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.
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
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.
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.
