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Semi-Riemannian cones

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.