Levi-Civita Connection

The Levi-Civita connection on a Riemannian manifold is a canonical connection (pronounced lē-vē shi-vit-e), sometimes also known as the Riemannian connection or covariant derivative. As a connection on the tangent bundle, it provides a well-defined method for differentiating vector fields, differential forms, or any other kind of tensor. The theorem asserting the existence of the Levi-Civita connection, which is the unique torsion-free connection on the tangent bundle compatible with the metric, is called the fundamental theorem of Riemannian geometry.

These properties can be described as follows. Let , , and be any vector fields, and denote the metric. Recall that vector fields act as derivations on the ring of smooth functions by the directional derivative, and that this action extends to an action on vector fields. The notation is the commutator of vector fields, . The Levi-Civita connection is torsion-free, meaning

(1)

and is compatible with the metric

(2)

In coordinates, the Levi-Civita connection can be described using the Christoffel symbols of the second kind . In particular, if , then

(3)

or in other words,

(4)

As a connection on the tangent bundle , it induces a connection on the dual bundle and on all their module tensor products . For a submanifold , the tangential orthogonal projection of , for tangent vector fields on , gives the Levi-Civita connection of the induced Riemannian metric on .

The Levi-Civita connection can be used to describe many intrinsic geometric objects. For instance, a path is a geodesic iff where is the path's tangent vector. On a more general path , the equation defines parallel transport for a vector field along . The second fundamental form of a submanifold is given by where is the tangent bundle of and is projection onto the normal bundle . The Riemann tensor of is given by

(5)