I know that if a connection is compatible with a Riemannian metric, we have $X\langle Y,Z \rangle = \langle \nabla_XY,Z \rangle + \langle Y,\nabla_XZ \rangle$ for any vector fields $X,Y,Z$. But does the converse hold here?
To be more specific to my problem. I was trying to do the computation which verifies that the second Christoffel identity produces a connection compatible to the given metric.