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
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 .
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© 1987 Springer-Verlag Berlin Heidelberg
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Gallot, S., Hulin, D., Lafontaine, J. (1987). Curvature. In: Riemannian Geometry. Universitext. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-97026-9_3
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DOI: https://doi.org/10.1007/978-3-642-97026-9_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-17923-8
Online ISBN: 978-3-642-97026-9
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