  In the context of the geometrical interpretation of the spin network states
of Loop Quantum Gravity, we look at the holonomies of the Ashtekar-Barbero
connection on loops embedded in space-like hyperboloids. We use this simple
setting to illustrate two points. First, the Ashtekar-Barbero connection is not
a space-time connection, its holonomies depend on the spacetime embedding of
the canonical hypersurface. This fact is usually interpreted as an
inconvenience, but we use it to extract the extrinsic curvature from the
holonomy and separate it from the 3d intrinsic curvature. Second, we show the
limitations of this reconstruction procedure, due to a periodicity of the
holonomy in the Immirzi parameter, which underlines the role of a real Immirzi
parameter as a cut-off for general relativity at the quantum level in contrast
with its role of a mere coupling constant at the classical level.
