Double divergence of a contravariant tensor
= Double divergence of a contravariant tensor
{title2=$T^{ab}{}_{;ab}=T^{ab}{}_{;ba}$}
For a <Levi-Civita connection>, a contravariant rank-two <tensor> satisfies $T^{ab}{}_{;ab}=T^{ab}{}_{;ba}$ without itself being symmetric. The <Ricci identity> gives their difference as $R_{pb}(T^{pb}-T^{bp})$, which vanishes because the <Ricci tensor> is symmetric. Zero torsion alone for a general affine connection does not ensure this symmetry.