  It has recently been shown by Goldberg et al that the holonomy group of the
chiral spin-connection is preserved under time evolution in vacuum general
relativity. Here, the underlying reason for the time-independence of the
holonomy group is traced to the self-duality of the curvature 2-form for an
Einstein space. This observation reveals that the holonomy group is
time-independent not only in vacuum, but also in the presence of a cosmological
constant. It also shows that once matter is coupled to gravity, the
"conservation of holonomy" is lost. When the fundamental group of space is
non-trivial, the holonomy group need not be connected. For each homotopy class
of loops, the holonomies comprise a coset of the full holonomy group modulo its
connected component. These cosets are also time-independent. All possible
holonomy groups that can arise are classified, and examples are given of
connections with these holonomy groups. The classification of local and global
solutions with given holonomy groups is discussed.
