= Double divergence of the Riemann tensor
{title2=$R^{ik}{}_{mn;ik}=0$}
For the <Levi-Civita connection>, the <second Bianchi identity> gives $\nabla_iR^{ik}{}_{mn}=\nabla_mR^k{}_n-\nabla_nR^k{}_m$. Applying another <covariant derivative> and the <contracted Bianchi identity> reduces its divergence to a commuting scalar Hessian plus curvature commutators on the <Ricci tensor>. The latter are the difference of symmetric tensors: a square of the <Ricci tensor> and $R^{ab}R_{anbm}$, which is symmetric in $m,n$ by curvature pair exchange and Ricci symmetry. Both differences vanish. This identity does not require vacuum field equations and is independent of the overall curvature sign convention.
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