Prescribed diagonal Schouten tensor in locally conformally flat manifolds

We consider the pseudo-euclidean space Rn, g), with n≥3 and gij = δij εi, εi = ± 1 and tensors of the form (Formula Presented.). In this paper, we obtain necessary and sufficient conditions for a diagonal tensor to admit a metric ḡ, conformal to g, so that Aḡ=T, where Aḡ is the Schouten Tensor of the metric ḡ. The solution to this problem is given explicitly for special cases for the tensor T, including a case where the metric ḡ is complete on ℝn. Similar problems are considered for locally conformally flat manifolds. As an application of these results we consider the problem of finding metrics g, conformal to g, such that σ2(ḡ) or σ2(ḡ)/σ1(ḡ) is equal to a given function. We prove that for some functions, f 1 and f 2, there exist complete metrics (Formula Presented), such that σ2(ḡ) or (Formula Presented.). © 2013 Springer Basel.