and A. A. Shaikh, On Lorentzian para-Sasakian manifolds, Rendiconti del Seminario Matematico di Messina 3 (1999), 149–158.

[10]   A. Ghosh and D. S. Patra, -Ricci soliton within the frame-work of Sasakian and (κ,μ)-contact manifold, Int. J. Geom. Methods in Mod. Phys. 15(7) (2018), 21 pages. https://doi.org/10.1142/S0219887818501207

[11]   T. Hamada, Real hypersurfaces of complex space forms in terms of Ricci -tensor, Tokyo J. Math. 25(2) (2002), 473–483. https://doi.org/10.3836/tjm/1244208866

[12]   R. S. Hamilton, The Ricci flow on surfaces, Contemp. Math. 71 (1988), 237–261. http://dx.doi.org/10.1090/conm/071

[13]   A. Haseeb and R. Prasad, η-Ricci solitons on ????-LP-Sasakian manifolds with a quarter symmetric metric connection, Honam Math. J. 41(3) (2019), 539–558. https://doi.org/10.5831/HMJ.2019.41.3.539

[14]   A. Haseeb and R. Prasad, On a Lorentzian para-Sasakian manifold with respect to the quarter-symmetric metric connection, Novi Sad J. Math. 46(2) (2016), 103–116. https://doi.org/10.30755/NSJOM.04279

[15]   A. Haseeb and R. Prasad, η-Ricci solitons in Lorentzian α-Sasakian manifolds, Facta Univ. Ser. Math. Inform. 35(3) (2020), 713–725.

[16]   G. Kaimakamis and K. Panagiotidou, -Ricci solitons of real hypersurfaces in non-flat complex space forms, J. Geom. Phys. 86 (2014), 408–413. https://doi.org/10.1016/j.geomphys.2014.09.004

[17]   P. Majhi, U. C. De and Y. J. Suh, -Ricci solitons on Sasakian 3-manifolds, Publ. Math. Debrecen 93(1–2) (2018), 241–252. https://doi.org/10.5486/PMD.2018.8245

[18]   K. Matsumoto, On Lorentzian paracontact manifolds, Bulletin of Yamagata University, Natural Science 12(2) (1989), 151–156.

[19]   I. Mihai and R. Rosca, On Lorentzian P-Sasakian Manifolds, Classical Analysis, World Scientific, Singapore, 1992, 155–169.

[20]   B. O. Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York, 1983.

[21]   D. G. Prakasha and B. S. Hadimani, η-Ricci solitons on para-Sasakian manifolds, J. Geom. 108(2) (2017), 383–392. https://doi.org/10.1007/s00022-016-0345-z

[22]   D. G. Prakasha and P. Veeresha, Para-Sasakian manifolds and -Ricci solitons, Afr. Mat. 30(7–8) (2018), 989–998. https://doi.org/10.1007/s13370-019-00698-9

[23]   S. Tachibana, On almost-analytic vectors in almost-Kahlerian manifolds, Tohoku Math. J. (2) 12(2) (1959), 247–265. https://doi.org/10.2748/tmj/1178244584

[24]   M. Turan, C. Yetim and S. K. Chaubey, On quasi-Sasakian 3-manifolds admitting η-Ricci solitons, Filomat 33(15) (2019), 4923–4930.

[25]   Venkatesha, D. M. Naik and H. A. Kumara, -Ricci solitons and gradient almost -Ricci solitons on Kenmotsu manifolds, Math. Slovaca 69(6) (2019), 1447–1458. https://doi.org/10.1515/ms-2017-0321

[26]   S. K. Yadav, S. K. Chaubey and S. K. Hui, On the perfect fluid Lorentzian para-Sasakian Spacetimes, Bulgarian Journal of Physics 46 (2019), 1–15.

[27]   K. Yano, Integral Formulas in Riemannian Geometry, Marcel Dekker, New York (1970).

[28]   M. C. Chaki, On pseudo Ricci symmetric manifolds, Bulgarian Journal of Physics 15(6) (1988), 526–531.