Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

3-Dimensional Trans-Sasakian Manifolds with Gradient Generalized Quasi-Yamabe and Quasi-Yamabe Metrics

  • Received : 2021.01.25
  • Accepted : 2021.07.06
  • Published : 2021.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

This paper examines the behavior of a 3-dimensional trans-Sasakian manifold equipped with a gradient generalized quasi-Yamabe soliton. In particular, It is shown that α-Sasakian, β-Kenmotsu and cosymplectic manifolds satisfy the gradient generalized quasi-Yamabe soliton equation. Furthermore, in the particular case when the potential vector field ζ of the quasi-Yamabe soliton is of gradient type ζ = grad(ψ), we derive a Poisson's equation from the quasi-Yamabe soliton equation. Also, we study harmonic aspects of quasi-Yamabe solitons on 3-dimensional trans-Sasakian manifolds sharing a harmonic potential function ψ. Finally, we observe that 3-dimensional compact trans-Sasakian manifold admits the gradient generalized almost quasi-Yamabe soliton with Hodge-de Rham potential ψ. This research ends with few examples of quasi-Yamabe solitons on 3-dimensional trans-Sasakian manifolds.

Keywords

Acknowledgement

The authors express their sincere thanks to the anonymous referees for the valuable suggestions and comments for the improvement of the paper.

References

  1. A. M. Blaga, A note on warped product almost quasi-Yamabe solitons, Filomat, 33(2019), 2009-2016. https://doi.org/10.2298/fil1907009b
  2. A. L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 10, Springer-Verlag, Berlin, 1987.
  3. D. E. Blair and J. A. Oubina, Conformal and related changes of metric on the product of two almost contact metric manifolds, Publ. Mat., 34(1990), 199-207. https://doi.org/10.5565/PUBLMAT_34190_15
  4. B. Y. Chen and S. Desahmukh, Yamabe and quasi-Yamabe soliton on euclidean submanifolds, Mediterranean Journal of Mathematics, August 2018, DOI: 10.1007/s00009-018-1237-2.
  5. S. Desahmukh and B. Y. Chen, A note on Yamabe solitons, Balk. J. Geom. Appl., 23(1)(2018), 37-43.
  6. D. Chinea and C. Gonzales, A classification of almost contact metric manifolds, Ann. Mat. Pura Appl., 156(1990) 15-30. https://doi.org/10.1007/BF01766972
  7. U. C. De, and A. Sarkar, On three-dimensional Trans-Sasakian Manifolds, Extracta Math., 23(2008) 265-277.
  8. C. Dey and U. C. De, A note on quasi-Yambe soliton on contact metric manifolds, J. of Geom., 111(11)(2020).
  9. A. Gray and L. M. Harvella, The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl., 123(1980), 35-58. https://doi.org/10.1007/BF01796539
  10. G. Huang and H. Li, On a classification of the quasi Yamabe gradient solitons, Methods Appl. Anal., 21(3)(2014) 379-389. https://doi.org/10.4310/MAA.2014.v21.n3.a7
  11. C. Aquino, A. Barros and E. jr. Riberio, Some applications of Hodge-de Rham decomposition to Ricci solitons, Results. Math., 60(2011), 235-246. https://doi.org/10.1007/s00025-011-0152-7
  12. R. S. Hamilton, The Ricci flow on surfaces, Mathematics and general relativity, Contemp. Math. Amer. Math. Soc., 71(1988), 237-262. https://doi.org/10.1090/conm/071/954419
  13. B. Leandro and H. Pina, Generalized quasi Yamabe gradient solitons, Differential Geom. Appl., 49(2016), 167-175. https://doi.org/10.1016/j.difgeo.2016.07.008
  14. K. Kenmotsu, A class of almost contact Riemannian manifolds, Tohoku Math. J., 24(1972), 93-103. https://doi.org/10.2748/tmj/1178241594
  15. J. C. Marrero, The local structure of Trans-Sasakian manifolds, Annali di Mat. Pura ed Appl., 162(1992), 77-86. https://doi.org/10.1007/BF01760000
  16. J. A. Oubina, New classes of almost contact metric structures, Publ. Math. Debrecen, 32(1985), 187-193. https://doi.org/10.5486/PMD.1985.32.3-4.07
  17. S. Pigola, M. Rigoli, M. Rimoldi and A. Setti, Ricci almost solitons, Ann. Sc. Norm. Super. Pisa Cl. Sci., 10(5)(2011) 757-799.
  18. A. A. Shaikh, M. H. Shahid and S. K. Hui, On weakly conformally symmetric manifolds. Matematicki Vesnik, 60(2008), 269-284.
  19. M. D. Siddiqi, Generalized Yamabe solitons on Trans-Sasakian manifolds, Matematika Instituti Byulleteni Bulletin of Institute of Mathematics, 3(2020), 77-85.
  20. J. B. Jun and M. D. Siddiqi, Almost Quasi-Yamabe Solitons on Lorentzian concircular structure manifolds- [(LCS)n], Honam Mathematical Journal, 42(3)(2020), 521-536. https://doi.org/10.5831/hmj.2020.42.3.521
  21. M. D. Siddiqi, Generalized Ricci Solitons on Trans-Sasakian Manifolds, Khayyam Journal of Math., 4(2)(2018), 178-186.
  22. M. D. Siddiqi, η-Ricci soliton in 3-diamensional normal almost contact metric manifolds, Bull. Transilvania Univ. Brasov, Series III: Math, Informatics, Physics, 11(60)(2018) 215-234.
  23. M. D. Siddiqi, η-Ricci soliton in (ε, δ)-trans-Sasakian manifolds, Facta. Univ. (Nis), Math. Inform., 34(1)(2019), 4556.
  24. S. T. Yau, Harmonic functions on complete Riemannian manifolds, Commu. Pure. Appl. Math., 28(1975), 201-228. https://doi.org/10.1002/cpa.3160280203
  25. M. El A. Mekki and A. M. Cherif, Generalised Ricci solitons on Sasakian manifolds, Kyungpook Math. J., 57(2017), 677-682. https://doi.org/10.5666/KMJ.2017.57.4.677