Indefinite Affine Hyperspheres Admitting a Pointwise Symmetry. Part 2

TL;DR

This paper classifies 3-dimensional indefinite affine hyperspheres with pointwise symmetries, revealing their structure as warped products of simpler geometric objects, extending previous classifications to new symmetry groups.

Contribution

It extends the classification of affine hyperspheres by analyzing those with pointwise ${ m Z}_3$- or ${ m SO}(2)$-symmetries, identifying their geometric structures as warped products.

Findings

Hyperspheres with ${ m Z}_3$-symmetry are warped products of affine spheres.

Hyperspheres with ${ m SO}(2)$-symmetry are warped products of quadrics.

The classification includes known cases with ${ m Z}_2 imes { m Z}_2$ and ${ m R}$-symmetry.

Abstract

An affine hypersurface $M$ is said to admit a pointwise symmetry, if there exists a subgroup $G$ of ${\rm Aut}(T_p M)$ 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.