Codazzi spinors and globally hyperbolic manifolds with special holonomy

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 $Spin(7)$.