Splitting theorems for spatially closed space-times

A Lorentzian splitting theorem is obtained for spatially closed spacetimes. The proof employs and extends some recent results of Bartnik and Gerhardt concerning the existence and rigid uniqueness of compact maximal hypersurfaces in spatially closed space-times. A splitting theorem for spatially closedtime-periodic space-times, which generalizes a result first considered by Avez, is derived as a corollary.