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Science China Mathematics

, Volume 61, Issue 10, pp 1825–1832 | Cite as

Complete Ricci solitons on Finsler manifolds

  • Behroz Bidabad
  • Mohamad Yar Ahmadi
Articles
  • 22 Downloads

Abstract

The geometric flow theory and its applications turned into one of the most intensively developing branches of modern geometry. Here, a brief introduction to Finslerian Ricci flow and their self-similar solutions known as Ricci solitons are given and some recent results are presented. They are a generalization of Einstein metrics and are previously developed by the present authors for Finsler manifolds. In the present work, it is shown that a complete shrinking Ricci soliton Finsler manifold has a finite fundamental group.

Keywords

quasi-Einstein shrinking Finsler metric Ricci soliton Ricci flow 

MSC(2010)

53C60 53C44 

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Notes

Acknowledgements

The second author thanks School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran for the support.

References

  1. 1.
    Akbar-Zadeh H. Sur les espaces de Finsler a courbures sectionnelles constantes. Acad Roy Belg Bull Cl Sci (6), 1988, 74: 199–202MathSciNetzbMATHGoogle Scholar
  2. 2.
    Akbar-Zadeh H. Initiation to Global Finslerian Geometry. Netherlands: Elsevier Science, 2006zbMATHGoogle Scholar
  3. 3.
    Bao D. On two curvature-driven problems in Riemann-Finsler geometry. Adv Stud Pure Math, 2007, 48: 19–71MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bao D, Chern S S, Shen Z. An Introduction to Riemann-Finsler Geometry. New York: Springer, 2000CrossRefzbMATHGoogle Scholar
  5. 5.
    Bidabad B, Joharinad P. Conformal vector fields on complete Finsler spaces of constant Ricci curvature. Differential Geom Appl, 2014, 33: 75–84MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bidabad B, Sedaghat M K. Ricciow on Finsler surfaces. J Geom Phys, 2018, 129: 238–254MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bidabad B, Shahi A, Yar Ahmadi M. Deformation of Cartan curvature on Finsler manifolds. Bull Korean Math Soc, 2017, 54: 2119–2139MathSciNetGoogle Scholar
  8. 8.
    Bidabad B, Yar Ahmadi M. On quasi-Einstein Finsler spaces. Bull Iranian Math Soc, 2014, 40: 921–930MathSciNetzbMATHGoogle Scholar
  9. 9.
    Bidabad B, Yar Ahmadi M. On complete Yamabe solitons. Adv Geom, 2018, 1: 101–104MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Derdzinski A. A Myers-type theorem and compact Ricci solitons. Proc Amer Math Soc, 2006, 134: 3645–3648MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hamilton R S. Three-manifolds with positive Ricci curvature. J Differential Geom, 1982, 17: 255–306MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hamilton R S. The Ricciow on surfaces. Contemp Math, 1988, 71: 237–361CrossRefGoogle Scholar
  13. 13.
    López M F, Ro E G. A remark on compact Ricci solitons. Math Ann, 2008, 340: 893–896MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lott J. Some geometric properties of the Bakry-Emery-Ricci solitons. Comment Math Helv, 2003, 78: 865–883MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Wylie W. Complete shrinking Ricci solitons have finite fundamental group. Proc Amer Math Soc, 2008, 136: 1803–1806MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Yar Ahmadi M, Bidabad B. On compact Ricci solitons in Finsler geometry. C R Acad Sci Paris Ser I, 2015, 353: 1023–1027MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Yar Ahmadi M, Bidabad B. Convergence of Finslerian metrics under Ricciow. Sci China Math. 2016, 59: 741–750MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceAmirkabir University of TechnologyTehranIran

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