Indefinite affine hyperspheres admitting a pointwise symmetry
| dc.creator | Scharlach, Christine | |
| dc.date | 2005-10-25 | |
| dc.date.accessioned | 2026-07-07T06:47:53Z | |
| dc.date.available | 2026-07-07T06:47:53Z | |
| dc.description | An affine hypersurface M is said to admit a pointwise symmetry, if there exists a subgroup G of Aut(T_p M) 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= H Id (and thus S is trivially preserved). First we solve an algebraic problem. We determine the non-trivial stabilizers G of a traceless cubic form on a Lorentz-Minkowski space R^3_1 under the action of the isometry group SO(1,2) and find a representative of each SO(1,2)/G-orbit. Since the affine cubic form is defined by h and K, this gives us the possible symmetry groups G and for each G a canonical form of K. Next, we classify hyperspheres admitting a pointwise G-symmetry for all non-trivial stabilizers G (apart from Z_2). Besides well-known hyperspheres (for Z_2 x Z_2 resp. R the hyperspheres have constant sectional curvature and Pick invariant J<0 resp. J=0 we obtain rich classes of new examples e.g. warped product structures of two-dimensional affine spheres (resp. quadrics) and curves. Moreover, we find a way to construct indefinite affine hyperspheres out of 2-dimensional quadrics or positive definite affine spheres. | |
| dc.description | 57 pages | |
| dc.identifier | https://arxiv.org/abs/math/0510531 | |
| dc.identifier | http://arxiv.org/abs/math/0510531 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/103802 | |
| dc.subject | Differential Geometry | |
| dc.subject | 53A15; (15A21; 53B30) | |
| dc.title | Indefinite affine hyperspheres admitting a pointwise symmetry | |
| dc.type | text |