Curvature commutator on a covariant tensor
= Curvature commutator on a covariant tensor
= Ricci identity
{c}
{synonym}
For a rank-two covariant tensor, the derivative commutator contributes one curvature action on each index:
$$
[\nabla_\alpha,\nabla_\beta]W_{\mu\nu}
=-R^\rho{}_{\mu\alpha\beta}W_{\rho\nu}
-R^\rho{}_{\nu\alpha\beta}W_{\mu\rho}.
$$