Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
권호 표지
DOI QR코드

DOI QR Code

CERTAIN SOLITONS ON GENERALIZED (𝜅, 𝜇) CONTACT METRIC MANIFOLDS

  • Received : 2020.09.15
  • Accepted : 2020.11.16
  • Published : 2020.12.30
Download PDF
⟨ Previous Next ⟩

Abstract

The aim of the present paper is to study some solitons on three dimensional generalized (𝜅, 𝜇)-contact metric manifolds. We study gradient Yamabe solitons on three dimensional generalized (𝜅, 𝜇)-contact metric manifolds. It is proved that if the metric of a three dimensional generalized (𝜅, 𝜇)-contact metric manifold is gradient Einstein soliton then ${\mu}={\frac{2{\kappa}}{{\kappa}-2}}$. It is shown that if the metric of a three dimensional generalized (𝜅, 𝜇)-contact metric manifold is closed m-quasi Einstein metric then ${\kappa}={\frac{\lambda}{m+2}}$ and 𝜇 = 0. We also study conformal gradient Ricci solitons on three dimensional generalized (𝜅, 𝜇)-contact metric manifolds.

Keywords

Acknowledgement

The authors are thankful to the referee for pointing out some typographical errors.

References

  1. A. Barros and E. Ribeiro Jr, Integral formulae on quasi-Einstein manifolds and its applications, Glasgow Math. J. 54 (2012), 213-223. https://doi.org/10.1017/S0017089511000565
  2. N., Basu and A. Bhattacharyya, conformal Ricci solition in Kenmotsu manifold, Golb. J. Adv. Res. class. Mod. Geom. 4 (2015), 15-21.
  3. D. E. Blair, Contact Manifolds in Riemannian Geometry, Lecture Notes in Math. 509 (1976), 199-207.
  4. D. E. Blair, T. Koufogiorgos and B. J. Papantoniou, Contact metric manifolds satisying nullity condition, Israel J. Math. 91 (1995), 189-214. https://doi.org/10.1007/BF02761646
  5. D. E. Blair, T. Koufogiorgos and R. Sharma, A classification of 3-dimensional contact tensor of a contact metric manifold with Qϕ=ϕQ, Kodai Math. J. 13 (2007), 391-401. https://doi.org/10.2996/kmj/1138039284
  6. G. Catino, L. Mazzieri, Gradient Einstein soliton, arXiv:1201.6620v5 [math.DG] 29 Nov 2013.
  7. U. C. De and S. Samui, Quasi-conformal curvature tensor on generalized (κ,�)-contact metric manifolds, Acta Univ. Apulensis Math. Inform. 40 (2014), 291-303. https://doi.org/10.17114/j.aua.2014.40.24
  8. A. Ghosh, Certain contact metric as Ricci almost solitons, Results Math. 65 (2014), 81-94. https://doi.org/10.1007/s00025-013-0331-9
  9. F. Gouli-Andreou and P. J. Xenos, A class of contact metric 3-manifolds with ξ ∈ (κ µ) and κ µ are functions, Algebras, Groups and Geom. 17 (2000), 401-407.
  10. R. S. Hamilton, Ricci flow on surfaces, Contemp. Math. 71 (1988), 237-261. https://doi.org/10.1090/conm/071/954419
  11. S. K. Hui, Almost conformal Ricci solitons on f-Kenmotsu manifolds, Khayyam Journal of Mathematics 5 (2019), 89-104.
  12. T. Koufogiorgos and C. Tsichlias, On the existance of new class of contact metric manifolds, Cand. Math. Bull., Vol. 43 (2000), 440-447. https://doi.org/10.4153/CMB-2000-052-6
  13. J. B. Jun, U. C. De and G. Pathak, On Kenmotsu manifolds, J. Korean Math. Soc. 42 (2005), 435-445. https://doi.org/10.4134/JKMS.2005.42.3.435
  14. M. Limoncu, Modification of the Ricci tensor and its applications, Arch. Math. 95 (2010), 191-199. https://doi.org/10.1007/s00013-010-0150-0
  15. P. Majhi, and G. Ghosh, Certain results on generalized (κ, µ)-contact manifolds, Bol. Soc. Parana. Mat. 37 (2019), 131-142.
  16. G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv: 0211159 mathDG, (2002)(Preprint).
  17. S. Pigola, et al., Ricci almost solitons, Ann. Sc. Norm. Sup. Pisa. Cl. Sci. 10 (2011), 757-799.
  18. A. Sarkar and P. Bhakta, Ricci almost soliton on (κ, µ) space forms, Acta Universitatis Apulensis, 57 (2019), 75-85.
  19. A. Sarkar and P. Bhakta, On certain soliton and Ricci tensor of generalized (κ, µ) manifolds, J. Adv. Math. Stud. 12 (2019), 314-323.
  20. A. Sarkar, A. Sil and A. K. Paul, Ricci almost solitons on three diemensional quasi-Sasakian manifolds, Proc. Nat. Acad. Sci, Ind., Sec A. Ph. Sc. 89 (2019), 705-710. https://doi.org/10.1007/s40010-018-0504-8
  21. A. Sarkar and R. Mandal, On N(κ)-para contact 3-manifolds with Ricci solitons, Math. Students. 88 (2019), 137-145.
  22. A. Sarkar and G. G. Biswas, Ricci solitons on three-dimensional generalized Sasakian space forms with quasi-Sasakian metric, Africa Mat. 31 (2020), 455-463. https://doi.org/10.1007/s13370-019-00735-7
  23. A. Sarkar, A. K. Paul and R. Mandal, On α-para Kenmotsu 3-manifolds with Ricci solitons, Balkan. J. Geom. Appl. 23 (2018), 100-112.
  24. A. Sarkar and G. G.Biswas, A Ricci soliton on three-dimensional trans-Sasakian manifolds, Mathematics students. 88 (2019), 153-164.
  25. R. Sharma, Almost Ricci solitons and K-contact geometry, Montash Math. 175 (2014), 621-628. https://doi.org/10.1007/s00605-014-0657-8
  26. T. Taniguchi, Charactrizations of real hypersurfaces of a complex hyperbolic space interms of holomorphic distribution, Tsukuba J. Math. 18 (1994), 469-482. https://doi.org/10.21099/tkbjm/1496162613
  27. S. Tanno, Ricci curvature of contact Riemannian manifolds, Tohoku Math. J. 40 (1988), 441-448. https://doi.org/10.2748/tmj/1178227985
  28. A. Yildiz, U. C. De, and A. Cetinkaya, On some classes of 3-dimensional generalized (κ, µ)-contact metric manifolds, Turkish J. Math. 39 (2015), 356-368. https://doi.org/10.3906/mat-1404-63