Fundamental Theorem of Riemannian Geometry - The Koszul Formula

The Koszul Formula

An alternative proof of the Fundamental theorem of Riemannian geometry proceeds by showing that a torsion-free metric connection on a Riemannian manifold is necessarily given by the Koszul formula:

This proves the uniqueness of the Levi-Civita connection. Existence is proven by showing that this expression is tensorial in X and Z, satisfies the Leibniz rule in Y, and that hence defines a connection. This is a metric connection, because the symmetric part of the formula in Y and Z is the first term on the first line; it is torsion-free because the anti-symmetric part of the formula in X and Y is the first term on the second line.