Constant Flag Curvature Spaces and Akbar-Zadeh’s Theorem

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

Abstract

In §3.9, we encountered the flag curvature. As the name suggests, this quantity (denoted K) involves a location x ϵ M, a flagpole ℓ:= with y ϵ T x M, and a transverse edge V ϵ T x M. The precise formula is quite elegantly given by (3.9.3):
$$K(\ell ,V): = \frac{{{V^i}({\ell ^j}{R_{jikl}}{\ell ^l}){V^k}}}{{g(\ell ,\ell )g(V,V) - {{[g(\ell ,V)]}^2}}} = \frac{{{V^i}{R_{ik}}{V^k}}}{{g(V,V) - {{[g(\ell ,V)]}^2}}}$$
.

Keywords

Manifold Assure Verse sinO 

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References

  1. [AZ]
    H. Akbar-Zadeh, Sur les espaces de Finsler à courbures sectionnelles constantes, Acad. Roy. Belg. Bull. Cl. Sci. (5) 74 (1988), 281–322.MathSciNetMATHGoogle Scholar
  2. [Br1]
    R. Bryant, Finsler structures on the 2-sphere satisfying K = 1, Cont. Math. 196 (1996), 27–42.CrossRefGoogle Scholar
  3. [Br2]
    R. Bryant, Projectively flat Finsler 2-spheres of constant curvature, Selecta Mathematica, N.S. 3 (1997), 161–203.MATHCrossRefGoogle Scholar
  4. [Br3]
    R. Bryant, Finsler surfaces with prescribed curvature conditions, Aisenstadt Lectures, in preparation.Google Scholar
  5. [F1]
    P. Funk, Über zweidimensionale Finslersche Räume, insbesondere über solche mit geradlinigen Extremalen und positiver konstanter Krümmung, Math. Zeitschr. 40 (1936), 86–93.MathSciNetCrossRefGoogle Scholar
  6. [F2]
    P. Funk, Eine Kennzeichnung der zweidimensionalen elliptischen Geometric, Österreichische Akad. der Wiss. Math., Sitzungsberichte Abteilung II 172 (1963), 251–269.MathSciNetMATHGoogle Scholar
  7. [Gr]
    A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed., CRC Press, 1998.Google Scholar
  8. [M5]
    M. Matsumoto, Randers spaces of constant curvature, Rep. on Math. Phys. 28 (1989), 249–261.MATHCrossRefGoogle Scholar
  9. [Num]
    S. Numata, On Landsberg spaces of scalar curvature, J. Korean Math. Soc. 12 (1975), 97–100.MathSciNetMATHGoogle Scholar
  10. [Ok]
    T. Okada, On models of projectively flat Finsler spaces of constant negative curvature, Tensor, N.S. 40 (1983), 117–124.MathSciNetMATHGoogle Scholar
  11. [On]
    B. O’Neill, Elementary Differential Geometry, 2nd ed., Academic Press, 1997.Google Scholar
  12. [Op]
    J. Oprea, Differential Geometry and its Applications, Prentice-Hall, 1997.Google Scholar
  13. [SK]
    C. Shibata and M. Kitayama, On Finsler spaces of constant positive curvature, Proceedings of the Romanian-Japanese Colloquium on Finsler Geometry, Braşov, 1984, pp. 139–156.Google Scholar
  14. [YS]
    H. Yasuda and H. Shimada, On Randers spaces of scalar curvature, Rep. on Math. Phys. 11 (1977), 347–360.MathSciNetMATHCrossRefGoogle 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

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