Parallel and symmetric $2$-tensor fields on pseudo-Riemannian cones

We study the complete pseudo-Riemannian manifolds whosecone admits a non-trivial parallel symmetric $2$-tensorfield. We are able to give an extensive description ofthese manifolds except when the tensor field is given by anilpotent endomorphism. In this case, we are able todescribe only a dense open subset of them. Moreover, weconstruct examples with non-constant curvature where thisopen set is proper.

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Published 21 March 2012