Compatible connection over a riemannian manifold

Question

How do I prove the following assertion:

Let $\nabla$ be a connection on a riemannian manifold, then $\nabla$ is compatible with the metric if and only if for all $X,Y,Z\in \mathfrak{X}(M)=\Gamma(TM)$ we have:

$X\langle Y,Z \rangle = \langle\nabla_XY,Z\rangle+\langle Y,\nabla_XZ\rangle$

Answer 1

Look at the Proposition 3.2 and Corollary 3.3