On parallel and symmetric 2-tensorfields on cones over pseudo-Riemannian manifolds

In this article, we study complete pseudo-Riemannian manifolds whose cone admits a parallel symmetric 2-tensorfield. The situation splits in three cases: nilpotent, decomposable or complex Riemannian. In the complex Riemannian and decomposable cases we provide a classification. In the nilpotent case, we are able to describe completely only a dense open subset of the manifold. To conclude, we give examples with non-constant curvature in the nilpotent case.