On the Ricci and Einstein equations on the pseudo-euclidean and hyperbolic spaces

We consider tensors T=fgT=fg on the pseudo-euclidean space RnRn and on the hyperbolic space HnHn, where n⩾3n⩾3, g is the standard metric and f   is a differentiable function. For such tensors, we consider, in both spaces, the problems of existence of a Riemannian metric g¯, conformal to g  , such that Ricg¯=T, and the existence of such a metric which satisfies Ricg¯−K¯g¯/2=T, where K¯ is the scalar curvature of g¯. We find the restrictions on the Ricci candidate for solvability and we construct the solutions g¯ when they exist. We show that these metrics are unique up to homothety, we characterize those globally defined and we determine the singularities for those which are not globally defined. None of the non-homothetic metrics g¯, defined on RnRn or HnHn, are complete. As a consequence of these results, we get positive solutions for the equation Δgu−n(n−2)4λu(n+2)(n−2)=0, where g is the pseudo-euclidean metric.