Skip to main content
SpringerLink
Log in

Kenmotsu Metric as Conformal \(\eta \)-Ricci Soliton

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

The object of the present paper is to characterize the class of Kenmotsu manifolds which admits conformal \(\eta \)-Ricci soliton. Here, we have investigated the nature of the conformal \(\eta \)-Ricci soliton within the framework of Kenmotsu manifolds. It is shown that an \(\eta \)-Einstein Kenmotsu manifold admitting conformal \(\eta \)-Ricci soliton is an Einstein one. Moving further, we have considered gradient conformal \(\eta \)-Ricci soliton on Kenmotsu manifold and established a relation between the potential vector field and the Reeb vector field. Next, it is proved that under certain condition, a conformal \(\eta \)-Ricci soliton on Kenmotsu manifolds under generalized D-conformal deformation remains invariant. Finally, we have constructed an example for the existence of conformal \(\eta \)-Ricci soliton on Kenmotsu manifold.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Availability of Data and Materials

Not applicable.

References

  1. Alegre, P., Carriazo, A.: Generalized Sasakian space form and conformal changes of the metric. Results Math. 59, 485–493 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Basu, N., Bhattacharyya, A.: Conformal Ricci soliton in Kenmotsu manifold. Glob. J. Adv. Res. Class. Mod. Geom. 4(1), 15–21 (2015)

    MathSciNet  Google Scholar 

  3. Blair, D.E.: Riemannian Geometry of Contact and Symplectic Manifolds, 2nd edn. Birkhauser, Basel (2010)

    Book  MATH  Google Scholar 

  4. Calin, C., Crasmareanu, M.: \(\eta \)-Ricci solitons on Hopf hypersurfaces in complex space forms. Revue Roum. Math. Pures Appl. 57(1), 53–63 (2012)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Fischer, A.E.: An introduction to conformal Ricci flow. Class. Quantum Gravity 21(3), S171–S218 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ganguly, D., Bhattacharyya, A.: A study on conformal Ricci solitons in the framework of \((LCS)_n\)-manifolds. Ganita 70(2), 201–216 (2020)

    MathSciNet  MATH  Google Scholar 

  8. Ganguly, D., Dey, S., Ali, A., Bhattacharyya, A.: Conformal Ricci soliton and quasi-Yamabe soliton on generalized Sasakian space form. J. Geom. Phys. 169, 104339 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ganguly, D., Dey, S., Bhattacharyya, A.: On trans-Sasakian3-manifolds as \(\eta \)-Einstein solitons. Carpathian Math. Publ. 13(2), 460–474 (2021). https://doi.org/10.15330/cmp.13.2.460-474

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Gupta, R.S., Rani, S.: Ricci soliton on GSSF with Sasakian metric. Note Math. 42(1), 95–108 (2022)

    MathSciNet  Google Scholar 

  12. Hamilton, R.S.: Three manifolds with positive Ricci curvature. J. Differ. Geom. 17, 255–306 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hamilton, R.S.: The Formation of Singularities in the Ricci Flow. Surveys in Differential Geometry, vol. II (Cambridge, MA, 1993), pp. 7–136. International Press, Cambridge (1995)

  14. Kenmotsu, K.: A class of almost contact Riemannian manifold. Tohoku Math. J. 24, 93–103 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  15. Li, Y., Ganguly, D., Dey, S., Bhattacharyya, A.: Conformal \(\eta \)-Ricci solitons within the framework of indefinite Kenmotsu manifolds. AIMS Math. 7(4), 5408–5430 (2022). https://doi.org/10.3934/math.2022300

    Article  MathSciNet  Google Scholar 

  16. Li, Y., Erdoğdu, M., Yavuz, A.: Differential geometric approach of Betchow-Da Rios soliton equation. Hacet. J. Math. Stat. (2022). https://doi.org/10.15672/hujms.1052831

    Article  MATH  Google Scholar 

  17. Li, Y., Prasad, R., Haseeb, A., Kumar, S., Kumar, S.: A study of Clairaut semi-invariant Riemannian maps from cosymplectic manifolds. Axioms 11, 503 (2022)

    Article  Google Scholar 

  18. Li, Y., Abdel-Salam, A.A., Saad, M.K.: Primitivoids of curves in Minkowski plane. AIMS Math. 8, 2386–2406 (2023)

    Article  MathSciNet  Google Scholar 

  19. Li, Y., Abolarinwa, A., Alkhaldi, A.H., Ali, A.: Some inequalities of Hardy type related to Witten-Laplace operator on smooth metric measure spaces. Mathematics 10, 4580 (2022)

    Article  Google Scholar 

  20. Li, Y., Maryam, T.A., Abdel-Baky, R.A.: Spacelike circular surfaces in Minkowski 3-space. Symmetry 15, 173 (2023)

    Article  Google Scholar 

  21. Li, Y., Alkhaldi, A., Ali, A., Abdel-Baky, R.A., Saad. M.K.: Investigation of ruled surfaces and their singularities according to Blaschke frame in Euclidean 3-space. AIMS Math. 8, 13875–13888 (2023). https://doi.org/10.3934/math.2023709

  22. Li, Y., Chen, Z., Nazra, S.H., Abdel-Baky, R.A.: Singularities for timelike developable surfaces in Minkowski 3-space. Symmetry 15, 277 (2023)

    Article  Google Scholar 

  23. Li, Y., Tuncer, O.O.: On (contra)pedals and (anti)orthotomics of frontals in de Sitter 2-space. Math. Methods Appl. Sci. (2023). https://doi.org/10.1002/mma.9173

    Article  MathSciNet  Google Scholar 

  24. Li, Y., Eren, K., Ayvacı, K.H., Ersoy, S.: The developable surfaces with pointwise 1-type Gauss map of Frenet type framed base curves in Euclidean 3-space. AIMS Math. 8, 2226–2239 (2023)

    Article  MathSciNet  Google Scholar 

  25. Nagaraja, H.G., Kiran Kumar, D.L.: Ricci solitons in Kenmotsu manifold under generalized \(D\)-conformal deformation. Lobachevskii J. Math. 40(2), 195–200 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  26. Naik, D.M.: Ricci solitons on Riemannian manifolds admitting certain vector field. Ricerche di Matematica (2021). https://doi.org/10.1007/s11587-021-00622-z

    Article  Google Scholar 

  27. Prakasha, D.G., Hui, S.K., Vikas, K.: On weakly \(\phi \)-Ricci symmetric Kenmotsu manifolds. Int. J. Pure Appl. Math. 95(4), 515–521 (2014)

    Article  Google Scholar 

  28. Qian, Y., Yu, D. : Rates of approximation by neural network interpolation operators. Appl. Math. Comput. 41, 126781 (2022)

  29. Rani, S., Gupta, R.S.: Ricci soliton on manifolds with cosymplectic metric. U. P. B. Sci. Bull. Ser. A 84(1), 89–98 (2022)

    MathSciNet  Google Scholar 

  30. Sharma, R., Ghosh, A.: Sasakian 3-manifold as a Ricci soliton represents the Heisenberg group. Int. J. of Geom. Methods Mod. Phys. 8, 149–154 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. Siddiqi, M.D.: Conformal \(\eta \)-Ricci solitons in \(\delta \)-Lorentzian trans Sasakian manifolds. Int. J. Maps Math. 1(1), 15–34 (2018)

    MathSciNet  Google Scholar 

  32. Tanno, S.: The topology of contact Riemannian manifolds. Ill. J. Math. 12, 700–717 (1968)

    MathSciNet  MATH  Google Scholar 

  33. Venkatesha, V., Naik, D.M., Kumara, H.A.: *-Ricci solitons and gradient almost *-Ricci solitons on Kenmotsu manifolds. Math. Slovaca 69, 1447–1458 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  34. Yano, K.: Integral Formulas in Riemannian Geometry. Marcel Dekker, New York (1970)

    MATH  Google Scholar 

  35. Zhang, P., Li, Y., Roy, S., Dey, S., Bhattacharyya, A.: Geometry of \(\alpha \)-Cosymplectic Metric as \(*\)-Conformal \(\eta \)-Ricci–Yamabe solitons admitting quarter-symmetric metric connection. Symmetry 13, 2189 (2021). https://doi.org/10.3390/sym13112189

    Article  Google Scholar 

  36. Zhang, P., Li, Y., Roy, S., Dey, S., Bhattacharyya, A.: Geometrical structure in a perfect fluid spacetime with conformal Ricci–Yamabe soliton. Symmetry 14, 594 (2022). https://doi.org/10.3390/sym14030594

    Article  Google Scholar 

Download references

Acknowledgements

We gratefully acknowledge the constructive comments from the editor and the anonymous referees.

Funding

This research was funded by National Natural Science Foundation of China (Grant no. 12101168) and Zhejiang Provincial Natural Science Foundation of China (Grant no. LQ22A010014).

Author information

Author notes

  1. Yanlin Li and Dipen Ganguly contributed equally to this work.

Authors and Affiliations

  1. School of Mathematics, Hangzhou Normal University, 311121, Hangzhou, Zhejiang, China

    Yanlin Li

  2. Department of Mathematics, Jadavpur University, Kolkata, 700032, West Bengal, India

    Dipen Ganguly

Authors
  1. Yanlin Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dipen Ganguly
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Conceptualization, YL, DG; methodology, YL, DG; investigation, YL, DG; writing—original draft preparation, YL, DG; writing—review and editing, YL, DG. All authors have read and agreed to the published version of the manuscript.

Corresponding author

Correspondence to Yanlin Li.

Ethics declarations

Conflict of Interest

The authors declare no conflict of interest.

Ethics Approval

Not applicable.

Consent to Participate

Not applicable.

Consent for Publication

Not applicable.

Code Available

Not applicable.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, Y., Ganguly, D. Kenmotsu Metric as Conformal \(\eta \)-Ricci Soliton. Mediterr. J. Math. 20, 193 (2023). https://doi.org/10.1007/s00009-023-02396-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-023-02396-0

Keywords

Mathematics Subject Classification

Search

Navigation