computation previously done in local coordinates can be written as This reduces immediately to the first Christoffel identity in the case that , , and are coordinate vector fields. The equations displayed above can be rearranged to produce the Koszul formula or identity This proves the uniqueness of a torsion-free and metric-compatible condition, since if is equal to for arbitrary , then must equal . This is a consequence of the non-degeneracy of the metric. In the local formulation above, this key property of the metric was implicitly used, in the same way, via the existence of . Furthermore, by the same reasoning, the Koszul formula can be used to define a vector field when given and , and it is routine to check that this defines a connection that is torsion-free and metric-compatible. Notes References Articles containing proofs Connection (mathematics) Riemannian geometry Riemannian manifolds Theorems in Riemannian geometry