Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 15 / 5 / a / i / Solution

An affine connection $\nabla$ is a metric connection for the Riemannian metric $g$ whenEquivalently the induced covariant derivative of the metric tensor satisfies $\nabla g=0$. This is metric compatibility; it imposes no condition on the torsion tensor. A Levi-Civita connection has the additional property of being a torsion-free connection.