Prescribed Schouten Tensor in Locally Conformally Flat Manifolds

Abstract

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