  In this paper we study the flat (n+1)-spacetimes admitting a Cauchy surface
diffeomorphic to a compact hyperbolic n-manifold. We show how to construct a
canonical future complete one among all such spacetimes sharing the same
holonomy. We study the geometry of such a spacetime in terms of its canonical
cosmological time. In particular we study the asymptotic behaviour of the level
surfaces of the cosmological time. The present work generalizes the case n=2
treated by Mess, taking from a work of Benedetti and Guadagnini the emphasis on
the fundamental role played by the canonical cosmological time.
