### 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]:

Theorem 2 (Generalized Wang Theorem) Let be an -symmetric principal fiber bundle classified by according to Theorem 1, and let be an -invariant connection on .

Then the connection is uniquely classified by a reduced connection on and a scalar field obeying Equation (69).

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 .