Tensor index antisymmetrization
= Tensor index antisymmetrization
{title2=$T_{[ab]}$}
Antisymmetrization over selected <tensor> indices averages their permutations with the <sign of a permutation>. Square brackets denote it, for example $T_{[ab]}=(T_{ab}-T_{ba})/2$. It projects onto the alternating part. Antisymmetrized second derivatives of a scalar vanish for a <torsion-free connection>, but antisymmetrized later derivatives act on a tensor and can produce <curvature>.