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 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
The curvature of a connected form is defined by
Here the commutator is Lie algebra valued differential forms by
and the outer derivative through
The curve form is -invariant and therefore defines a 2-form on .
Relationship and curvature form satisfy the equation
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.
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
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.
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