Ricci Almost Solitons on Three-Dimensional Quasi-Sasakian Manifolds

  • Avijit SarkarEmail author
  • Amit Sil
  • Avijit Kumar Paul
Research Article


In this paper it is shown that a three-dimensional non-cosymplectic quasi-Sasakian manifold admitting Ricci almost soliton is locally \(\phi \)-symmetric. It is proved that a Ricci almost soliton on a three-dimensional quasi-Sasakian manifold reduces to a Ricci soliton. It is also proved that if a three-dimensional non-cosymplectic quasi-Sasakian manifold admits gradient Ricci soliton, then the potential function is invariant in the orthogonal distribution of the Reeb vector field \(\xi\). We also improve some previous results regarding gradient Ricci soliton on three-dimensional quasi-Sasakian manifolds. An illustrative example is given to support the obtained results.


Ricci flow Ricci solitons Locally \(\phi \) symmetric Gradient Ricci solitons Quasi-Sasakian manifolds 

Mathematics Subject Classification

53 C 15 53 D 15 


  1. 1.
    Acharya BS, Figurea F, Hull CM, Spence BJ (1999) Branes at canonical singularities holography. Adv Theor Math Phys 2:1249–1286CrossRefGoogle Scholar
  2. 2.
    Agricola I, Friedrich T (2003) Killing spinors in super gravity with 4-fluxes. Class Quantum Gravity 20:4707–4717ADSCrossRefzbMATHGoogle Scholar
  3. 3.
    Agricola I, Friedrich T, Nagy PA, Phule C (2005) On the Ricci tensor in the common sector of type II, string theory. Class Quantum Gravity 22:2569–2577ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Blair DE (2002) Riemannian geometry of contact and symplectic manifolds, vol 203. Birkhauser, BaselCrossRefzbMATHGoogle Scholar
  5. 5.
    Blair DE (1967) The theory of quasi-Sasakian structures. J Differ Geom 1:331–345MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chow B, Knopf D (2004) The Ricci flow, an introduction, mathematical surveys and monograph, vol 110. AMS, ProvidenceCrossRefzbMATHGoogle Scholar
  7. 7.
    De UC, Sarkar A (2008) On three dimensional locally \(\phi \)-recurrent quasi-Sasakian manifolds. Demonstr Math 41:677–684MathSciNetzbMATHGoogle Scholar
  8. 8.
    De UC, Sarkar A (2009) On three dimensional quasi-Sasakian manifolds. SUT J Math 45:59–71MathSciNetzbMATHGoogle Scholar
  9. 9.
    De UC, Mondal AK (2011) 3-dimensional quasi-Sasakian manifolds and Ricci solitons. SUT J Math 48:71–81MathSciNetzbMATHGoogle Scholar
  10. 10.
    Fridan D (1985) Non linear models in \(2+\varepsilon \) dimensions. Ann Phys 163:318–419ADSCrossRefGoogle Scholar
  11. 11.
    Friedrich T, Ivanov S (2003) Almost contact manifolds, connections with torsion and parallel spinor. J Rein Angew Math 559:217–236MathSciNetzbMATHGoogle Scholar
  12. 12.
    Ghosh A (2011) Kenmotsu 3-metric as a Ricci soliton. Chaos Solitons Fractals 44:647–650ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ghosh A (2014) Certain contact metric as Ricci almost solitons. Results Math 65:81–94MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gonzalez JC, Chinea D (1989) Quasi-Sasakian homogeneous structures on the generalized Heisenberg group H (p, 1). Proc Am Math Soc 105:173–184MathSciNetzbMATHGoogle Scholar
  15. 15.
    Hamilton RS (1988) The Ricci flow on surfaces, mathematics and general relativity, vol 71. AMS, ProvidenceGoogle Scholar
  16. 16.
    Kim BH (1990) Fibered Riemannian spaces with quasi-Sasakian structure. Hiroshima Math J 20:477–513MathSciNetzbMATHGoogle Scholar
  17. 17.
    Olszak Z (1982) Curvature properties of quasi-Sasakian manifolds. Tensor N S 38:19–28MathSciNetzbMATHGoogle Scholar
  18. 18.
    Olszak Z (1986) Normal almost contact metric manifolds of dimension three. Ann Pol Math 47:41–50MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Olszak Z (1996) On three-dimensional conformally flat quasi-Sasakian manifolds. Period Math Hung 33:105–113MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Perelman G (2002) The entropy formula for the Ricci flow and its geometric applications. arxiv:MathDG/0211159 (Preprint)
  21. 21.
    Pigola S, Rigoli M, Rimoldi M, Setti M (2011) Ricci almost solitons. Ann Sc Norm Sup Pisa cl Sci 10:757–799MathSciNetzbMATHGoogle Scholar
  22. 22.
    Sharma R (2008) Certain results on K-contact and \((\kappa,\mu )\)-contact manifolds. J Geom 89:138–147MathSciNetCrossRefGoogle Scholar
  23. 23.
    Sharma R (2014) Almost Ricci solitons and K-contact geometry. Montash Math 175:621–628MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Takahashi T (1977) Sasakian \(\phi \)-symmetric spaces. Tohoku Math J 29:91–113MathSciNetCrossRefGoogle Scholar
  25. 25.
    Tanno S (1971) Quasi-Sasakian structure of rank 2p + 1. J Differ Geom 5:317–324MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Wang Y (2016) Gradient Ricci almost solitons on two classes of almost Kenmotsu manifolds. J Korean Math Soc 53:1101–1114MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Yano K (1970) Integral formulas on Riemannian geometry. Marcel Dekker, New YorkzbMATHGoogle Scholar

Copyright information

© The National Academy of Sciences, India 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KalyaniKalyani, NadiaIndia

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