Prescribed curvature tensor in locally conformally flat manifolds

Abstract

Abstract. In the euclidean space (R n , g), with n ≥ 3, g ij = δ ij , we consider a (0,4)-tensor R = T ⊙ g where T = i f i (x)dx 2 i is a diagonal (0,2)-tensor. We obtain necessary and sufficient conditions for the existence of a metricḡ, conformal to g, such thatR = R, whereR is the Riemannian curvature tensor of the metricḡ. The solution to this problem is given explicitly for the special cases of the tensor R, including a case where the metricḡ is complete on R n . Similar problems are considered for locally conformally flat manifolds. As applications of these results we exhibit explicit examples of metricsḡ, conformal to g they are solutions for this problem.