A class of solutions of the Ricci and Einstein equations

JOURNAL OF GEOMETRY AND PHYSICS

卷:57

Article

Pina, Romildo ; Tenenblat, Keti

关键词: Ricci tensor;    conformal metric;    Ricci equation;    Einstein equation;   

DOI  :  10.1016/j.geomphys.2006.06.009

来源: Elsevier

【 摘 要 】

We consider the pseudo-Euclidean space (R-n, g), with n >= 3 and g(ij) = delta(ij)epsilon(i), epsilon(i) = +/- 1 and tensors of the form T = Sigma i f(i)(x(k))epsilon(i)dx(i)(2) for a fixed k, 1 <= k <= n. We provide necessary and sufficient conditions for such a tensor to admit metrics (g) over bar, conformal to g, that solve the Ricci equation or the Einstein equation. The solution to this problem is given explicitly and it depends on an arbitrary differentiable function of one variable. Similar problems are considered for locally conformally flat manifolds. Examples are provided of complete metrics on R-n, whose Ricci curvature is negative. Complete metrics are also given on the cylinder or on the n-dimensional torus, that solve the Ricci equation or the Einstein equation. Examples of metrics with positive Ricci curvatures are given on half-spaces of R-n. (c) 2006 Published by Elsevier B.V.

【 授权许可】

Free