Fundamental Theorem of Riemannian Geometry

In Riemannian geometry, the fundamental theorem of Riemannian geometry states that on any Riemannian manifold (or pseudo-Riemannian manifold) there is a unique torsion-free metric connection, called the Levi-Civita connection of the given metric. Here a metric (or Riemannian) connection is a connection which preserves the metric tensor.

More precisely:

Let be a Riemannian manifold (or pseudo-Riemannian manifold). Then there is a unique connection which satisfies the following conditions:

1. for any vector fields we have, where denotes the derivative of the function along vector field .
2. for any vector fields, ,
   where denotes the Lie bracket for vector fields .

(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 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.