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Conformal metrics and Ricci tensors in the pseudo-Euclidean space

Authors:	Romildo Pina Keti Tenenblat

Institution:	Instituto de Matemática e Estatística, Universidade Federal de Goiás, 74001-970 Goiânia, GO, Brazil ; Departamento de Matemática, Universidade de Brasília, 70910-900, Brasília, DF, Brazil

Abstract:	We consider constant symmetric tensors on , $n\geq 3$ , and we study the problem of finding metrics $\bar{g}$ conformal to the pseudo-Euclidean metric such that $\mbox{Ric}\,\bar{g} = T$ . We show that such tensors are determined by the diagonal elements and we obtain explicitly the metrics $\bar{g}$ . As a consequence of these results we get solutions globally defined on for the equation $- \varphi \Delta_g \varphi +n \vert\vert\nabla_g \varphi\vert\vert^2/2 + \lambda \varphi^2 = 0.$ Moreover, we show that for certain unbounded functions $\overline{K}$ defined on , there are metrics conformal to the pseudo-Euclidean metric with scalar curvature $\overline{K}$ .

Keywords:	Conformal metrics Ricci tensor

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