Existence and uniqueness of the Levi-Civita connection

The Koszul formula uniquely determines $g({\nabla }_{X}Y,Z)$ for all vector fields $X,Y,Z$, so nondegeneracy of the Riemannian metric determines ${\nabla }_{X}Y$. The resulting operation is $C^{\infty }$-linear in $X$, obeys the Leibniz rule in $Y$, is compatible with the metric, and is torsion-free. It is therefore the unique Levi-Civita connection.