Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

SOME RESULTS ON 𝜂-RICCI SOLITONS IN QUASI-SASAKIAN 3-MANIFOLDS

  • Haseeb, Abdul (Department of Mathematics Faculty of Science Jazan University) ;
  • Pandey, Shashikant (Department of Mathematics and Adtronomy University of Lucknow) ;
  • Prasad, Rajendra (Department of Mathematics and Adtronomy University of Lucknow)
  • Received : 2020.06.08
  • Accepted : 2020.11.23
  • Published : 2021.04.30
Download PDF
⟨ Previous Next ⟩

Abstract

In the present paper, we characterize quasi-Sasakian 3-manifolds admitting 𝜂-Ricci solitons. Finally, the existence of 𝜂-Ricci soliton in a quasi-Sasakian 3-manifold has been proved by a concrete example.

Keywords

Acknowledgement

The authors are thankful to the editor and anonymous referees for their valuable suggestions in the improvement of the paper.

References

  1. A. M. Blaga, η-Ricci solitons on Lorentzian para-Sasakian manifolds, Filomat 30 (2016), no. 2, 489-496. https://doi.org/10.2298/FIL1602489B
  2. D. E. Blair, The theory of quasi-Sasakian structures, J. Differential Geometry 1 (1967), 331-345. http://projecteuclid.org/euclid.jdg/1214428097
  3. D. E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Mathematics, Vol. 509, Springer-Verlag, Berlin, 1976.
  4. D. E. Blair, Riemannian geometry of contact and symplectic manifolds, Progress in Mathematics, 203, Birkhauser Boston, Inc., Boston, MA, 2002. https://doi.org/10.1007/978-1-4757-3604-5
  5. C. Calin and M. Crasmareanu, From the Eisenhart problem to Ricci solitons in f-Kenmotsu manifolds, Bull. Malays. Math. Sci. Soc. (2) 33 (2010), no. 3, 361-368.
  6. J. T. Cho and M. Kimura, Ricci solitons and real hypersurfaces in a complex space form, Tohoku Math. J. (2) 61 (2009), no. 2, 205-212. https://doi.org/10.2748/tmj/1245849443
  7. U. C. De and P. Majhi, ϕ-semisymmetric generalized Sasakian space-forms, Arab J. Math. Sci. 21 (2015), no. 2, 170-178. https://doi.org/10.1016/j.ajmsc.2015.01.002
  8. U. C. De and A. K. Mondal, 3-dimensional quasi-Sasakian manifolds and Ricci solitons, SUT J. Math. 48 (2012), no. 1, 71-81.
  9. U. C. De and A. Sarkar, On three-dimensional quasi-Sasakian manifolds, SUT J. Math. 45 (2009), no. 1, 59-71.
  10. U. C. De, A. Yildiz, M. Turan, and B. E. Acet, 3-dimensional quasi-Sasakian manifolds with semi-symmetric non-metric connection, Hacet. J. Math. Stat. 41 (2012), no. 1, 127-137.
  11. U. K. Gautam, A. Haseeb, and R. Prasad, Some results on projective curvature tensor in Sasakian manifolds, Commun. Korean Math. Soc. 34 (2019), no. 3, 881-896. https://doi.org/10.4134/CKMS.c180235
  12. S. Ghosh, η-Ricci solitons on quasi-Sasakian manifolds, An. Univ. Vest Timi,s. Ser. Mat.-Inform. 56 (2018), no. 1, 73-85. https://doi.org/10.2478/awutm-2018-0006
  13. J. C. Gonzalez and D. Chinea, Quasi-Sasakian homogeneous structures on the generalized Heisenberg group H(p, 1), Proc. Amer. Math. Soc. 105 (1989), no. 1, 173-184. https://doi.org/10.2307/2046753
  14. A. Haseeb and R. Prasad, η-Ricci solitons on ⲉ-LP-Sasakian manifolds with a quartersymmetric metric connection, Honam Math. J. 41 (2019), no. 3, 539-558. https://doi.org/10.5831/hmj.2019.41.3.539
  15. A. Haseeb and R. Prasad, η-Ricci solitons in Lorentzian α-Sasakian manifolds, Facta Universitatis (NIS), Ser. Math. Inform. 35 (2020), 713-725.
  16. S. Kanemaki, Quasi-Sasakian manifolds, Tohoku Math. J. 29 (1977), no. 2, 227-233. https://doi.org/10.2748/tmj/1178240654
  17. S. Kanemaki, On quasi-Sasakian manifolds, in Differential geometry (Warsaw, 1979), 95-125, Banach Center Publ., 12, PWN, Warsaw, 1984.
  18. P. Majhi, U. C. De, and D. Kar, η-Ricci solitons on Sasakian 3-manifolds, An. Univ. Vest Timis. Ser. Mat.-Inform. 55 (2017), no. 2, 143-156. https://doi.org/10.1515/awutm2017-0019
  19. Z. Olszak, Normal almost contact metric manifolds of dimension three, Ann. Polon. Math. 47 (1986), no. 1, 41-50. https://doi.org/10.4064/ap-47-1-41-50
  20. Z. Olszak, On three-dimensional conformally flat quasi-Sasakian manifolds, Period. Math. Hungar. 33 (1996), no. 2, 105-113. https://doi.org/10.1007/BF02093508
  21. D. G. Prakasha and B. S. Hadimani, η-Ricci solitons on para-Sasakian manifolds, J. Geom. 108 (2017), no. 2, 383-392. https://doi.org/10.1007/s00022-016-0345-z
  22. M. Turan, C. Yetim, and S. K. Chaubey, On quasi-Sasakian 3-manifolds admitting η-Ricci solitons, Filomat 33 (2019), no. 15, 4923-4930. https://doi.org/10.2298/fil1915923t
  23. A. G. Walker, On Ruse's spaces of recurrent curvature, Proc. London Math. Soc. (2) 52 (1950), 36-64. https://doi.org/10.1112/plms/s2-52.1.36
  24. K. Yano and M. Kon, Structures on Manifolds, Series in Pure Mathematics, 3, World Scientific Publishing Co., Singapore, 1984.