Suppose \(T\) is a covariant \(k\)-tensor field on a Riemannian manifold and \(R\) the Riemann curvature endomorphism. Then for all vector fields \(X,Y,V_1,\ldots, V_k\) the second covariant derivative of \(T\) satisfies the following identity

\begin{align*} \nabla^2_{X,Y}T(V_1,\ldots, V_k)=&\nabla^2_{Y,X}T(V_1,\ldots ,V_k) -T(R(X,Y)V_1,V_2,\ldots ,V_k)- \cdots -\\ &-T(V_1,\ldots ,V_{k-1}, R(X,Y)V_k) \end{align*}

[1, Theorem 7.14].

There is a more general version for \((l,k)\)-tensor fields in [1].

Links Link to heading

• curvature tensor
• second covariant derivative

See also Link to heading

• Ricci identity in local coordinates
• visualization using Penrose notation
• \(R\) measures non-flatness of a manifold

References Link to heading

1. J. Lee, Introduction to Riemannian Manifolds. Cham: Springer International Publishing, 2018. doi:10.1007/978-3-319-91755-9