There Are No Conformal Einstein Rescalings of Pseudo-Riemannian Einstein Spaces with n Complete Light-Like Geodesics
Josef MikešDepartment of Algebra and Geometry, Palacky University, 17. listopadu 12, 77146 Olomouc, Czech RepublicIrena HinterleitnerDepartment of Mathematics, Faculty of Civil Engineering, Brno University of Technology, 60190 Brno, Czech RepublicN. I. GusevaDepartment of Geometry, Moscow Pedagogical State University, 1/1 M. Pirogovskaya Str., 119991 Moscow, Russian
2019en
ABI
Abstract
In the present paper, we study conformal mappings between a connected n-dimension pseudo-Riemannian Einstein manifolds. Let g be a pseudo-Riemannian Einstein metric of indefinite signature on a connected n-dimensional manifold M. Further assume that there is a point at which not all sectional curvatures are equal and through which in linearly independent directions pass n complete null (light-like) geodesics. If, for the function ψ the metric ψ − 2 g is also Einstein, then ψ is a constant, and conformal mapping is homothetic. Note that Kiosak and Matveev previously assumed that all light-lines were complete. If the Einstein manifold is closed, the completeness assumption can be omitted (the latter result is due to Mikeš and Kühnel).
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