Curvature

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 .