Indefinite Affine Hyperspheres Admitting a Pointwise Symmetry. Part 2 - Scharlach, C. (Symmetry, Integrability and Geometry: Methods and Applications, 2009)
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.