Yamabe solitons on three-dimensional normal almost paracontact metric manifolds

  • Irem Küpeli ErkenEmail author


The purpose of the paper is to study Yamabe solitons on three-dimensional para-Sasakian, paracosymplectic and para-Kenmotsu manifolds. Mainly, we prove that the following:
  • If the semi-Riemannian metric of a three-dimensional para-Sasakian manifold is a Yamabe soliton, then it is of constant scalar curvature, and the flow vector field V is Killing. In the next step, we prove that either the manifold has constant curvature \(-\,1\) or V is an infinitesimal automorphism of the paracontact metric structure on the manifold.

  • If the semi-Riemannian metric of a three-dimensional paracosymplectic manifold is a Yamabe soliton, then it has constant scalar curvature. Furthermore either the manifold is \(\eta \)-Einstein, or Ricci flat.

  • If the semi-Riemannian metric on a three-dimensional para-Kenmotsu manifold is a Yamabe soliton, then the manifold is of constant sectional curvature \(-\,1\). Furthermore, Yamabe soliton is expanding with \(\lambda =-6\).

Finally, we construct examples to illustrate the results obtained in previous sections.


Para-Sasakian manifold Paracosymplectic manifold Para-Kenmotsu manifold Yamabe soliton Ricci soliton Infinitesimal automorphism Constant scalar curvature 

Mathematics Subject Classification

53C25 53C21 53C44 53D15 



  1. 1.
    G. Calvaruso, D. Perrone, Geometry of H-paracontact metric manifolds. Publ. Math. Debrecen 86(3–4), 325–346 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    G. Calvaruso, A. Perrone, Ricci solitons in three-dimensional paracontact geometry. J. Geom. Phys. 98, 1–12 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    G. Calvaruso, A. Zaeim, A complete classification of Ricci and Yamabe solitons of non-reductive homogeneous 4-spaces. J. Geom. Phys. 80, 15–25 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    B. Cappelletti-Montano, I.K. Erken, C. Murathan, Nullity conditions in paracontact geometry. Differ. Geom. Appl. 30, 665–693 (2012)MathSciNetCrossRefGoogle Scholar
  5. 5.
    B. Chow, P. Lu, L. Ni, Hamilton’s Ricci flow, Graduate Studies in Mathematics, vol. 77 (American Mathematical Society, Providence, RI, 2006)Google Scholar
  6. 6.
    P. Dacko, On almost para-cosymplectic manifolds. Tsukuba J. Math. 28, 193–213 (2004)MathSciNetCrossRefGoogle Scholar
  7. 7.
    I.K. Erken, C. Murathan, A complete study of three-dimensional paracontact (\(\kappa,\mu,\nu \))-spaces. Int. J. Geom. Methods Mod. Phys. (2017). CrossRefGoogle Scholar
  8. 8.
    I.K. Erken, P. Dacko, C. Murathan, Almost \(\alpha \) paracosymplectic manifolds. J. Geom. Phys. 88, 30–51 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    R.S. Hamilton, in Mathematics and General Relativity, Contemporary Mathematics, Santa Cruz, CA, 1986, vol. 71 (American Mathematical Society, Providence, RI, 1988), pp. 237–262Google Scholar
  10. 10.
    R.S. Hamilton, Lectures on Geometric Flows (Unpublished manuscript, 1989)Google Scholar
  11. 11.
    S. Kaneyuki, F.L. Williams, Almost paracontact and parahodge structures on manifolds. Nagoya Math. J. 99, 173–187 (1985)MathSciNetCrossRefGoogle Scholar
  12. 12.
    B. O’neill, Semi-Riemannian Geometry (Academic Press, New York, 1983)zbMATHGoogle Scholar
  13. 13.
    R. Pina, K. Tenenblat, On solitons of the Ricci curvature and the Einstein equation. Isr. J. Math. 171, 61–76 (2009)CrossRefGoogle Scholar
  14. 14.
    R. Sharma, A 3-dimensional Sasakian metric as a Yamabe soliton. Int. J. Geom. Methods Mod. Phys. 9, 1220003 (2012)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Y. Wang, Yamabe solitons on three-dimensional Kenmotsu manifolds. Bull. Belg. Math. Soc. Simon Stevin 23, 345–355 (2016)MathSciNetzbMATHGoogle Scholar
  16. 16.
    J. Wełyczko, On Basic Curvature Identities for Almost (para) Contact Metric Manifolds. arXiv:1209.4731v1 [math.DG]
  17. 17.
    K. Yano, Integral Formulas in Riemannian Geometry (Marcel Dekker, New York, 1970)zbMATHGoogle Scholar
  18. 18.
    S. Zamkovoy, Canonical connections on paracontact manifolds. Ann. Glob. Anal. Geom. 36, 37–60 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Engineering and Natural SciencesBursa Technical UniversityBursaTurkey

Personalised recommendations