Fundamental theorem of Riemannian geometry

Fundamental Theorem of Riemannian Geometry. Let be a Riemannian manifold. Then there is a unique connection ∇ which satisfies the following conditions:

• for any vector fields X, Y, Z we have
• for any vector fields X, Y,

The first condition means that the metric tensor is preserved by parallel transport, while the second condition expresses the fact that the torsion of ∇ is zero.
An extension of the fundamental theorem states that given a pseudo-Riemannian manifold there is a unique connection preserving the metric tensor with any given vector-valued 2-form as its torsion. The difference between an arbitrary connection and the corresponding Levi-Civita connection is the contorsion tensor.
The following technical proof presents a formula for Christoffel symbols of the connection in a local coordinate system. For a given metric this set of equations can become rather complicated. There are quicker and simpler methods to obtain the Christoffel symbols for a given metric, e.g. using the action integral and the associated Euler-Lagrange equations.

Geodesics defined by a metric or a connection

A metric defines the curves which are geodesics ; but a connection also defines the geodesics. A connection is said to be equal to another in two different ways:

• obviously if for every vectors fields
• if and define the same geodesics and have the same torsion

This means that two different connections can lead to the same geodesics while giving different results for some vector fields.
Because a metric also defines the geodesics of a differential manifold, for some metric there is not only one connection defining the same geodesics , and given a metric, the only connection which defines the same geodesics and which is torsion-free is the Levi-Civita connection.

Proof of the theorem

Let m be the dimension of M and, in some local chart, consider the standard coordinate vector fields
Locally, the entry g_ij 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
This is the first Christoffel identity.
Since
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.

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.