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(TpM) for all p ∈ 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 = H...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2009 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2009
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/149115 |
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| Cite this: | Indefinite Affine Hyperspheres Admitting a Pointwise Symmetry. Part 2 / C. Scharlach // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 14 назв. — англ. |
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Scharlach, C. 2019-02-19T17:29:18Z 2019-02-19T17:29:18Z 2009 Indefinite Affine Hyperspheres Admitting a Pointwise Symmetry. Part 2 / C. Scharlach // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 14 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 53A15; 53B30 https://nasplib.isofts.kiev.ua/handle/123456789/149115 An affine hypersurface M is said to admit a pointwise symmetry, if there exists a subgroup G of Aut(TpM) for all p ∈ 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 Z₂ × Z₂ 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 Z₃- or SO(2)-symmetry. They turn out to be warped products of affine spheres (Z₃) or quadrics (SO(2)) with a curve. This paper is a contribution to the Special Issue “Elie Cartan and Differential Geometry”. Partially supported by the DFG-Project PI 158/4-5 ‘Geometric Problems and Special PDEs’. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Indefinite Affine Hyperspheres Admitting a Pointwise Symmetry. Part 2 Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Indefinite Affine Hyperspheres Admitting a Pointwise Symmetry. Part 2 |
| spellingShingle |
Indefinite Affine Hyperspheres Admitting a Pointwise Symmetry. Part 2 Scharlach, C. |
| title_short |
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_sort |
indefinite affine hyperspheres admitting a pointwise symmetry. part 2 |
| author |
Scharlach, C. |
| author_facet |
Scharlach, C. |
| publishDate |
2009 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
An affine hypersurface M is said to admit a pointwise symmetry, if there exists a subgroup G of Aut(TpM) for all p ∈ 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 Z₂ × Z₂ 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 Z₃- or SO(2)-symmetry. They turn out to be warped products of affine spheres (Z₃) or quadrics (SO(2)) with a curve.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/149115 |
| citation_txt |
Indefinite Affine Hyperspheres Admitting a Pointwise Symmetry. Part 2 / C. Scharlach // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 14 назв. — англ. |
| work_keys_str_mv |
AT scharlachc indefiniteaffinehyperspheresadmittingapointwisesymmetrypart2 |
| first_indexed |
2025-12-07T16:37:12Z |
| last_indexed |
2025-12-07T16:37:12Z |
| _version_ |
1850868166010339328 |