Prescribed diagonal Schouten tensor in locally conformally flat manifolds

We consider the pseudo-euclidean space ${(mathbb{R}^n, g)}$ , with n ≥  3 and ${g_{ij} = delta_{ij} varepsilon_i, varepsilon_i = pm 1}$ and tensors of the form ${T = sum nolimits_i varepsilon_i f_i (x) dx_i^2}$ . In this paper, we obtain necessary and sufficient conditions for a diagonal tensor to admit a metric ${bar{g}}$ , conformal to g, so that ${A_{bar g}=T}$ , where ${A_{bar g}}$ is the Schouten Tensor of the metric ${bar g}$ . The solution to this problem is given explicitly for special cases for the tensor T, including a case where the metric ${bar g}$ is complete on ${mathbb{R}^n}$ . Similar problems are considered for locally conformally flat manifolds. As an application of these results we consider the problem of finding metrics ${bar g}$ , conformal to g, such that ${sigma_2 ({bar g })}$ or ${frac{sigma_2 ({bar g })}{sigma_1 ({bar g })}}$ is equal to a given function. We prove that for some functions, f ₁ and f ₂, there exist complete metrics ${bar{g} = g/{varphi^2}}$ , such that ${sigma_2 ({bar g }) = f_1}$ or ${frac{sigma_2 ({bar g })}{sigma_1 ({bar g })} = f_2}$ .