Prescribed diagonal Schouten tensor in locally conformally flat manifolds

Pieterzack, Mauricio; Pina, Romildo

doi:10.1007/s00022-013-0159-1

Cited by 1 publication

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“…Before proving our results, it follows from [15], that if (R n , g) is a pseudo-Euclidean space andḡ = g/ϕ 2 Is a conformal metric, then the Ricci tensorḡ is given by…”

Section: Proof Of the Main Resultsmentioning

confidence: 87%

“…In [15], the authors considered the pseudo-Euclidean space (R n , g), with n ≥ 3 and g ij = δ ij ε i , ε i = ±1, and tensors of the form T = i ε i f i (x)dx 2 i , and found necessary and sufficient conditions for the existence of a metricḡ, conformal to g, such that Aḡ = T . The solution to this problem was explicitly given for special cases of the tensor T , including a case where the metricḡ is complete in R n .…”

Section: Introductionmentioning

confidence: 99%

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Prescribed Schouten Tensor in Locally Conformally Flat Manifolds

2019

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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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“…Before proving our results, it follows from [15], that if (R n , g) is a pseudo-Euclidean space andḡ = g/ϕ 2 Is a conformal metric, then the Ricci tensorḡ is given by…”

Section: Proof Of the Main Resultsmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%