Riemannian Manifolds and Two of Hopf’s Theorems

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


A Riemannian metric g on a manifold M is a family of inner products {g x}xM such that the quantities
$${g_{ij}}(x): = g\left( {\frac{\partial }{{\partial {x^i}}},\frac{\partial }{{\partial {x^i}}}} \right)$$
are smooth in local coordinates. The Finsler function F(x, y) of a Riemannian manifold has the characteristic structure
$$F(x,y) = \sqrt {{g_{ij(x)}}{y^i}{y^j}} $$


Manifold Assure Univer 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AL]
    P. L. Antonelli and B. Lackey (eds.), The Theory of Finslerian Laplacians and Applications, MAIA 459, Kluwer Academic Publishers, 1998.Google Scholar
  2. [BL1]
    D. Bao and B. Lackey, Randers surfaces whose Laplacians have completely positive symbol, Nonlinear Analysis 38 (1999), 27–40.MathSciNetMATHCrossRefGoogle Scholar
  3. [BL3]
    D. Bao and B. Lackey, A geometric inequality and a Weitzenböck formula for Finsler surfaces, The Theory of Finslerian Laplacians and Applications, P. Antonelli and B. Lackey (eds.), MAIA 459, Kluwer Academic Publishers, 1998, pp. 245–275.Google Scholar
  4. [CE]
    J. Cheeger and D. Ebin, Comparison Theorems in Riemannian Geometry, North Holland/American Elsevier, 1975.Google Scholar
  5. [GHL]
    S. Gallot, D. Hulin, and J. Lafontaine, Riemannian Geometry, Universitext, 2nd ed., Springer-Verlag, 1990.Google Scholar
  6. [J]
    J. Jost, Riemannian Geometry and Geometric Analysis, Universitext, Springer-Verlag, 1995.Google Scholar
  7. [KN1]
    S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, vol. I, Wiley-Interscience, 1963 (1996).Google Scholar
  8. [On]
    B. O’Neill, Elementary Differential Geometry, 2nd ed., Academic Press, 1997.Google Scholar
  9. [SS]
    J. Schouten and D. Struik, On some properties of general manifolds relating to Einstein’s theory of gravitation, Amer. J. Math. 43 (1921), 213–216.MathSciNetMATHCrossRefGoogle Scholar
  10. [SY]
    R. Schoen and S. T. Yau, Lectures on Differential Geometry, Conference Proceedings and Lecture Notes in Geometry and Topology, vol. I, International Press, 1994.Google Scholar
  11. [Wa]
    F. Warner, Foundations of Differentiable Manifolds and Lie Groups, Scott-Foresman, 1971.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