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


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
$$\left( {D_{x,y}^2} \right){Z_m}\; = \;{D_x}{\left( {{D_y}Z} \right)_m} - \left( {{D_{{D_z}Y}}} \right){Z_m}$$
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 .


Vector Field Riemannian Manifold Covariant Derivative Sectional Curvature Curvature Tensor 
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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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