Conformal and Quasi-Einstein Metrics on Pseudo-Euclidean Space

Abstract

We consider constant symmetric tensors T on R n , n ≥ 3, and we study the problem of finding metrics ¯ g conformal to the pseudo-Euclidean metric g such that Ric ¯ g = T. We show that such tensors are determined by the diagonal elements and we obtain explicitly the metrics ¯ g. As a consequence of these results we get solutions globally defined on R n for the equation −ϕ∆gϕ + n|||gϕ|| 2 /2 + λϕ 2 = 0. Moreover, we show that for certain unbounded functions K defined on R n , there are metrics conformal to the pseudo-Euclidean metric with scalar curvature K.