  We extend the Gallot-Tanno Theorem to closed pseudo-Riemannian manifolds. It
is done by showing that if the cone over a manifold admits a parallel symmetric
$(0,2)-$tensor then it is Riemannian. Applications of this result to the
existence of metrics with distinct Levi-Civita connections but having the same
unparametrized geodesics and to the projective Obata conjecture are given. We
also apply our result to show that the holonomy group of a closed
$(O(p+1,q),S^{p,q})$-manifold does not preserve any nondegenerate splitting of
$\R^{p+1,q}$.
