Connection (principal bundle)

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In differential geometry , the relationship is a concept that can be used to explain parallel transport between the fibers of a principal bundle . In physics, such relationships are used to describe fields in the Yang-Mills theories .


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


Also denote the Lie algebra of the Lie group .

A relationship is then a -value 1-form , which is -equivariant and whose restriction to the fibers corresponds to the Maurer-Cartan form . So the following two conditions should be met:

for all


for everyone .

Here is defined by . denotes the differential of . is the adjoint effect and is the so-called fundamental vector field . It will go through


on defined.


The curvature of a connected form is defined by

Here the commutator is Lie algebra valued differential forms by

and the outer derivative through

Are defined.

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

Bianchi identity

Relationship and curvature form satisfy the equation


Horizontal subspaces

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


The horizontal subspaces are transverse to the tangent spaces of the fibers of , and they are -invariant, i.e. H. for everyone .

From the horizontal subspaces can form the connection recover (for identification of the tangent of the fiber ) by projection of along the tangent of the fiber.

Parallel transport

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

In particular, you have one through each way

defined figure


the so-called parallel transport along the way .

At one point , the holonomy group is defined as a subset of the diffeomorphisms of the fiber as follows. To a closed path with and one there is a clear elevation with and we define . The group that is for everyone is the holonomy group.

Riemannian connection

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

Let be the matrix that by using a local base

is defined, where the Levi-Civita context is, so is by

defines the Riemannian form of connection. It applies



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