### A.3 Classification of invariant connections

Now let be an -invariant connection on the symmetric bundle classified by
, i.e., for any . After restriction, induces a connection on
the reduced bundle . Because of the -invariance of , the reduced connection is
a 1-form on with values in the Lie algebra of the reduced structure group. To see this,
fix a point and a vector in , such that , where is the
embedding of into . Such a vector, which does not have components along symmetry orbits,
is fixed by the action of the isotropy group: . The pull back of by
applied to is by definition . Now using the fact that
acts as gauge transformation in the fibers and observing the definition of and the adjoint
transformation of , we obtain . By assumption, the connection is
-invariant implying for all . This shows that
, and can be restricted to a connection on the bundle with structure group
.
Furthermore, using we can construct the linear map for any .
Here, is the vector field on given by for any
and . For the vector field is a vertical vector field, and we have
, where is the derivative of the homomorphism defined above. This
component of is therefore already given by the classifying structure of the principal fiber
bundle. Using a suitable gauge, can be held constant along . The remaining components
yield information about the invariant connection . They are subject to the condition

which follows from the transformation of under the adjoint representation and which provides a set of
equations determining the form of the components .
Keeping only the information characterizing we have, besides , the scalar field ,
which is determined by and can be regarded as having components of -valued
scalar fields. The reduced connection and the scalar field suffice to characterize an invariant
connection [116]:

In general, transforms under some representation of the reduced structure group ; its values lie
in the subspace of determined by Equation (84) and form a representation space for all group
elements of (which act on ), whose action preserves the subspace. These are, by definition, precisely
elements of the reduced group.

The connection can be reconstructed from its classifying structure as follows. According to
the decomposition we have

where is given by in a gauge depending on the (local) embedding . Here
is the Maurer–Cartan form on taking values in . Through , depends on and
.