I'm asked to prove the following statement in my physics book:
A vector field with covariant components $v^b$, in order to have a vanishing covariant derivative everywhere in a manifold, must satisfy: $$(\partial_b\Gamma^{d}{}_{ac}-\partial_c\Gamma^{d}{}_{ab}+\Gamma^{e}{}_{ac}\Gamma^{d}{}_{eb}-\Gamma^{e}{}_{ab}\Gamma^{d}{}_{ec})v^a=0$$
I've tried taking the covariant derivative $\nabla_av^b=\partial_a v^b+\Gamma^b{}_{ca}v^c$ and equating this to zero, and I think I might be able to get something by taking double derivatives, since it would be a way to get the gamma coefficients derived... But I haven't been able to get anything clear. Any help will be appreciated!