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
In differential geometry by a composite bundle is meant the composition of fiber bundles It is provided with bundle coordinates
, i.e., transition functions of coordinates
The following fact provides the above-mentioned physical applications of composite bundles.
be a global section of a fiber bundle
is a subbundle of a fiber bundle
be a principal bundle with a structure Lie group
which is reducible to its closed subgroup
is a principal bundle with a structure group
is a reduced principal subbundle of
In gauge theory, sections of
are treated as classical Higgs fields.
They are provided with the adapted coordinates
There is the canonical map This canonical map defines the relations between connections on fiber bundles
define a connection on a composite bundle
It is called the composite connection.
This is a unique connection such that the horizontal lift
by means of the composite connection
(1), there is the following exact sequence of vector bundles over
yields the splitting of the exact sequence (2).
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.
be a section of a fiber bundle
induces the pullback connection on
Then the restriction of a vertical covariant differential
coincides with the familiar covariant differential
relative to the pullback connection