On $\widetilde{J}$-tangent affine hyperspheres

TL;DR

This paper classifies $ ilde{J}$-tangent affine hyperspheres in various dimensions, explores their relation to Calabi products, and shows most odd-dimensional proper flat affine hyperspheres are $ ilde{J}$-tangent after affine transformations.

Contribution

It provides a comprehensive classification of $ ilde{J}$-tangent affine hyperspheres and establishes their connection to Calabi products, including in the 3-dimensional case.

Findings

Classified all $ ilde{J}$-tangent affine hyperspheres with involutive distribution.

Established a relation between $ ilde{J}$-tangent hyperspheres and Calabi products.

Most odd-dimensional proper flat affine hyperspheres are $ ilde{J}$-tangent after affine transformation.

Abstract

In this paper we study $\widetilde{J}$-tangent affine hyperspheres, where $\widetilde{J}$ is the canonical para-complex structure on $\mathbb{R}^{2n+2}$. The main purpose of this paper is to give a classification of $\widetilde{J}$-tangent affine hyperspheres of an arbitrary dimension with an involutive distribution $\mathcal{D}$. In particular, we classify all such hyperspheres in the $3$-dimensional case. We also show that there is a direct relation between $\widetilde{J}$-tangent affine hyperspheres and Calabi products. As an application we obtain certain classification results. In particular, we show that, with one exception, all odd dimensional proper flat affine hyperspheres are, after a suitable affine transformation, $\widetilde{J}$-tangent. Some examples of $\widetilde{J}$-tangent affine hyperspheres are also given.