Connection (principal bundle)

In the differential geometry of the connection is a concept that can be explained between the fibers of a principal bundle with the parallel transport. In physics such relationships for the description of fields in the Yang-Mills theories are used.

Definition

Be a principal bundle with the structure group. The group works through

Furthermore, the Lie algebra of the Lie group call.

A connection is then a one -valent form, which is invariant and their restriction corresponds to the fibers by the Mason Cartan form. So there are the following two conditions be met:

And

This is defined by. denotes the differential of. is the adjoint action and is called the fundamental vector field. It is

To define.

Curvature

The curvature of a connection form is defined by

Here, the commutator Lie algebra -valued differential forms is by

And the exterior derivative by

Defined.

The curvature form is invariant and therefore defines a 2- form on.

Bianchi identity

Connection and curvature form satisfy the equation

Horizontal subspaces

For a connection form on a principal bundle the horizontal subspaces are defined by

The horizontal sub-spaces are transverse to the tangent of the fibers, and they are invariant, that is, for everyone.

From the horizontal subspaces can be the connection form to recover ( after identification of the tangent space of the fiber ) by projecting along the tangent space of the fiber.

Parallel transport

For each path and each is available with a path and. ( This follows from the existence and uniqueness theorem for ordinary differential equations. )

In particular, it is every way by a

Defined mapping

The so-called parallel transport along the path.

At one point we define the holonomy group as a subgroup of diffeomorphisms of the fiber as follows. There is a closed path with and one with a unique high elevation and we define. The group of all is the holonomy group.

Riemannian connection

For a Riemannian manifold, the frame bundle is a principal bundle with the linear group.

Be the matrix by using a local base

Is defined, where the Levi- Civita connection is, so is

The Riemannian connection form defined. It is