Indefinite Affine Hyperspheres Admitting a Pointwise Symmetry. Part 2
An affine hypersurface M is said to admit a pointwise symmetry, if there exists a subgroup G of Aut(T_pM) for all p in M, which preserves (pointwise) the affine metric h, the difference tensor K and the affine shape operator S. Here, we consider 3-dimensional indefinite affine hyperspheres, i.e. S =...
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| Format: | Article |
| Language: | English |
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National Academy of Science of Ukraine
2009-10-01
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| Series: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Online Access: | http://dx.doi.org/10.3842/SIGMA.2009.097 |
| _version_ | 1818526362158235648 |
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| author | Christine Scharlach |
| author_facet | Christine Scharlach |
| author_sort | Christine Scharlach |
| collection | DOAJ |
| description | An affine hypersurface M is said to admit a pointwise symmetry, if there exists a subgroup G of Aut(T_pM) for all p in M, which preserves (pointwise) the affine metric h, the difference tensor K and the affine shape operator S. Here, we consider 3-dimensional indefinite affine hyperspheres, i.e. S = HId (and thus S is trivially preserved). In Part 1 we found the possible symmetry groups G and gave for each G a canonical form of K. We started a classification by showing that hyperspheres admitting a pointwise Z_2 × Z_2 resp. R-symmetry are well-known, they have constant sectional curvature and Pick invariant J < 0 resp. J = 0. Here, we continue with affine hyperspheres admitting a pointwise Z_3- or SO(2)-symmetry. They turn out to be warped products of affine spheres (Z_3) or quadrics (SO(2)) with a curve. |
| first_indexed | 2024-12-11T06:21:56Z |
| format | Article |
| id | doaj.art-1da218053bf04deb85fef2626feb9282 |
| institution | Directory Open Access Journal |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2024-12-11T06:21:56Z |
| publishDate | 2009-10-01 |
| publisher | National Academy of Science of Ukraine |
| record_format | Article |
| series | Symmetry, Integrability and Geometry: Methods and Applications |
| spelling | doaj.art-1da218053bf04deb85fef2626feb92822022-12-22T01:17:47ZengNational Academy of Science of UkraineSymmetry, Integrability and Geometry: Methods and Applications1815-06592009-10-015097Indefinite Affine Hyperspheres Admitting a Pointwise Symmetry. Part 2Christine ScharlachAn affine hypersurface M is said to admit a pointwise symmetry, if there exists a subgroup G of Aut(T_pM) for all p in M, which preserves (pointwise) the affine metric h, the difference tensor K and the affine shape operator S. Here, we consider 3-dimensional indefinite affine hyperspheres, i.e. S = HId (and thus S is trivially preserved). In Part 1 we found the possible symmetry groups G and gave for each G a canonical form of K. We started a classification by showing that hyperspheres admitting a pointwise Z_2 × Z_2 resp. R-symmetry are well-known, they have constant sectional curvature and Pick invariant J < 0 resp. J = 0. Here, we continue with affine hyperspheres admitting a pointwise Z_3- or SO(2)-symmetry. They turn out to be warped products of affine spheres (Z_3) or quadrics (SO(2)) with a curve.http://dx.doi.org/10.3842/SIGMA.2009.097affine hyperspheresindefinite affine metricpointwise symmetryaffine differential geometryaffine sphereswarped products |
| spellingShingle | Christine Scharlach Indefinite Affine Hyperspheres Admitting a Pointwise Symmetry. Part 2 Symmetry, Integrability and Geometry: Methods and Applications affine hyperspheres indefinite affine metric pointwise symmetry affine differential geometry affine spheres warped products |
| title | Indefinite Affine Hyperspheres Admitting a Pointwise Symmetry. Part 2 |
| title_full | Indefinite Affine Hyperspheres Admitting a Pointwise Symmetry. Part 2 |
| title_fullStr | Indefinite Affine Hyperspheres Admitting a Pointwise Symmetry. Part 2 |
| title_full_unstemmed | Indefinite Affine Hyperspheres Admitting a Pointwise Symmetry. Part 2 |
| title_short | Indefinite Affine Hyperspheres Admitting a Pointwise Symmetry. Part 2 |
| title_sort | indefinite affine hyperspheres admitting a pointwise symmetry part 2 |
| topic | affine hyperspheres indefinite affine metric pointwise symmetry affine differential geometry affine spheres warped products |
| url | http://dx.doi.org/10.3842/SIGMA.2009.097 |
| work_keys_str_mv | AT christinescharlach indefiniteaffinehyperspheresadmittingapointwisesymmetrypart2 |