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 (84).
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
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