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Riemannian metrics

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

Abstract

Pythagoras theorem says that the squared length of an infinitesimal vector, say in R3, whose components are dx, dy and dz, is dx2 +dy2 +dz2. Thus, the length of a parametrized curve c(t) = (x(t), y(t), z(t)) is given by the integral
$$ \int {ds = \int {(x'^2 + } } y'^2 + z'^2 )^{1/2} dt. $$

Keywords

Riemannian Manifold Isometry Group Riemannian Metrics Complete Riemannian Manifold Klein Bottle 
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 2004

Authors and Affiliations

  • Sylvestre Gallot
    • 1
  • Dominique Hulin
    • 2
  • Jacques Lafontaine
    • 3
  1. 1.Institut Fourier, C.N.R.S., UMR 5582Université Grenoble 1Saint-Martin d’HèresFrance
  2. 2.Département de MathématiquesUniversité Paris XIOrsay CXFrance
  3. 3.Département de Mathématiques, C.N.R.S., UMR 5149Université Montpellier IIMontpellier CX 05France

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