
SIGMA 5 (2009), 097, 22 pages arXiv:0910.3609
https://doi.org/10.3842/SIGMA.2009.097
Contribution to the Special Issue “Élie Cartan and Differential Geometry”
Indefinite Affine Hyperspheres Admitting a Pointwise Symmetry. Part 2
Christine Scharlach
Technische Universität Berlin,
Fak. II, Inst. f. Mathematik, MA 83, 10623 Berlin, Germany
Received May 08, 2009, in final form October 06, 2009; Published online October 19, 2009
Abstract
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 ∈ M, which
preserves (pointwise) the affine metric h, the difference tensor K
and the affine shape operator S. Here, we consider 3dimensional
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. Rsymmetry are wellknown, 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.
Key words:
affine hyperspheres; indefinite affine metric; pointwise symmetry; affine differential geometry; affine spheres; warped products.
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