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

We consider tensors T = fg oil the pseudo-euclidean space R-n and on the hyperbolic space H-n, where n >= 3, K 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) over bar, conformal to (g) over bar, such that Ric (g) over bar = T. and the existence Of such a metric which satisfies Ric (g) over bar - (K) over bar (g) over bar /2 = T, where (k) over bar is the scalar curvature of (g) over bar. We find the restrictions on the Ricci candidate for solvability and we construct the solutions (g) over bar when they exist. We show that these metrics are unique up to homothety, we characterize those globally defined and we determine the singularities tor those which are not globally defined. None of the non-homothetic metrics (g) over bar, defined on R-n or H-n, are complete. As a consequence of these results, we get positive solutions for the equation Delta(g)u - (n-2)/4 (lambda u)(n+2)/(n-2) = 0, where g is the pseudo-eucliclean metric. (C) 2005 Elsevier B.V. All rights reserved.