  The three fundamental geometric components of Yang-Mills theory -gauge field,
gauge fixing and ghost field- are unified in a new object: an extended
connection in a properly chosen principal fiber bundle. To do this, it is
necessary to generalize the notion of gauge fixing by using a gauge fixing
connection instead of a section. From the equations for the extended
connection's curvature, we derive the relevant BRST transformations without
imposing the usual horizontality conditions. We show that the gauge field's
standard BRST transformation is only valid in a local trivialization and we
obtain the corresponding global generalization. By using the Faddeev-Popov
method, we apply the generalized gauge fixing to the path integral quantization
of Yang-Mills theory. We show that the proposed gauge fixing can be used even
in the presence of a Gribov's obstruction.
