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
[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
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