  In this paper we examine the structure of Riemannian manifolds with a special
kind of Codazzi tensors. We use them to construct globally hyperbolic
Lorentzian manifolds with complete Cauchy hypersurfaces for any weakly
irreducible holonomy representation with parallel spinors, i.e. with a holonomy
group which is a semidirect product between $\R^{n-2}$ and one of $\1, SU(k),
Sp(1), G_2$ and $Spin(7)$.
