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Curvature

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

Abstract

Let Z be a vector field on a Riemannian manifold (M,g). From 2.58 the covariant derivative of the (1,1)-tensor DZ is the (1,2) tensor defined by
$$\left( {D_{x,y}^2} \right){Z_m}\; = \;{D_x}{\left( {{D_y}Z} \right)_m} - \left( {{D_{{D_z}Y}}} \right){Z_m}$$
(3.1)
where Y is a vector field such that Y m = y. We already met in 2.64 the second covariant derivative of a function, which is a symmetric 2-tensor. This property is no more true for the second derivative of a tensor. However, \({\left( {D_{x,y}^2Z - D_{y,z}^2Z} \right)_m}\) only depends on Z m .

Keywords

Vector Field Riemannian Manifold Covariant Derivative Sectional Curvature Curvature Tensor 
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 1987

Authors and Affiliations

  • Sylvestre Gallot
    • 1
  • Dominique Hulin
    • 2
  • Jacques Lafontaine
    • 3
  1. 1.Université de SavoieChambéry CedexFrance
  2. 2.Centre d’Orsay, MathématiqueUniversité Paris11Orsay CedexFrance
  3. 3.U.F.R. de MathématiquesUniversité Paris 7Paris Cedex 05France

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