Finsler Surfaces and a Generalized Gauss-Bonnet Theorem

  • D. Bao
  • S.-S. Chern
  • Z. Shen
Part of the Graduate Texts in Mathematics book series (GTM, volume 200)


So far, our treatment has emphasized the use of natural coordinates. At the beginning of Chapter 2, we stated our policy that in important computations, we only use objects which are invariant under positive rescaling in y. Consequently, our treatment using natural coordinates on TM \0 can be regarded as occurring on the (projective) sphere bundle SM, in the context of homogeneous coordinates.


Bianchi Identity Orthonormal Frame Finsler Space Minkowski Plane Unit Speed 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AIM]
    P. L. Antonelli, R. S. Ingarden, and M. Matsumoto, The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology, FTPH 58, Kluwer Academic Publishers, 1993.Google Scholar
  2. [BC2]
    D. Bao and S. S. Chern, A note on the Gauss-Bonnet theorem for Finsler spaces, Ann. Math. 143 (1996), 233–252.MathSciNetMATHCrossRefGoogle Scholar
  3. [BCS3]
    D. Bao, S. S. Chern, and Z. Shen, Rigidity issues on Finsler surfaces, Rev. Roumaine Math. Pures Appl. 42 (1997), 707–735.MathSciNetGoogle Scholar
  4. [BL1]
    D. Bao and B. Lackey, Randers surfaces whose Laplacians have completely positive symbol, Nonlinear Analysis 38 (1999), 27–40.MathSciNetMATHCrossRefGoogle Scholar
  5. [BS]
    D. Bao and Z. Shen, On the volume of unit tangent spheres in a Finsler manifold, Results in Math. 26 (1994), 1–17.MathSciNetGoogle Scholar
  6. [Ber1]
    L. Berwald, Atti Congresso Internal dei Mate., Bologna 3-10, Sept. (1928).Google Scholar
  7. [Ch2]
    S. S. Chern, Historical remarks on Gauss-Bonnet, Analysis et Cetera, volume dedicated to Jürgen Moser, Academic Press, 1990, pp. 209–217.Google Scholar
  8. [doC1]
    M. P. do Carmo, Differential Forms and Applications, Springer-Verlag, 1994.Google Scholar
  9. [M2]
    M. Matsumoto, Foundations of Finsler Geometry and Special Finsler Spaces, Kaiseisha Press, Japan, 1986.MATHGoogle Scholar
  10. [M3]
    M. Matsumoto, Theory of curves in tangent planes of two-dimensional Finsler spaces, Tensor, N.S. 37 (1982), 35–42.MathSciNetMATHGoogle Scholar
  11. [On]
    B. O’Neill, Elementary Differential Geometry, 2nd ed., Academic Press, 1997.Google Scholar
  12. [R]
    H. Rund, The Differential Geometry of Finsler Spaces, Springer-Verlag, 1959.Google Scholar

Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • D. Bao
    • 1
  • S.-S. Chern
    • 2
  • Z. Shen
    • 3
  1. 1.Department of MathematicsUniversity of HoustonUniversity Park, HoustonUSA
  2. 2.Department of MathematicsUniversity of California at BerkeleyBerkeleyUSA
  3. 3.Department of Mathematical SciencesIndiana University-Purdue University IndianapolisIndianapolisUSA

Personalised recommendations