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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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Інститут математики НАН України
2009
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| Series: | Symmetry, Integrability and Geometry: Methods and Applications |
| Online Access: | http://dspace.nbuv.gov.ua/handle/123456789/149115 |
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irk-123456789-1491152019-02-20T01:26:06Z Indefinite Affine Hyperspheres Admitting a Pointwise Symmetry. Part 2 Scharlach, C. 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. 2009 Article 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 http://dspace.nbuv.gov.ua/handle/123456789/149115 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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English |
| 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. |
| format |
Article |
| author |
Scharlach, C. |
| spellingShingle |
Scharlach, C. Indefinite Affine Hyperspheres Admitting a Pointwise Symmetry. Part 2 Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Scharlach, C. |
| author_sort |
Scharlach, C. |
| title |
Indefinite Affine Hyperspheres Admitting a Pointwise Symmetry. Part 2 |
| 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 |
| publisher |
Інститут математики НАН України |
| publishDate |
2009 |
| url |
http://dspace.nbuv.gov.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. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT scharlachc indefiniteaffinehyperspheresadmittingapointwisesymmetrypart2 |
| first_indexed |
2023-05-20T17:32:10Z |
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2023-05-20T17:32:10Z |
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1796153521695883264 |