We consider the pseudo-Euclidean space \((\mathbb {R}^n,g)\), \(n \ge 3\), with coordinates \(x=\left( x_1,...,x_n\right) \) and metric components \(g_{ij} = \delta _{ij}\epsilon _i\), \(1\le i, j\le n\), where \(\varepsilon _i=\pm 1\), with at least one \(\varepsilon _i=1\) and one diagonal (0,2)-tensors of the form \(T=\sum _i\epsilon _i{h_i(x)dx_i^2}\). We obtain necessary and sufficient conditions for the existence of a metric \(\bar{g}\), conformal to g, such that \(\hbox {Ric}_{\bar{g}}-\displaystyle \frac{\bar{K}}{2} \bar{g} =T\), where \(\hbox {Ric}_{\bar{g}}\) and \(\bar{K}\) are the Ricci tensor and scalar curvature of the metric \(\bar{g}\), respectively. Using the results obtained, we construct an example of a static perfect fluid spacetime. Similar problems are considered for locally conformally flat manifolds.

我们考虑伪欧式空间\((\mathbb {R}^n,g)\) , \(n \ge 3\)，坐标为\(x=\left( x_1,...,x_n\right ) \)和度量分量\(g_{ij} = \delta _{ij}\epsilon _i\) , \(1\le i, j\le n\)，其中\(\varepsilon _i=\pm 1\ )，至少有一个\(\varepsilon _i=1\)和一个形式为\(T=\sum _i\epsilon _i{h_i(x)dx_i^2}\) 的对角线 (0,2)-张量。我们得到的度量的存在的充分必要条件\（\巴{G} \） ，共形克，使得\（\ hbox中{里克} _ {\酒吧{G}} - \的DisplayStyle \压裂{\ bar{K}}{2} \bar{g} =T\)，其中\(\hbox {Ric}_{\bar{g}}\)和\(\bar{K}\)分别是度量\(\bar{g}\)的 Ricci 张量和标量曲率。使用获得的结果，我们构建了一个静态完美流体时空的例子。对于局部共形扁平流形，也考虑了类似的问题。