There Are No Conformal Einstein Rescalings of Pseudo-Riemannian Einstein Spaces with n Complete Light-Like Geodesics

Josef Mikeš, Irena Hinterleitner and Nadezda Guseva
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Josef Mikeš: Department of Algebra and Geometry, Palacky University, 17. listopadu 12, 77146 Olomouc, Czech Republic
Irena Hinterleitner: Department of Mathematics, Faculty of Civil Engineering, Brno University of Technology, 60190 Brno, Czech Republic
Nadezda Guseva: Department of Geometry, Moscow Pedagogical State University, 1/1 M. Pirogovskaya Str., 119991 Moscow, Russian

Mathematics, 2019, vol. 7, issue 9, 1-6

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).

Keywords: pseudo-Riemannian manifold; Einstein manifold; concircular vector field; conformal mapping; light-like geodesic; complete geodesic (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
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