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In this article we extend the Gallot-Tanno theorem to closedpseudo-Riemannian manifolds. It is done by showing that if the cone over such amanifold admits a parallel symmetric 2-tensor then it is incomplete and has nonzero constant curvature. An application of this result to the existence ofmetrics with distinct Levi-Civita connections but having the sameunparametrized geodesics is given.

Gallot-Tanno theorem for closed incomplete pseudo-Riemannian manifolds and application