  The Barbero-Immirzi (BI) connection, as usually introduced out of a spin
connection, is a global object though it does not transform properly as a
genuine connection with respect to generic spin transformations, unless quite
specific and suitable gauges are imposed. We shall here investigate whether and
under which global conditions a (properly transforming and hence global)
SU(2)-connection can be canonically defined in a gauge covariant way in such a
way that SU(2)-connection locally agrees with the usual BI connection and can
be defined on pretty general bundles (in particular triviality is not assumed).
As a by-product we shall also introduce a global covariant SU(2)-connection
over the whole spacetime (while for technical reasons the BI connection in the
standard formulation is just introduced on a space slice) which restricts to
the usual BI connection on a space slice.
