Existence and uniqueness of the Levi-Civita connection
= Existence and uniqueness of the Levi-Civita connection
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The <Koszul formula> uniquely determines $g(\nabla_XY,Z)$ for all <vector fields> $X,Y,Z$, so nondegeneracy of the <Riemannian metric> determines $\nabla_XY$. 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>.