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.
We consider the pseudo-Euclidean space $$({\mathbb {R}}^n,g)$$(Rn,g), with $$n \ge 3$$n≥3 and $$g_{ij} = \delta _{ij} \varepsilon _{i}$$gij=δijεi, where $$\varepsilon _{i} = \pm 1$$εi=±1, with at least one positive $$\varepsilon _{i}$$εi and non-diagonal symmetric tensors $$T = \sum \nolimits _{i,j}f_{ij}(x) dx_i \otimes dx_{j} $$T=∑i,jfij(x)dxi⊗dxj. Assuming that the solutions are invariant by the action of a translation $$(n-1)$$(n-1)- dimensional group, we find the necessary and sufficient conditions for the existence of a metric $$\bar{g}$$g¯ conformal to g, such that the Schouten tensor $$\bar{g}$$g¯, is equal to T. From the obtained results, we show that for certain functions h, defined in $$\mathbb {R}^{n}$$Rn, there exist complete metrics $$\bar{g}$$g¯, conformal to the Euclidean metric g, whose curvature $$\sigma _{2}(\bar{g}) = h$$σ2(g¯)=h.
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