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Riemannian Metrics

  • Sylvestre Gallot
  • Dominique Hulin
  • Jacques Lafontaine
Part of the Universitext book series (UTX)

Abstract

The Pythagorus theorem just says that the squared length of an infinitesimal vector, say in R 3, whose components are dx, dy and dz, is dx 2 + dy 2 + dz 2. Thus, the length of a parameterized curve c(t) = (x(t), y(t), z(t)) is given by the integral
$$ \int {ds = \int {{{\left( {{x^{{'2}}} + {y^{{'2}}} + {z^{{'2}}}} \right)}^{{1/2}}}} } dt $$
.

Keywords

Vector Field Riemannian Manifold Homogeneous Space Parallel Transport Riemannian Metrics 
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-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  • Sylvestre Gallot
    • 1
  • Dominique Hulin
    • 1
  • Jacques Lafontaine
    • 2
  1. 1.Ecole Polytechnique, Unité de Recherche Associée du CNRS D 0169Centre de MathématiquesPalaiseau CedexFrance
  2. 2.Départment de Mathématiques, GETODIM - Unité de Recherche Associée du CNRS 1407Université de MontpellierMontpellier Cedex 5France

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