Indefinite Affine Hyperspheres Admitting a Pointwise Symmetry. Part 2

Abstract

An affine hypersurface $M$ is said to admit a pointwise symmetry, if there exists a subgroup $G$ of $\mathrm{Aut}(T_{p}M)$ 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. $S=HId$ (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 ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ resp. $\mathbb{R}$-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 ${\mathbb{Z}}_3$- or SO(2)-symmetry. They turn out to be warped products of affine spheres (${\mathbb{Z}}_3$) or quadrics (SO(2)) with a curve.