### 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 -invariance of the reduced connection is a
one-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
[82]:

In general, transforms under some representation of the reduced structure group : Its values lie
in the subspace of determined by Equation (69) 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
.