Fundamental Theorem of Riemannian Geometry

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:

$\begin{matrix} 2 g(\nabla_XY, Z) =& \partial_X (g(Y,Z)) + \partial_Y (g(X,Z)) - \partial_Z (g(X,Y))\\ {} & {}+ g(,Z) - g(,Y) - g(,X). \end{matrix}$

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.

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