Gallot-Tanno theorem for pseudo-Riemannian metrics and a proof that decomposable cones over closed complete pseudo-Riemannian manifolds do not exist

The equation also appears in investigation of geodesically equivalent metrics. Recall that two metrics on one manifold are geodesically equivalent, if every geodesic of one metric is a reparametrized geodesic of the second metric. Solodovnikov [9] has shown that Riemannian metrics on (n > 3)−dimensional manifolds admitting nontrivial 3-parameter family of geodesically equivalent metrics allow nontrivial solutions of (a certain generlaization of) (1). Recently, this result was generalised for pseudo-Riemannian metrics [6, Corollary 4]. Moreover, as it was shown in [5, Corollary 3] (see also [4]), an Einstein manifold of nonconstant scalar curvature admitting nontrival geodesic equivalence, after a proper scaling, admits a nonconstant solution of (1). Tanno [10] (see also [4]) related the equation (1) to projective vector fields, i.e., to vector fields whose local flows take unparametrized geodesics to geodesics. He has shown that every nonconstant solution λ of this equation allows to construct a nontrivial projective vector field.