covariant derivative vs. exterior derivative

Question

I have the following question. Let $M$ be a Riemannian manifold with metric $g$ and $\nabla$ the Levi-Civita connection. Let furthermore $\alpha \in \Omega^{k}(M)$ be a $k$-form such that $\nabla \alpha = 0$. Why is then $d \alpha = 0$?

thanks, jan

Answer 1

I'm not sure what sort of approach you're looking for, but in coordinates, it is easy to see that (ignoring possible combinatoric factors) the antisymmetrization of a covariant derivative kills off everything except the partial derivative which corresponds to the exterior derivative in coordinates. This works for any torsion free connection.