m-quasi-Einstein metric and contact geometry

  • Amalendu GhoshEmail author
Original Paper


We study m-quasi-Einstein metric in the framework of contact metric manifolds. The existence of such metric has been confirmed on the class of \( \eta \)-Einstein K-contact manifold, in which the potential vector field V is a constant multiple of the Reeb vector field \(\xi \). Next, we consider closed m-quasi-Einstein metric on a complete K-contact manifold and prove that it is Sasakian and Einstein provided \(m\ne 1\). We also proved that, if a K-contact manifold M admits an m-quasi-Einstein metric such that the potential vector field V is conformal, then V becomes Killing and M is \( \eta \)-Einstein. Finally, we obtain a couple of results on a contact metric manifold M admitting an m-quasi-Einstein metric, whose potential vector field is a point wise collinear with the Reeb vector field.


m-quasi-Einstein metric Ricci soliton Gradient Ricci soliton Contact metric manifold K-contact manifold 

Mathematics Subject Classification

53C25 53C15 53D10 



The author expresses his sincere thanks to the referee for offering many valuable suggestions towards the improvement of the paper.


  1. 1.
    Aquino, C., Barros, A., Ribeiro Jr., E.: Some applications of the Hodge-de Rahm decomposition to Ricci solitons. Res. Math. 60, 245–254 (2011)CrossRefzbMATHGoogle Scholar
  2. 2.
    Barros, A., Ribeiro Jr., E.: Integral formulae on quasi-Einstein manifolds and applications. Glasgow Math. J 54, 213–223 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Barros, A., Ribeiro Jr., E.: Characterizations and integral formulae for generalized m-quasi-Einstein metrics. Bull. Braz. Math. Soc. (N.S.) 45, 325–341 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Barros, A., Gomes, J.N.: Triviality of compact \(m\)-quasi-Einstein manifolds. Res. Math. 71(1), 241–250 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Barros, A., Ribeiro Jr., E., Silva, J.F.: Uniqueness of quasi-Einstein metrics on 3-dimensional homogeneous Riemannian manifolds. Differ. Geom. Appl. 35, 60–73 (2014)CrossRefzbMATHGoogle Scholar
  6. 6.
    Besse, A.L.: Einstein Manifolds. Springer, Berlin (1987)CrossRefzbMATHGoogle Scholar
  7. 7.
    Blair, D.E.: Riemannian Geometry of Contact and Symplectic Manifolds. Birkhauser, Boston (2002)CrossRefzbMATHGoogle Scholar
  8. 8.
    Boyer, C.P., Galicki, K.: Einstein manifolds and contact geometry. Proc. Am. Math. Soc. 129, 2419–2430 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Boyer, C.P., Galicki, K.: Sasakian Geometry. Oxford University Press, Oxford (2008)zbMATHGoogle Scholar
  10. 10.
    Boyer, C.P., Galicki, K., Matzeu, P.: On \(\eta \)-Einstein Sasakian geometry. Cmmun. Math. Phys. 262, 177–208 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cao, H.D.: Recent progress on Ricci soliton. Adv. Lect. Math. 11, 1–38 (2009)MathSciNetGoogle Scholar
  12. 12.
    Case, J.: On the nonexistence of quasi-Einstein metrics. Pac. J. Math. 248, 227–284 (2010)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Case, J., Shu, Y., Wei, G.: Rigidity of quasi-Einstein metrics. Diff. Geom. Appl. 29, 93–100 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Cho, J.T., Sharma, R.: Contact geometry and Ricci solitons. Int. J. Geom. Methods Mod. Phys. 7, 951–960 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Filho, J.F.S.: Quasi Einstein manifolds endowed with a parallel vector fields. Monatsh. Math. 179, 305–320 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ghosh, A.: \((m, \rho )\)-quasi-Einstein metrics in the frame-work of \(K\)-contact manifolds. Math. Phys. Anal. 17, 369–376 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ghosh, A.: Generalized m-quasi-Einstein metric within the framework of Sasakian and K-contact manifolds. Ann. Polonici Math. 115(1), 33–41 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ghosh, A., Sharma, R., Cho, J.T.: Contact metric manifolds with \( \eta \)-parallel torsion tensor. Ann. Glob. Anal. Geom. 34, 287–299 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hu, Z., Li, D., Xu, J.: On generalized m-quasi-Einstein manifolds with constant scalar curvature. J. Math. Anal. Appl. 432, 733–743 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hasegawa, I., Seino, M.: Some remarks on Sasakian geometry-applications of Myers’ theorem and the canonical affine connection. J. Hokkaido Univ. Educ 32(section IIA), 1–7 (1981)MathSciNetGoogle Scholar
  21. 21.
    Limoncu, M.: Modification of the Ricci tensor and applications. Arch. Math. 95, 191–199 (2010)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Myers, S.B.: Connections between differential geometry and topology. Duke Math. J. 1, 376–391 (1935)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Perelman, G.: The entropy formula for the Ricci flow and its geometric applications, Preprint. arXiv:math.DG/02111159
  24. 24.
    Perrone, D.: Contact metric manifolds whose characteristic vector field is a harmonic vector field. Diff. Geom. Appl. 20, 367–378 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Qian, Z.: Estimates for weighted volumes and applications. Q. J. Math. Oxford Ser. (2) 48(190), 235–242 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Sharma, R.: Certain results on \(K\)-contact and (\(k,\mu \))-contact manifolds. J. Geom. 89, 138–147 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Tanno, S.: The topology of contact Riemannian manifolds. Illinois J. Math. 12, 700–717 (1968)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Wang, L.F.: On noncompact \(\tau \)-quasi-Einstein metrics. Pac. J Math. 254(2), 449–464 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Yano, K.: Integral Formulas in Riemannian Geometry. Marcel Dekker, New York (1970)zbMATHGoogle Scholar
  30. 30.
    Yano, K., Kon, M.: Structures on Manifolds. World Scientific, Singapore (1984)zbMATHGoogle Scholar

Copyright information

© The Royal Academy of Sciences, Madrid 2019

Authors and Affiliations

  1. 1.Department of MathematicsChandernagore CollegeChandannagarIndia

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