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Almost \(\eta \)-Ricci solitons on Kenmotsu manifolds

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Abstract

We characterize the Einstein metrics in such broad classes of metrics as almost \(\eta \)-Ricci solitons and \(\eta \)-Ricci solitons on Kenmotsu manifolds, and generalize some known results. First, we prove that a Kenmotsu metric as an \(\eta \)-Ricci soliton is Einstein metric if either it is \(\eta \)-Einstein or the potential vector field V is an infinitesimal contact transformation or V is collinear to the Reeb vector field. Further, we prove that if a Kenmotsu manifold admits a gradient almost \(\eta \)-Ricci soliton with a Reeb vector field leaving the scalar curvature invariant, then it is an Einstein manifold. Finally, we present new examples of \(\eta \)-Ricci solitons and gradient \(\eta \)-Ricci solitons, which illustrate our results.

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References

  1. Barros, A., Ribeiro Jr., E.: Some characterizations for compact almost Ricci solitons. Proc. Amer. Math. Soc. 140(3), 1033–1040 (2012)

    Article  MathSciNet  Google Scholar 

  2. Blaga, A.M.: Almost \(\eta \)-Ricci solitons in \((LCS)_n\)-manifolds. Bull. Belg. Math. Soc. Simon Stevin 25(5), 641–653 (2018)

    Article  MathSciNet  Google Scholar 

  3. Blaga, A.M.: \(\eta \)-Ricci solitons on Lorentzian para-Sasakian manifolds. Filomat 30(2), 489–496 (2016)

    Article  MathSciNet  Google Scholar 

  4. Blaga, A.M., Özgür, C.: Almost \(\eta \)-Ricci and almost \(\eta \)-Yamabe solitons with torseforming potential vector field (2020). arXiv:2003.12574

  5. Blair, D.E.: Riemannian Geometry of Contact and Symplectic Manifolds. Progress in Mathematics, vol. 203. Birkhäuser, Boston (2002)

    Book  Google Scholar 

  6. Călin, C., Crâşmăreanu, M.: \(\eta \)-Ricci solitons on Hopf hypersurfaces in complex space forms. Rev. Roumaine Math. Pures Appl. 57(1), 55–63 (2012)

    MathSciNet  MATH  Google Scholar 

  7. Chen, B.-Y.: Differential Geometry of Warped Product Manifolds and Submanifolds. World Scientific, Hackensack (2017)

    Book  Google Scholar 

  8. Cho, J.T., Kimura, M.: Ricci solitons and real hypersurfaces in a complex space form. Tohoku Math. J. 61(2), 205–212 (2009)

    Article  MathSciNet  Google Scholar 

  9. Cho, J.T., Sharma, R.: Contact geometry and Ricci solitons. Int. J. Geom. Meth. Mod. Phys. 7(6), 951–960 (2010)

    Article  MathSciNet  Google Scholar 

  10. Chow, B., Knopf, D.: The Ricci Flow. Mathematical Surveys and Monographs, vol. 110. American Mathematical Society, Providence (2004)

    MATH  Google Scholar 

  11. Dileo, G., Pastore, A.M.: Almost Kenmotsu manifolds and nullity distributions. J. Geom. 93(1–2), 46–61 (2009)

    Article  MathSciNet  Google Scholar 

  12. Dileo, G., Pastore, A.M.: Almost Kenmotsu manifolds and local symmetry. Bull. Belg. Math. Soc. Simon Stevin 14(2), 343–354 (2007)

    Article  MathSciNet  Google Scholar 

  13. Duggal, K.L.: Almost Ricci solitons and physical applications. Int. Electron. J. Geom. 10(2), 1–10 (2017)

    MathSciNet  MATH  Google Scholar 

  14. Eyasmin, S., Chowdhury, P.R., Baishya, K.K.: \(\eta \)-Ricci solitons on Kenmotsu manifolds. Honam Math. J. 40(2), 367–376 (2018)

    MathSciNet  MATH  Google Scholar 

  15. Ghosh, A.: Kenmotsu 3-metric as a Ricci soliton. Chaos Solitons Fractals 44(8), 647–650 (2011)

    Article  MathSciNet  Google Scholar 

  16. Ghosh, A.: An \(\eta \)-Einstein Kenmotsu metric as a Ricci soliton. Publ. Math. Debrecen 82(3–4), 591–598 (2013)

    Article  MathSciNet  Google Scholar 

  17. Ghosh, A.: Certain contact metrics as Ricci almost solitons. Results Math. 65(1–2), 81–94 (2014)

    Article  MathSciNet  Google Scholar 

  18. Ghosh, A.: Ricci soliton and Ricci almost soliton within the framework of Kenmotsu manifold. Carpathian Math. Publ. 11(1), 59–69 (2019)

    Article  MathSciNet  Google Scholar 

  19. Ghosh, A., Patra, D.S.: The \(k\)-almost Ricci solitons and contact geometry. J. Korean Math. Soc. 55(1), 161–174 (2018)

    MathSciNet  MATH  Google Scholar 

  20. Hamilton, R.S.: The Ricci flow on surfaces. In: Isenberg, J.A. (ed.) Mathematics and General Relativity. Contemporary Mathematics, vol. 71, pp. 237–262. American Mathematical Society, Providence (1988)

    Chapter  Google Scholar 

  21. Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Space-Time. Cambridge Monographs on Mathematical Physics, vol. 1. Cambridge University Press, Cambridge (1973)

    Book  Google Scholar 

  22. Kenmotsu, K.: A class of almost contact Riemannian manifolds. Tôhoku Math. J. 24, 93–103 (1972)

    Article  MathSciNet  Google Scholar 

  23. Manev, M.: Ricci-like solitons on almost contact B-metric manifolds. J. Geom. Phys. 154, 103734 (2020)

    Article  MathSciNet  Google Scholar 

  24. Naik, D.M., Venkatesha, V.: \(\eta \)-Ricci soliton and almost \(\eta \)-Ricci soliton on para-Sasakian manifolds. Int. J. Geom. Methods Mod. Phys. 16(9), 1950134 (2019)

    Article  MathSciNet  Google Scholar 

  25. Pigola, S., Rigoli, M., Rimoldi, M., Setti, A.G.: Ricci almost solitons. Ann. Sci. Norm. Sup. Pisa Cl. Sci. 10(4), 757–799 (2011)

  26. Shanmukha, B., Venkatesh, V.: Some Ricci solitons on Kenmotsu manifold. J. Anal. 28(4), 1155–1164 (2020)

    Article  MathSciNet  Google Scholar 

  27. Sharma, R.: Certain results on \(K\)-contact and \((\kappa,\mu )\)-contact manifolds. J. Geom. 89(1–2), 138–147 (2008)

    Article  MathSciNet  Google Scholar 

  28. Sharma, R.: Almost Ricci solitons and \(K\)-contact geometry. Monatsh Math. 175(4), 621–628 (2015)

    Article  MathSciNet  Google Scholar 

  29. Yano, K.: Integral Formulas in Riemannian Geometry. Pure and Applied Mathematics, vol. 1. Dekker, New York (1970)

    Google Scholar 

  30. Yano, K., Kon, M.: Structures on Manifolds. Series in Pure Mathematics, vol. 3. World Scientific, Singapore (1984)

    MATH  Google Scholar 

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Correspondence to Vladimir Rovenski.

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Patra, D.S., Rovenski, V. Almost \(\eta \)-Ricci solitons on Kenmotsu manifolds. European Journal of Mathematics 7, 1753–1766 (2021). https://doi.org/10.1007/s40879-021-00474-9

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