Advertisement

Curvature

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

Abstract

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.

Keywords

Riemannian Manifold Sectional Curvature Curvature Tensor Constant Curvature Ricci Curvature 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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

Personalised recommendations