Skip to main content
SpringerLink
Log in

A note on almost Ricci solitons

  • Published:
Analysis and Mathematical Physics Aims and scope Submit manuscript

Abstract

We find several sufficient conditions on a compact almost Ricci soliton under which it is a trivial Ricci soliton. We also find a sufficient condition under which a compact almost Ricci soliton is isometric to a sphere.

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

References

  1. Aquino, C., Barros, A., Ribeiro Jr., E.: Some applications of the Hodgede Rham decomposition to Ricci solitons. Results Math. 60, 235–246 (2011)

    Article  MathSciNet  Google Scholar 

  2. Barros, A., Gomes, J.N., Ribeiro Jr., E.: A note on rigidity of the almost Ricci soliton. Arch. Math. 100, 481–490 (2013)

    Article  MathSciNet  Google Scholar 

  3. Barros, A., Batista, R., Ribeiro Jr., E.: Compact almost Ricci solitons with constant scalar curvature are gradient. Monatsh. Math. 174, 29–39 (2014)

    Article  MathSciNet  Google Scholar 

  4. Cao, H.D.: Recent progress on Ricci solitons. Adv. Lect. Math. (ALM) 11, 1–38 (2009)

    MathSciNet  MATH  Google Scholar 

  5. Chow, B., Lu, P., Ni, L.: Hamilton’s Ricci Flow. Graduate Studies in Mathematics, vol. 77. AMS Scientific Press, Providence (2010)

    Google Scholar 

  6. Deshmukh, S., Al-Eid, A.: Curvature bounds for the spectrum of a compact Riemannian manifold of constant scalar curvature. J. Geom. Anal. 15(4), 589–606 (2005)

    Article  MathSciNet  Google Scholar 

  7. Deshmukh, S.: Almost Ricci solitons isometric to spheres. Int. J. Geom. Methods Mod. Phys. 16(5), 9 (2019)

    Article  MathSciNet  Google Scholar 

  8. Deshmukh, S., Alsodais, H.: A note on Ricci solitons. Symmetry (MDPI) 12, 289 (2020)

    Article  Google Scholar 

  9. Hamilton, R.S.: The Ricci flow on surfaces, mathematics and general relativity. Contemp. Math. 71, 237–262 (1988)

    Article  Google Scholar 

  10. Hinterleitner, I., Kiosak, V.: \(\varphi (Ric)\)-vector fields on conformally flat spaces. AIP Conf. Proc. 1191, 98–103 (2009)

    Article  Google Scholar 

  11. Hinterleitner, I., Kiosak, V.: \(Ric\)-vector fields in Riemannian spaces. Arch. Math. Brno 44(5), 385–390 (2008)

    MathSciNet  MATH  Google Scholar 

  12. Mikes, J., Stepanova, E., Vanzurova, E.: Differential Geometry of Special Mappings. Palacky University Press, Olomouc (2019)

    Book  Google Scholar 

  13. Obata, M.: Certain conditions for a Riemannian manifold to be isometric with a sphere. J. Math. Soc. Jpn. 14, 333–340 (1962)

    Article  MathSciNet  Google Scholar 

  14. Pigola, S., Rigoli, M., Rimoldi, M., Setti, A.G.: Ricci almost solitons. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) X, 757–799 (2011)

    MathSciNet  MATH  Google Scholar 

  15. Sharma, R.: Almost Ricci Solitons and K-contact geometry. Monatsh. Math. 175, 621–628 (2014)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

Download references

Acknowledgements

The authors extend their appreciations to the Deanship of Scientific Research, King Saud University for funding this work through research group no. (RG-1441-P182). We sincerely thank Referee for helpful comments in improving the paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia

    Sharief Deshmukh & Hana Al-Sodais

Authors
  1. Sharief Deshmukh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hana Al-Sodais
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sharief Deshmukh.

Ethics declarations

Conflict of interest

Authors declare that there is no conflict of interest.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Deshmukh, S., Al-Sodais, H. A note on almost Ricci solitons. Anal.Math.Phys. 10, 76 (2020). https://doi.org/10.1007/s13324-020-00424-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13324-020-00424-w

Keywords

Mathematics Subject Classification

Search

Navigation