= Solution
An <affine connection> $\nabla$ is a <metric connection> for the <Riemannian metric> $g$ when
$$
\boxed{X\bigl(g(Y,Z)\bigr)=g(\nabla_XY,Z)+g(Y,\nabla_XZ)
\quad\text{for all vector fields }X,Y,Z.}
$$
Equivalently 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>.
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