Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Some Geometric Properties of η∗-Ricci Solitons on α-Lorentzian Sasakian Manifolds

  • Shashikant, Pandey (Department of Mathematics and Astronomy, University of Lucknow) ;
  • Abhishek, Singh (Department of Mathematics and Astronomy, University of Lucknow) ;
  • Rajendra, Prasad (Department of Mathematics and Astronomy, University of Lucknow)
  • Received : 2021.12.20
  • Accepted : 2022.05.03
  • Published : 2022.12.31
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Abstract

We investigate the geometric properties of 𝜂*-Ricci solitons on α-Lorentzian Sasakian (α-LS) manifolds, and show that a Ricci semisymmetric 𝜂*-Ricci soliton on an α-LS manifold is an 𝜂*-Einstein manifold. Further, we study 𝜑*-symmetric 𝜂*-Ricci solitons on such manifolds. We prove that 𝜑*-Ricci symmetric 𝜂*-Ricci solitons on an α-LS manifold are also 𝜂*-Einstein manifolds and provide an example of a 3-dimensional α-LS manifold for the existence of such solitons.

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References

  1. C. S. Baghewadi and G. Ingalahalli, Ricci solitons in Lorentzian α-Sasakian manifolds, Acta Math. Acad. Paedagog. Nyhzi., 28(2012), 59-68.
  2. A. M. Blaga, η-Ricci solitons on Lorentzian para-Sasakian manifolds, Filomat, 30(2)(2016), 489-496. https://doi.org/10.2298/FIL1602489B
  3. A. M. Blaga, On warped product gradient η-Ricci solitons, Filomat, 31(18)(2017), 5791-5801. https://doi.org/10.2298/FIL1718791B
  4. A. M. Blaga and S. Y. Perktas, Remarks on almost η-Ricci solitons in 𝛜-para Sasakian manifolds, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 68(2)(2019), 1621-1628.
  5. C. Calin and M. Crasmareanu, η-Ricci solitons on Hopf hypersurfaces in complex space forms, Rev. Roumaine Math. Pures Appl., 57(1)(2012), 55-63.
  6. C. Calin and M. Crasmareanu, From the Eisenhart problem to Ricci solitons in f -Kenmotsu manifolds, Bull. Malays. Math. Sci. Soc., 33(3)(2010), 361-368.
  7. T. Chave and G. Valent , Quasi-Einstein metrics and their renormalizability properties, Helv. Phys. Acta., 69(1996), 344-347.
  8. T. Chave and G. Valent, On a class of compact and non-compact quasi-Einstein metrics and their renormalizability properties, Nuclear Phys. B., 478(3)(1996), 758-778. https://doi.org/10.1016/0550-3213(96)00341-0
  9. J. T. Cho and M. Kimura, Ricci solitons and real hypersurfaces in a complex space form, Tohoku Math. J., 61(2)(2009), 205-212. https://doi.org/10.2748/tmj/1245849443
  10. B. Chow, P. Lu and L. Ni, Hamilton's Ricci flow, Graduate Studies in Mathemtics, AMS, Providence, RI, USA, 77(2006).
  11. U. C. De, Ricci solitons and gradient Ricci solitons in a P -Sasakian manifold, Aligarh Bull. Math., 29(1)(2010), 29-33.
  12. U. C. De and A. K. Mondal, 3-dimensional quasi-Sasakian manifolds and Ricci solitons, SUT J. Math., 48(1)(2012), 71-81.
  13. U. C. De and A. Sarkar, On ϕ*-Ricci symmetric Sasakian manifolds, Proc. Jangjeon Math. Soc., 11(1)(2008), 47-52.
  14. U. C. De, M. Turan, A. Yildiz and A. De, Ricci solitons and gradient Ricci solitons on 3-dimensional normal almost contact metric manifolds, Publ. Math. Debrecen, 80)(1-2)(2012), 127-142. https://doi.org/10.5486/PMD.2012.4947
  15. S. Deshmukh, H. Alodan and H. Al-Sodais, A Note on Ricci Solitons, Balkan J. Geom. Appl., 16(1)(2011), 48-55.
  16. S. Dey, B. Pal and A. Bhattacharyya, Some classes of Lorentzian α-Sasakian manifolds admitting quarter symmetric metric connection, Acta Univ. Palacki. Olomuc. Fac. Rer. Nat., Mathematica, 55(2)(2016), 41-55.
  17. D. Friedan, Non linear models in 2 + 𝛜 dimensions, Ann. Physics, 163(1985), 318-419. https://doi.org/10.1016/0003-4916(85)90384-7
  18. A. Gray, Einstein-like manifolds which are not Einstein, Geom. Dedicata, 7(1978), 259-280. https://doi.org/10.1007/BF00151525
  19. R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geometry., 17(2)(1982), 255-306. https://doi.org/10.4310/jdg/1214436922
  20. R. S. Hamilton, The Ricci flow on surfaces, Contemp. Math., American Math. Soc.(1988).
  21. A. Haseeb, S. Pandey and R. Prasad, Some results on η-Ricci solitons in quasiSasakian 3-manifolds, Commun. Korean Math. Soc., 36(2)(2021), 377-387. https://doi.org/10.4134/CKMS.C200196
  22. A. Haseeb and R. Prasad, η-Ricci solitons in Lorentzian α-Sasakian manifolds, Facta Univ. Ser. Math. Inform., 35(3)(2020), 713-725.
  23. T. Ivey, Ricci solitons on compact 3-manifolds, Differential Geom. Appl., 3(1993), 301-307. https://doi.org/10.1016/0926-2245(93)90008-O
  24. S. Kishor and A. Singh, η-Ricci solitons on 3-dimensional Kenmotsu manifolds, Bull. Transilv. Univ. Braov Ser. III. Math. Comput. Sci., 13(62)(2020), 209-218.
  25. P. Majhi, U. C. De and D. Kar, η-Ricci solitons on Sasakian 3-Manifolds,An. Univ. Vest Timi. Ser. Mat.-Inform., 55(2)(2017), 143-156.
  26. D. G. Prakasha and B. S. Hadimani, η-Ricci solitons on para-Sasakian manifolds, J. Geom., 108(2017), 383-392. https://doi.org/10.1007/s00022-016-0345-z
  27. R. Prasad, S. Pandey and A. Haseeb, On a Lorentzian Sasakian manifold endowed with a quarter-symmetric metric connection, An. Univ. Vest Timi. Ser. Mat.-Inform., 57(2)(2019), 61-76.
  28. A. Singh and S. Kishor, Some types of η-Ricci solitons on Lorentzian para-Sasakian manifolds, Facta Univ. Ser. Math. Inform., 33(2)(2018), 217-230.
  29. A. Singh and S. Kishor, Curvature properties of η-Ricci solitons on ParaKenmotsu manifolds, Kyungpook Math. J., 59(2019), 149-161. https://doi.org/10.5666/KMJ.2019.59.1.149
  30. T. Takahashi, Sasakian ϕ-symmetric spaces, Tohoku Math. J., 29(1977), 91-113. https://doi.org/10.2748/tmj/1178240699