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 [116]:
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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