Equivalent conditions for a linear connection $\nabla$ to be compatible with Riemannian metric $g$

Question

I am reading John M. Lee's Riemannian Manifolds: An Introduction to Curvature. In Lemma $5.2$, it is said that the following conditions are equivalent for a linear connection $\nabla$ on a Riemannian manifold:

(a) $\nabla$ is compatible with $g$. i.e., for any vector fields $X,Y,Z$, $$ \nabla_X g(Y,Z) = g(\nabla_X Y,Z) + g(Y,\nabla_X Z) $$ (b) $\nabla g\equiv 0.$

How do we go from (a) to (b) (and (b) to (a))?

Answer 1

Hint: Use Lemma $4.6$ (ii) (i.e. the formula displayed here) to deduce that

$$(\nabla_X g)(Y, Z) = \nabla_Xg(Y, Z) - g(\nabla_XY, Z) - g(Y, \nabla_XZ).$$

---

As $g(Y, Z)$ is a smooth function, $\nabla_Xg(Y, Z) = Xg(Y, Z)$ so the metric compatibility condition can also be written as

$$Xg(Y, Z) = g(\nabla_XY, Z) + g(Y, \nabla_XZ).$$