  In this article we extend the Gallot-Tanno theorem to closed
pseudo-Riemannian manifolds. It is done by showing that if the cone over such a
manifold admits a parallel symmetric 2-tensor then it is incomplete and has non
zero constant curvature. An application of this result to the existence of
metrics with distinct Levi-Civita connections but having the same
unparametrized geodesics is given.
