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Conformal metrics and Ricci tensors in the pseudo-Euclidean space

Authors: Romildo Pina and Keti Tenenblat
Journal: Proc. Amer. Math. Soc. 129 (2001), 1149-1160
MSC (1991): Primary 53C21, 53C50, 58G30
Published electronically: October 4, 2000
MathSciNet review: 1814152
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We consider constant symmetric tensors $T$ on $R^n$, $n\geq 3$, and we study the problem of finding metrics $\bar{g}$ conformal to the pseudo-Euclidean metric $g$ 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 $R^n$ 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 $R^n$, there are metrics conformal to the pseudo-Euclidean metric with scalar curvature $\overline{K}$.

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Additional Information

Romildo Pina
Affiliation: Instituto de Matemática e Estatística, Universidade Federal de Goiás, 74001-970 Goiânia, GO, Brazil

Keti Tenenblat
Affiliation: Departamento de Matemática, Universidade de Brasília, 70910-900, Brasília, DF, Brazil

Keywords: Conformal metrics, Ricci tensor
Received by editor(s): June 14, 1999
Published electronically: October 4, 2000
Additional Notes: The first author was supported in part by CAPES
The second author was supported in part by CNPq and FAPDF
Communicated by: Christopher Croke
Article copyright: © Copyright 2000 American Mathematical Society

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