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


3.1 A parallel vector field in R2 is just a constant field. Now, on a surface, there are generally no (even local) parallel vector fields. How much the parallel transport of a field along a small closed curve differ from the identity is measured in terms of the curvature of the surface, a function k: MR. Now, on an n-dimensional manifold, the effect of the parallel transport along small closed curves lying in different “2-planes” depends on these very planes, and actually involves a (1, 3)-tensor, the curvature tensor of the Riemannian manifold.


Riemannian Manifold Sectional Curvature Curvature Tensor Constant Curvature Ricci Curvature 
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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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