THE SPIN HOLONOMY GROUP IN GENERAL-RELATIVITY

It has recently been shown by Goldberg et al. that the holonomy group of the chiral spin-connection is preserved under time evolution in vac uum general relativity. Here, the underlying reason for the time-indep endence of the holonomy group is traced to the self-duality of the cur vature 2-form for an Einstein space. This observation reveals that the holonomy group is time-independent not only in vacuum, but also in th e 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 co mprise a coset of the full holonomy group modulo its connected compone nt. These cosets are also time-independent. All possible holonomy grou ps that can arise are classified, and examples are given of connection s with these holonomy groups. The classification of local and global s olutions with given holonomy groups is discussed.