There are no conformal Einstein rescalings of complete pseudo-Riemannian Einstein metrics

摘要

Let g be an Einstein metric of indefinite signature such that the conformally-equivalent metric psi(-2) g is also Einstein. We show that if the metric g is light-line complete, then the conformal coefficient psi is constant. If the manifold is closed, the completeness assumption can be omitted (the latter result is due to Mikes-Radulovich and Kuhnel, but our proof is much simpler). The proof is based on the investigation of the behavior of the function psi along light-line geodesics: we show that for every light-line geodesic gamma(t) we have psi(gamma(t)) = const(1) . t + const(2). Since the function psi cannot vanish, the light-line completeness of the metric implies psi = const(2). If the manifold is closed, the function psi accepts its maximal value psi(max) at a certain point. Then, for every light-line geodesic gamma through this point we have const(1) = 0 implying psi = psi(max) at every point of this geodesic. Repeating the argumentation, we obtain that for every light-line geodesic gamma(1) intersecting gamma we have psi = psi(max) at every point of gamma(1) as well and so on. Since every two points can be connected by a sequence of light-line geodesics, psi is constant on the whole manifold. To cite this article: V. Kiosak, VS. Matveev, C R. Acad. Sci. Paris, Ser. 1347 (2009).