Fundamental Theorem of Riemannian Geometry - Proof

Proof

Let m be the dimension of M and, in some local chart, consider the standard coordinate vector fields

Locally, the entry gi j of the metric tensor is then given by

To specify the connection it is enough to specify, for all i, j, and k,

We also recall that, locally, a connection is given by m3 smooth functions {}, where

The torsion-free property means

On the other hand, compatibility with the Riemannian metric implies that

For a fixed, i, j, and k, permutation gives 3 equations with 6 unknowns. The torsion free assumption reduces the number of variables to 3. Solving the resulting system of 3 linear equations gives unique solutions

\langle \nabla_{ \partial_i }\partial_j, \partial_k \rangle = \frac{1}{2}( \partial_i g_{jk}- \partial_k g_{ij} + \partial_j g_{ik}).

This is the first Christoffel identity.

Since

\langle \nabla_{ \partial_i }\partial_j, \partial_k \rangle = \Gamma^l _{ij} g_{lk},

Where we use Einstein summation convention. That is, an index repeated subscript and superscript implies that it is summed over all values. inverting the metric tensor gives the second Christoffel identity:

Once again, with Einstein summation convention. The resulting unique connection is called the Levi-Civita connection.

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