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Yamabe and Quasi-Yamabe Solitons on Euclidean Submanifolds

  • Bang-Yen Chen
  • Sharief Deshmukh
Article
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Abstract

In this paper, we initiate the study of Yamabe and quasi-Yamabe solitons on Euclidean submanifolds whose soliton fields are the tangential components of their position vector fields. Several fundamental results of such solitons were proved. In particular, we classify such Yamabe and quasi-Yamabe solitons on Euclidean hypersurfaces.

Keywords

Yamabe soliton Quasi-Yamabe soliton Euclidean hypersurface Euclidean submanifolds Position vector field Torse-forming vector field 

Mathematics Subject Classification

53C25 53C40 

Notes

Acknowledgements

This work is supported by King Saud University, Deanship of Scientific Research, College of Science Research Center.

References

  1. 1.
    Chen, B.-Y.: Geometry of Submanifolds. Marcer Dekker, New York (1973)zbMATHGoogle Scholar
  2. 2.
    Chen, B.-Y.: Pseudo-Riemannian Geometry, \(\delta \)-invariants and Applications. World Scientific, Hackensack (2011)CrossRefGoogle Scholar
  3. 3.
    Chen, B.-Y.: Total Mean Curvature and Submanifolds of Finite Type, 2nd edn. World Scientific, Hackensack (2015)zbMATHGoogle Scholar
  4. 4.
    Chen, B.-Y.: Differential geometry of rectifying submanifolds. Int. Electron. J. Geom. 9(2), 1–8 (2016)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Chen, B.-Y.: Addendum to: differential geometry of rectifying submanifolds. Int. Electron. J. Geom. 10(1), 81–82 (2017)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Chen, B.-Y.: Topics in differential geometry associated with position vector fields on Euclidean submanifolds. Arab J. Math. Sci. 23(1), 1–17 (2017)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Chen, B.-Y.: Euclidean submanifolds and the tangential components of their position vector fields. Mathematics 5, 17 (2017). Art. 51CrossRefGoogle Scholar
  8. 8.
    Chen, B.-Y., Deshmukh, S.: Classification of Ricci solitons on Euclidean hypersurfaces. Intern. J. Math. 25(11), 22 (2014). Art. 1450104MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chen, B.-Y., Deshmukh, S.: Ricci solitons and concurrent vector fields. Balkan J. Geom. Appl. 20(1), 14–25 (2015)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Chen, B.-Y., Verstraelen, L.: A link between torse-forming vector fields and rotational hypersurfaces. Int. J. Geom. Methods Mod. Phys. 14(12), 10 (2017). Art. 1750177MathSciNetCrossRefGoogle Scholar
  11. 11.
    Chen, B.-Y., Wei, S.W.: Differential geometry of concircular submanifolds of Euclidean spaces. Serdica Math. J. 43(1), 36–48 (2017)MathSciNetGoogle Scholar
  12. 12.
    Chen, B.-Y., Yano, K.: Integral formulas for submanifolds and their applications. J. Differ. Geom. 5, 467–477 (1971)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Chen, B.-Y., Yano, K.: Umbilical submanifolds with respect to a nonparallel normal direction. J. Differ. Geom. 8, 589–597 (1973)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Hamilton, R.: S.: The Ricci flow on surfaces. Math. Gen. Relativ. (Santa Cruz, CA, 1986). Contemp. Math. 71, 237–262 (1998)CrossRefGoogle Scholar
  15. 15.
    Huang, G., Li, H.: On a classification of the quasi Yamabe gradient solitons. Methods Appl. Anal. 21(3), 379–389 (2014)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Leandro, B., Pina, H.: Generalized quasi Yamabe gradient solitons. Differ. Geom. Appl. 49, 167–175 (2016)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Mihai, A., Mihai, I.: Torse forming vector fields and exterior concurrent vector fields on Riemannian manifolds and applications. J. Geom. Phys. 73, 200–208 (2013)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Weyl, H.: Reine infinitesimalgeometrie. Math. Z. 26, 384–411 (1918)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Yano, K.: On torse forming direction in a Riemannian space. Proc. Imp. Acad. Tokyo 20, 340–346 (1944)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsMichigan State UniversityEast LansingUSA
  2. 2.Department of Mathematics, College of scienceKing Saud UniversityRiyadhSaudi Arabia

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