There Are No Conformal Einstein Rescalings of Pseudo-Riemannian Einstein Spaces with <i>n</i> Complete Light-Like Geodesics
In the present paper, we study conformal mappings between a connected <i>n</i>-dimension pseudo-Riemannian Einstein manifolds. Let <i>g</i> be a pseudo-Riemannian Einstein metric of indefinite signature on a connected <i>n</i>-dimensional manifold <i>M</i...
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MDPI AG
2019-09-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/7/9/801 |
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| author | Josef Mikeš Irena Hinterleitner Nadezda Guseva |
| author_facet | Josef Mikeš Irena Hinterleitner Nadezda Guseva |
| author_sort | Josef Mikeš |
| collection | DOAJ |
| description | In the present paper, we study conformal mappings between a connected <i>n</i>-dimension pseudo-Riemannian Einstein manifolds. Let <i>g</i> be a pseudo-Riemannian Einstein metric of indefinite signature on a connected <i>n</i>-dimensional manifold <i>M</i>. Further assume that there is a point at which not all sectional curvatures are equal and through which in linearly independent directions pass <i>n</i> complete null (light-like) geodesics. If, for the function <inline-formula> <math display="inline"> <semantics> <mi>ψ</mi> </semantics> </math> </inline-formula> the metric <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>ψ</mi> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mi>g</mi> </mrow> </semantics> </math> </inline-formula> is also Einstein, then <inline-formula> <math display="inline"> <semantics> <mi>ψ</mi> </semantics> </math> </inline-formula> 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). |
| first_indexed | 2024-12-20T17:07:39Z |
| format | Article |
| id | doaj.art-963ddf61670e4e73b677539d238aca16 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-12-20T17:07:39Z |
| publishDate | 2019-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-963ddf61670e4e73b677539d238aca162022-12-21T19:32:14ZengMDPI AGMathematics2227-73902019-09-017980110.3390/math7090801math7090801There Are No Conformal Einstein Rescalings of Pseudo-Riemannian Einstein Spaces with <i>n</i> Complete Light-Like GeodesicsJosef Mikeš0Irena Hinterleitner1Nadezda Guseva2Department of Algebra and Geometry, Palacky University, 17. listopadu 12, 77146 Olomouc, Czech RepublicDepartment of Mathematics, Faculty of Civil Engineering, Brno University of Technology, 60190 Brno, Czech RepublicDepartment of Geometry, Moscow Pedagogical State University, 1/1 M. Pirogovskaya Str., 119991 Moscow, RussianIn the present paper, we study conformal mappings between a connected <i>n</i>-dimension pseudo-Riemannian Einstein manifolds. Let <i>g</i> be a pseudo-Riemannian Einstein metric of indefinite signature on a connected <i>n</i>-dimensional manifold <i>M</i>. Further assume that there is a point at which not all sectional curvatures are equal and through which in linearly independent directions pass <i>n</i> complete null (light-like) geodesics. If, for the function <inline-formula> <math display="inline"> <semantics> <mi>ψ</mi> </semantics> </math> </inline-formula> the metric <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>ψ</mi> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mi>g</mi> </mrow> </semantics> </math> </inline-formula> is also Einstein, then <inline-formula> <math display="inline"> <semantics> <mi>ψ</mi> </semantics> </math> </inline-formula> 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).https://www.mdpi.com/2227-7390/7/9/801pseudo-Riemannian manifoldEinstein manifoldconcircular vector fieldconformal mappinglight-like geodesiccomplete geodesic |
| spellingShingle | Josef Mikeš Irena Hinterleitner Nadezda Guseva There Are No Conformal Einstein Rescalings of Pseudo-Riemannian Einstein Spaces with <i>n</i> Complete Light-Like Geodesics Mathematics pseudo-Riemannian manifold Einstein manifold concircular vector field conformal mapping light-like geodesic complete geodesic |
| title | There Are No Conformal Einstein Rescalings of Pseudo-Riemannian Einstein Spaces with <i>n</i> Complete Light-Like Geodesics |
| title_full | There Are No Conformal Einstein Rescalings of Pseudo-Riemannian Einstein Spaces with <i>n</i> Complete Light-Like Geodesics |
| title_fullStr | There Are No Conformal Einstein Rescalings of Pseudo-Riemannian Einstein Spaces with <i>n</i> Complete Light-Like Geodesics |
| title_full_unstemmed | There Are No Conformal Einstein Rescalings of Pseudo-Riemannian Einstein Spaces with <i>n</i> Complete Light-Like Geodesics |
| title_short | There Are No Conformal Einstein Rescalings of Pseudo-Riemannian Einstein Spaces with <i>n</i> Complete Light-Like Geodesics |
| title_sort | there are no conformal einstein rescalings of pseudo riemannian einstein spaces with i n i complete light like geodesics |
| topic | pseudo-Riemannian manifold Einstein manifold concircular vector field conformal mapping light-like geodesic complete geodesic |
| url | https://www.mdpi.com/2227-7390/7/9/801 |
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