Skip to main content
SpringerLink
Log in

On symmetric Finsler spaces

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we study symmetric Finsler spaces. We first study some geometric properties of globally symmetric Finsler spaces and prove that any such space can be written as a coset space of a Lie group with an invariant Finsler metric. Then we prove that a globally symmetric Finsler space is a Berwald space. As an application, we use the notion of Minkowski symmetric Lie algebras to give an algebraic description of symmetric Finsler spaces and obtain the formulas for flag curvature and Ricci scalar. Finally, some rigidity results of locally symmetric Finsler spaces related to the flag curvature are also given.

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

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. W. Ambrose and I. M. Singer, A theorem on holonomy, Transactions of the American Mathematical Society 75 (1953), 428–443.

    Article  MathSciNet  MATH  Google Scholar 

  2. P. L. Antonelli, R. S. Ingarden and M. Matsumoto, The Theory of Sprays and Finsler spaces with applications in Physics and Biology, Kluwer Academic Publishers, Dordrecht, 1993.

    MATH  Google Scholar 

  3. D. Bao and S. S. Chern, On a notable connection in Finsler geometry, Houston Journal of Mathematics 19 (1993), 135–180.

    MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  5. D. Bao, C. Robles and Z. Shen, Zermelo navigation on Riemannian manifolds, Journal of Differential Geometry 66 (2004), 377–435.

    MATH  MathSciNet  Google Scholar 

  6. H. Busemann and B. Phadke, Two theorems on general symmetric spaces, Pacific Journal of Mathematics 92 (1981), 39–48.

    MATH  MathSciNet  Google Scholar 

  7. S. S. Chern, Z. Shen,, Riemann-Finsler Geometry, WorldScientific, Singapore, 2004.

    MATH  Google Scholar 

  8. S. Deng and Z. Hou, The group of isometries of a Finsler space, Pacific Journal of Mathematics 207 (2002), 149–155.

    Article  MATH  MathSciNet  Google Scholar 

  9. S. Deng and Z. Hou, Invariant Finsler metrics on homogeneous manifolds, Journal of Physics A: Mathematical and General 37 (2004), 8245–8253.

    Article  MATH  MathSciNet  Google Scholar 

  10. S. Deng and Z. Hou, On locally and globally symmetric Berwald spaces, Journal of Physics A: Mathematical and General 38 (2005), 1691–1697.

    Article  MathSciNet  Google Scholar 

  11. S. Deng and Z. Hou, Minkowski symmetric Lie algebras and symmetric Berwald spaces, Geometriae Dedicata 113 (2005), 95–105.

    Article  MATH  MathSciNet  Google Scholar 

  12. P. Foulon, Curvature and global rigidity in Finsler manifolds, Houston Journal of Mathematics 28 (2002), 263–292.

    MATH  MathSciNet  Google Scholar 

  13. S. Helgason, Differential Geometry, Lie groups and Symmetric Spaces, 2nd ed., Academic Press, 1978.

  14. S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Interscience Publishers, Vol. 1, 1963, Vol. 2, 1969.

  15. Z. Shen, Differential Geometry of Spray and Finsler Spaces, Kluwer Academic Publishers, Dordrecht, 2001.

    MATH  Google Scholar 

  16. Z. I. Szabó, Positive Definite Berwald Spaces, The Tensor Society, Tensor New Series 38 (1981), 25–39.

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Nankai University, Tianjin, 300071, P. R. China

    Shaoqiang Deng & Zixin Hou

Authors
  1. Shaoqiang Deng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zixin Hou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shaoqiang Deng.

Additional information

Members of LPMC and supported by NSFC (No. 10671096) and NCET of China.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Deng, S., Hou, Z. On symmetric Finsler spaces. Isr. J. Math. 162, 197–219 (2007). https://doi.org/10.1007/s11856-007-0095-6

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-007-0095-6

Keywords

Search

Navigation