Problem 2, chapter 3 of Wald's General Relativity. The details don't matter much, but it is given a totally anti-symetric tensor field Eab such that EabEab=2(-1)^(s), s being the signature of the metric. I have checked a solution to the exercise, and somewhere during the development there is the following reasoning:
[tex]\nabla[/tex]cEabEab=0 (because EabEab is contant);
This implies that:
2Eab[tex]\nabla[/tex]cEab = 0 (applying Leibniz rule and noting that [tex]\nabla[/tex]gab = 0);
And then, the reasoning goes on saying that, because Eab is totally anti-symetric and non-vanishing, we can conclude that:
[tex]\nabla[/tex]cEab = 0
Which doesn't make sense to me. Can anyone explain to me the last line, please?
The Attempt at a Solution