Connection (composite bundle)

Composite bundles play a prominent role in gauge theory with symmetry breaking, e.g., gauge gravitation theory, non-autonomous mechanics where is the time axis, e.g., mechanics with time-dependent parameters, and so on. There are the important relations between connections on fiber bundles, and.

Composite bundle

In differential geometry by a composite bundle is meant the composition
of fiber bundles
It is provided with bundle coordinates, where are bundle coordinates on a fiber bundle, i.e., transition functions of coordinates are independent of coordinates.
The following fact provides the above-mentioned physical applications of composite bundles. Given the composite bundle, let be a global section
of a fiber bundle, if any. Then the pullback bundle
over is a subbundle of a fiber bundle.

Composite principal bundle

For instance, let be a principal bundle with a structure Lie group which is reducible to its closed subgroup. There is a composite bundle where is a principal bundle with a structure group and is a fiber bundle associated with. Given a global section of, the pullback bundle is a reduced principal subbundle of with a structure group. In gauge theory, sections of are treated as classical Higgs fields.

Jet manifolds of a composite bundle

Given the composite bundle, consider the jet manifolds,, and of the fiber bundles,, and, respectively. They are provided with the adapted coordinates,, and
There is the canonical map

Composite connection

This canonical map defines the relations between connections on fiber bundles, and. These connections are given by the corresponding tangent-valued connection forms
A connection on a fiber bundle
and a connection on a fiber bundle define a connection
on a composite bundle. It is called the composite connection. This is a unique connection such that the horizontal lift onto of a vector field on by means of the composite connection coincides with the composition of horizontal lifts of onto by means of a connection and then onto by means of a connection.

Vertical covariant differential

Given the composite bundle, there is the following exact sequence of vector bundles over :
where and are the vertical tangent bundle and the vertical cotangent bundle of. Every connection on a fiber bundle yields the splitting
of the exact sequence. Using this splitting, one can construct a first order differential operator
on a composite bundle. It is called the vertical covariant differential.
It possesses the following important property.
Let be a section of a fiber bundle, and let be the pullback bundle over. Every connection induces the pullback connection
on. Then the restriction of a vertical covariant differential to coincides with the familiar covariant differential
on relative to the pullback connection.