The fundamental theorem of Riemannian geometry states that there is a unique connection which satisfies these properties.
In the theory of Riemannian and pseudo-Riemannian manifolds the term covariant derivative is often used for the Levi-Civita connection. The components of this connection with respect to a system of local coordinates are called Christoffel symbols.
The Levi-Civita connection is named for Tullio Levi-Civita, although originally discovered by Elwin Bruno Christoffel. Levi-Civita, along with Gregorio Ricci-Curbastro, used Christoffel's connection to define a means of parallel transport and explore the relationship of parallel transport with the curvature, thus developing the modern notion of holonomy.
The unique connection satisfying these conditions has the form:
The Levi-Civita connection (like any affine connection) defines also a derivative along curves, sometimes denoted by .
Given a smooth curve on and a vector field along its derivative is defined by
In particular, is a vector field along the curve itself. If vanishes, the curve is called a geodesic of the covariant derivative. If the covariant derivative is the Levi-Civita connection of a certain metric, then the geodesics for the connection are precisely those geodesics of the metric that are parametrised proportionally to their arc length.
In general, parallel transport along a curve with respect to a connection defines isomorphisms between the tangent spaces at the points of the curve. If the connection is a Levi-Civita connection, then these isomorphisms are orthogonal – that is, they preserve the inner products on the various tangent spaces.
Denote by the differential of such a map. Then we have:
Lemma The formula
defines an affine connection on with vanishing torsion.
It is straightforward to prove that satisfies the Leibniz identity and is linear in the first variable. It is also a straightforward computation to show that this connection is torsion free.
So all that needs to be proved here is that the formula above does indeed define a vector field. That is, we need to prove that for all in
Consider the map
The map is constant, hence its differential vanishes. In particular
The equation (1) above follows.
In fact, this connection is the Levi-Civita connection for the metric on inherited from . Indeed, one can check that this connection preserves the metric.