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

Author

Listed:
  • 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)

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).

Suggested Citation

  • Josef Mikeš & Irena Hinterleitner & Nadezda Guseva, 2019. "There Are No Conformal Einstein Rescalings of Pseudo-Riemannian Einstein Spaces with n Complete Light-Like Geodesics," Mathematics, MDPI, vol. 7(9), pages 1-6, September.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:9:p:801-:d:262944
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