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:

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 .