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: 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 (with torsion) 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. A metric defines the curves which are geodesics ; but a connection also defines the geodesics (see also parallel transport). A connection ∇ ¯ {displaystyle {ar { abla }}} is said to be equal to another ∇ {displaystyle abla } in two different ways: