Suppose is a differentiable manifold of dimension
.
Connection on vector bundles
Definition 1. A connection on a vector bundle
is a map
which satisfies the following conditions
- For any
,
.
- For
and any
,
.
If is a tangent vector field on
(i.e. a section of the tangent bundle
) one can define a covariant derivative along
, denoted by
, as follows
where represents the pairing between
and
.
Locally, a connection is given by a set of differential 1-forms. Suppose is a coordinate neighborhood of
with local coordinates
,
. Choose
smooth sections
of
on
such that they are linearly independent everywhere. Such a set of
sections is called a local frame field of
on
. It is obvious that at every point
forms a basis for the tensor space . Because
is a local section on
, we can write
where are smooth functions on
. Denote
then
.
Definition 2 (curvature operator). Suppose
are two arbitrary smooth tangent vector fields on the manifold
. Then
is the curvature operator of the connection
.
Obviously, has the following properties
,
,
,
where ,
and
.
Connection on tangent bundles (affine connections)
A tangent bundle is an
-dimensional vector bundle determined intrinsically by the differentiable structure of an
-dimensional smooth manifold
. A connection of
is called an affine connection on
. Affine connection is usually denoted by
.
Definition 3 (curvature tensor). The curvature tensor is a (1,3)-tensor defined by
.
In local coordinates, the curvature tensor is given by
.
A simple calculation shows us that
.
Definition 4 (torsion tensor). The torsion tensor is a (1,2)-tensor defined by
.
In local coordinates, the torsion tensor is given by
.
A simple calculation shows us that
.
Definition 5 (torsion free). If the torsion tensor of an affine connection
is zero, then the connection is said to be torsion free.
When is a Riemannian manifold with metric
then we have the following definition
Definition 6 (Levi-Civita connection). An affine connection
is called a Levi-Civita connection if:
- It preserves the metric, i.e., for any vector fields
,
,
we have
where
denotes the derivative of the function
along the vector field
.
- It is torsion free.
The first condition above is called metric connection condition. Thus, the Levi-Civita connection is the torsion free metric connection, i.e., the torsion free connection on the tangent bundle (an affine connection) preserving a given Riemannian metric.
There is a theorem in the literature saying that the Levi-Civita connection is unique and it is given by the following identity
.
In local coordinate, the Levi-Civita connection is given by
where are called Christoffel symbols which are determined by
where .