A characterisation of the Calabi product of hyperbolic affine spheres

TL;DR

This paper investigates the inverse problem of identifying when a hyperbolic affine sphere can be decomposed into a Calabi product of two lower-dimensional hyperbolic affine spheres based on properties of its difference tensor.

Contribution

It provides criteria to determine the decomposability of a hyperbolic affine sphere into a Calabi product using the difference tensor's properties.

Findings

Characterization criteria for decomposability of hyperbolic affine spheres

Conditions on the difference tensor indicating Calabi product structure

Insights into the structure of hyperbolic affine spheres

Abstract

There exists a well known construction which allows to associate with two hyperbolic affine spheres $f_i: M_i^{n_i} \to \mathbb R^{n_i+1}$ a new hyperbolic affine sphere immersion of $I \times M_1 \times M_2$ into $\mathbb R^{n_1+n_2+3}$. In this paper we deal with the inverse problem: how to determine from properties of the difference tensor whether a given hyperbolic affine sphere immersion of a manifold $M^n \to \mathbb R^{n+1}$ can be decomposed in such a way.