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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

Riemannian Manifold Sectional Curvature Bianchi Identity Conformal Factor Warped Product 
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.

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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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