Gradient \(\rho \)-Einstein soliton on almost Kenmotsu manifolds

  • V. VenkateshaEmail author
  • H. Aruna Kumara
Original Research


In this paper, we prove that if the metric of an almost Kenmotsu manifold with conformal Reeb foliation admits a gradient \(\rho \)-Einstein soliton, then either \(M^{2n+1}\) is Einstein or the potential function is pointwise collinear with the Reeb vector field \(\xi \) on an open set \({\mathcal {O}}\) of \(M^{2n+1}\). Moreover, we prove that if the metric of a \((\kappa ,-2)'\)-almost Kenmotsu manifold with \(h'\ne 0\) admits a gradient \(\rho \)-Einstein soliton, then the manifold is locally isometric to the Riemannian product \({\mathbb {H}}^{n+1}(-4)\times {\mathbb {R}}^n\) and potential vector field is tangential to the Euclidean factor \({\mathbb {R}}^n\). We show that there does not exist gradient \(\rho \)-Einstein soliton on generalized \((\kappa ,\mu )\)-almost Kenmotsu manifold of constant scalar curvature. Finally, we construct an example for gradient \(\rho \)-Einstein soliton.


Gradient \(\rho \)-Einstein soliton Einstein manifolds Ricci solitons Almost Kenmotsu manifolds 

Mathematics Subject Classification

53C25 53C20 53C15 



The authors are immensely grateful to the referee for many valuable suggestions.


  1. 1.
    Blair, D.E.: Riemannian Geometry of Contact and Symplectic Manifolds. Progress in Mathematics, vol. 203. Birkhäuser, Basel (2010)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bourguignon, J.P.: Ricci curvature and Einstein metrics. In: Global differential geometry and global analysis (Berlin, 1979), Lecture Notes in Math. , vol. 838, pp. 42–63 (1981)Google Scholar
  3. 3.
    Catino, G., Mazzieri, L.: Gradient Einstein solitons. Nonlinear Anal. 132, 66–94 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Catino, G.: Rigidity of gradient Einstein shrinkers. Commun. Contemp. Math. 17(6), 1550046 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    De, U.C., Mandal, K.: On a type of almost Kenmotsu manifolds with nullity distributions. Arab J. Math. Sci. 23(2), 109–123 (2017)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Dileo, G., Pastore, A.M.: Almost Kenmotsu manifolds and local symmetry. Bull. Belg. Math. Soc. Simon Stevin 14(2), 343–354 (2007)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Dileo, G., Pastore, A.M.: Almost Kenmotsu manifolds and nullity distributions. J. Geom. 93, 46–61 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ghosh, A.: \((m,\rho )\)-quasi Einstein metrics within the frame-work of \(K\)-contact manifolds. J. Math. Phys. Anal. Geom. 17, 369–376 (2014)CrossRefzbMATHGoogle Scholar
  9. 9.
    Ghosh, A.: Quasi-Einstein contact metric manifolds. Glasgow Math. J. 57, 569–577 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hamilton, R.S.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17, 255–306 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Huang, G.: Integral pinched gradient shrinking \(\rho \)-Einstein solitons. J. Math. Anal. Appl. 451(2), 1025–1055 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Huang, G., Wei, Y.: The classification of \((m,\rho )\)-quasi-Einstein manifolds. Ann. Global Anal. Geom. 44, 269–282 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Janssens, D., Vanhecke, L.: Almost contact structures and curvature tensors. Kodai Math. J. 4(1), 1–27 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kenmotsu, K.: A class of almost contact Riemannian manifolds. Tohoku Math. J. 24, 93–103 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kim, T.W., Pak, H.K.: Canonical foliations of certain classes of almost contact metric structures. Acta Math. Sin. Engl. Ser. 21, 841–846 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Pastore, A.M., Saltarelli, V.: Almost Kenmotsu manifolds with conformal Reeb foliation. Bull. Belg. Math. Soc. Simon Stevin. 21, 343–354 (2012)zbMATHGoogle Scholar
  17. 17.
    Pastore, A.M., Saltarelli, V.: Generalized nullity distributions on almost Kenmotsu manifolds. Int. Electron. J. Geom. 4(2), 168–183 (2011)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Wang, Y., Liu, X.: Ricci solitons on three-dimensional \(\eta \)-Einstein almost Kenmotsu manifolds. Taiwan. J. Math. 19, 91–100 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Wang, Y., De, U.C., Liu, X.: Gradient Ricci solitons on almost Kenmotsu manifolds. Publ. De L’Inst. Math. 98(112), 227–235 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Wang, Y.: Gradient Ricci almost solitons on two classes of almost Kenmotsu manifolds. J. Korean Math. Soc. 53(5), 1101–1114 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Wang, Y., Liu, X.: On almost Kenmotsu manifolds satisfying some nillity distributions. Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci 86(3), 347–353 (2016)CrossRefzbMATHGoogle Scholar

Copyright information

© Università degli Studi di Ferrara 2019

Authors and Affiliations

  1. 1.Department of MathematicsKuvempu UniversityShivamoggaIndia

Personalised recommendations