How do I prove the following assertion:
Let $\nabla$ be a connection on a riemannian manifold. $\nabla$ is compatible with the metric if and only if for all vector fields $X,Y,Z$ we must have:
$X\langle Y,Z \rangle = \langle\nabla_X^Y,Z\rangle+\langle Y,\nabla_X^Z\rangle$