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
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 :
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
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