Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 115 / 4 / b / Solution

The connection is orthogonal, or metric-compatible, when $\nabla g=0$; equivalently, its parallel transport preserves the Riemannian metric. Compatibility of the induced connections with tensor contraction givesfor all vector fields $U,V$. Taking $U={\partial }_j$ and $V={\partial }_k$ yieldsConversely, this coordinate identity makes every component of $\nabla g$ vanish, so it is equivalent to orthogonality.