= Commutator of covariant derivatives
{title2=$[\nabla_\alpha,\nabla_\beta]V^\mu=R^\mu{}_{\nu\alpha\beta}V^\nu$}
= Curvature commutator
{synonym}
For a <torsion-free connection>, the <commutator> of coordinate <covariant derivatives> vanishes on a <scalar field> and acts on a <vector field> through the displayed <Riemann curvature tensor>. On a <covector field> it instead gives $[\nabla_\alpha,\nabla_\beta]A_\mu=-R^\nu{}_{\mu\alpha\beta}A_\nu$. The <curvature commutator is a tensor derivation>, so each index of a general <tensor> contributes independently with the corresponding sign. Including the lower derivative index in the second <covariant derivative> is essential for this identity.
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