  Due to a result by Gallot a Riemannian cone over a complete Riemannian
manifold is either flat or has an irreducible holonomy representation. This is
false in general for indefinite cones but the structures induced on the cone by
holonomy invariant subspaces can be used to study the geometry on the base of
the cone. The purpose of this paper is twofold: first we will give a survey of
general results about semi-Riemannian cones with non irreducible holonomy
representation and then, as the main result, we will derive improved versions
of these general statements in the case when the cone admits a parallel vector
field. We will show that if the base manifold is complete and the fibre of the
cone and the parallel vector field have the same causal character, then the
cone is flat, and that otherwise, the base manifold admits a certain global
warped product structure. We will use these results to give a new proof of the
classification results for Riemannian manifolds with imaginary Killing spinors
and Lorentzian manifolds with real Killing spinors which are due to Baum and
Bohle.
