Skip to main content

Some results on h-almost Ricci-Bourguignon soliton

Abstract

In this paper, we consider an h-almost Ricci-Bourguignon soliton and we provide some useful formulas of structure for this soliton. Then for an application of these formulas we show that a noncompact, complete nontrivial gradient h-Ricci-Bourguignon soliton is isometric to Euclidean space and a compact nontrivial h-almost Ricci-Bourguignon soliton is isometric to a Euclidean sphere with some conditions.

This is a preview of subscription content, access via your institution.

References

  1. Cao, H.-D., Chen, B.-L., Zhu, X.-P.: Recent developments on Hamilton’s Ricci flow, Surveys in differential geometry. Vol. XII. Geometric flows, vol. 12, pp. 47–112. Int. Press, Somerville (2008)

  2. Catino, G., Mazzieri, L.: Gradient Einstein solitons. Nonlinear Anal. 132, 66–94 (2016)

    MathSciNet  Article  Google Scholar 

  3. Catino, G., Mazzieri, L., Mongodi, S.: Rigidity of gradient Einstein shrinkers. Commun. Contemp. Math. 17(6), 18 (2015)

    MathSciNet  Article  Google Scholar 

  4. Dwivedi, S.: Some results on Ricci-Bourguignon and Ricci-Bourguignon almost solitons. Can. Math. Bull. 2020, 1–15 (2020)

    Google Scholar 

  5. Ghahremani-Gol, H.: Some results on \(h\)-almost Ricci solitons. J. Geom. Phys. 137, 212–216 (2019)

    MathSciNet  Article  Google Scholar 

  6. Gomes, J.N., Wang, Q., Xia, C.: On the \(h\)-almost Ricci soliton. J. Geom. Phys. 114, 216–222 (2017)

    MathSciNet  Article  Google Scholar 

  7. Ivey, T.: Ricci solitons on compact three-manifolds. Differ. Geom. Appl. 3(4), 301–307 (1993)

    MathSciNet  Article  Google Scholar 

  8. Lichnerowicz, A.: Géométrie des Groupes de Tranformations, Travaux et Recherches Mathématiques. III. Dunod, Paris (1958)

    Google Scholar 

  9. Naber, A.: Noncompact shrinking four solitons with nonnegative curvature. J. Reine Angew. Math. 645, 125–153 (2010)

    MathSciNet  MATH  Google Scholar 

  10. Ni, L., Wallach, N.: On a classification of gradient shrinking solitons. Math. Res. Lett. 15(5), 941–955 (2008)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  12. Perelman, G.: The entropy formula for the Ricci flow and its geometric applications (2002)

  13. Petersen, P., Wylie, W.: Rigidity of gradient Ricci solitons. Pac. J. Math. 241(2), 329–345 (2009)

    MathSciNet  Article  Google Scholar 

  14. Petersen, P., Wylie, W.: On gradient Ricci solitons with symmetry. Proc. Am. Math. Soc. 137(6), 2085–2092 (2009)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  16. Tashiro, Y.: Complete Riemannian manifolds and some vector fields. Trans. Am. Math. Soc. 117, 251–275 (1965)

    MathSciNet  Article  Google Scholar 

  17. Yano, K.: Integral Formulas in Riemannian Geometry, Pure and Applied Mathematics, no. 1. Marcel Dekker, Inc., New York (1970)

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Shahroud Azami.

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 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.

Springer Nature or its licensor 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

Verify currency and authenticity via CrossMark

Cite this article

Azami, S. Some results on h-almost Ricci-Bourguignon soliton. Afr. Mat. 33, 88 (2022). https://doi.org/10.1007/s13370-022-01025-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13370-022-01025-5

Keywords

  • h-almost Ricci-Bourguignon soliton
  • Gradient h-almost Ricci-Bourguignon soliton
  • Ricci-Bourguignon flow

Mathematics Subject Classification

  • 53C25
  • 53C21
  • 53E20