There Are No Conformal Einstein Rescalings of Pseudo-Riemannian Einstein Spaces with n Complete Light-Like Geodesics
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Department of Algebra and Geometry, Palacky University, 17. listopadu 12, 77146 Olomouc, Czech Republic
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Department of Mathematics, Faculty of Civil Engineering, Brno University of Technology, 60190 Brno, Czech Republic
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Department of Geometry, Moscow Pedagogical State University, 1/1 M. Pirogovskaya Str., 119991 Moscow, Russian
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(9), 801; https://doi.org/10.3390/math7090801
Received: 31 July 2019
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Revised: 26 August 2019
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Accepted: 29 August 2019
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Published: 1 September 2019
(This article belongs to the Special Issue Differential Geometry of Special Mappings)
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 the metric is also Einstein, then 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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Keywords:
pseudo-Riemannian manifold; Einstein manifold; concircular vector field; conformal mapping; light-like geodesic; complete geodesic
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MDPI and ACS Style
Mikeš, J.; Hinterleitner, I.; Guseva, N. There Are No Conformal Einstein Rescalings of Pseudo-Riemannian Einstein Spaces with n Complete Light-Like Geodesics. Mathematics 2019, 7, 801. https://doi.org/10.3390/math7090801
AMA Style
Mikeš J, Hinterleitner I, Guseva N. There Are No Conformal Einstein Rescalings of Pseudo-Riemannian Einstein Spaces with n Complete Light-Like Geodesics. Mathematics. 2019; 7(9):801. https://doi.org/10.3390/math7090801
Chicago/Turabian StyleMikeš, Josef, Irena Hinterleitner, and Nadezda Guseva. 2019. "There Are No Conformal Einstein Rescalings of Pseudo-Riemannian Einstein Spaces with n Complete Light-Like Geodesics" Mathematics 7, no. 9: 801. https://doi.org/10.3390/math7090801
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