On a Riemannian manifold , there is a canonical connection called the Levi-Civita 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, 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 derivation algebras 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
.
Also, given a submanifold
it restricts to
to give the Levi-Civita connection from the restriction of
the metric to
.
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 curvature of
is given by
.