On the equiaffine symmetric hyperspheres

TL;DR

This paper classifies equiaffine symmetric hyperspheres, focusing on locally strongly convex cases, and offers new proofs for existing classification theorems using different methods, enriching the understanding of affine differential geometry.

Contribution

It provides a direct classification of equiaffine symmetric hyperspheres and offers alternative proofs for known theorems on affine hypersurfaces with parallel Fubini-Pick forms.

Findings

Complete classification of locally strongly convex equiaffine symmetric hyperspheres.

Alternative proof for the classification of affine hypersurfaces with parallel Fubini-Pick forms.

Extension of classification results to a broader class of hyperspheres.

Abstract

We introduce and study the equiaffine symmetric {\bf hyperspheres}. For the first step we consider the locally strongly convex ones. In fact, by the idea used by Naitoh, we provide in this paper a direct proof of the complete classification for those affine symmetric hyperspheres. Then, via an earlier result of the first author, we are able to provide an alternative proof for the classification theorem of the affine hypersurface with parallel Fubini-Pick forms, which has already been established by Z.J. Hu et al in a totally different way.