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)

Abstract

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
$$F(x,y) = \sqrt {{g_{ij(x)}}{y^i}{y^j}} $$
.

Keywords

Manifold Assure Univer 

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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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