Skip to main content
SpringerLink
Log in

Convergence of Finslerian metrics under Ricci flow

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

In this work, we study the convergence of evolving Finslerian metrics first in a general flow and next under Finslerian Ricci flow. More intuitively it is proved that a family of Finslerian metrics g(t) which are solutions to the Finslerian Ricci flow converges in C ∞ to a smooth limit Finslerian metric as t approaches the finite time T. As a consequence of this result one can show that in a compact Finsler manifold the curvature tensor along the Ricci flow blows up in a short time.

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. Akbar-Zadeh H. Sur les espaces de Finsler a courbures sectionnelles constantes. Acad Roy Belg Bull Cl Sci, 1988, 74: 281–322

    MathSciNet  MATH  Google Scholar 

  2. Akbar-Zadeh H. Initiation to Global Finslerian Geometry. Netherlands: Elsevier Science, 2006

    MATH  Google Scholar 

  3. Bao D. On two curvature-driven problems in Riemann-Finsler geometry. Adv Stud Pure Math, 2007, 48: 19–71

    MathSciNet  MATH  Google Scholar 

  4. Bao D, Chern S S, Shen Z. An Introduction to Riemann-Finsler Geometry. New York: Springer, 2000

    Book  MATH  Google Scholar 

  5. Bidabad B, Yar Ahmadi M. On quasi-Einstein Finsler spaces. Bull Iranian Math Soc, 2014, 40: 921–930

    MathSciNet  Google Scholar 

  6. Brendle S. Ricci Flow and the Sphere Theorem. Providence, RI: Amer Math Soc, 2010

    Book  MATH  Google Scholar 

  7. Chow B, Knopf D. The Ricci Flow: An Introduction. Providence, RI: Amer Math Soc, 2004

    Book  MATH  Google Scholar 

  8. Hamilton R S. Three-manifolds with positive Ricci curvature. J Differential Geom, 1982, 17: 255–306

    MathSciNet  MATH  Google Scholar 

  9. Hopper C, Andrews B. The Ricci Flow in Riemannian Geometry. New York: Springer, 2010

    MATH  Google Scholar 

  10. Sesum N. Curvature tensor under the Ricci flow. Amer J math, 2005, 127: 1315–1324

    Article  MathSciNet  MATH  Google Scholar 

  11. Shen Z. Lectures on Finsler Geometry. Singapore: World Scientific, 2001

    Book  MATH  Google Scholar 

  12. Shi W X. Deforming the metric on complete Riemannian manifolds. J Differential Geom, 1989, 30: 223–301

    MathSciNet  MATH  Google Scholar 

  13. Yar Ahmadi M, Bidabad B. On compact Ricci solitons in Finsler geometry. C R Acad Sci Paris Ser I, 2015, 353: 1023–1027

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, 15914, Iran

    Mohamad Yar Ahmadi & Behroz Bidabad

Authors
  1. Mohamad Yar Ahmadi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Behroz Bidabad
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Behroz Bidabad.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yar Ahmadi, M., Bidabad, B. Convergence of Finslerian metrics under Ricci flow. Sci. China Math. 59, 741–750 (2016). https://doi.org/10.1007/s11425-015-5092-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-015-5092-3

Keywords

MSC(2010)

Search

Navigation