A new characterization of Calabi composition of hyperbolic affine hyperspheres

TL;DR

This paper provides a new, direct characterization of the Calabi composition of hyperbolic affine hyperspheres, complementing previous methods and offering a clearer understanding of their structure.

Contribution

It introduces a novel, direct proof for characterizing Calabi compositions, expanding on prior dual correspondence approaches and related characterizations.

Findings

Established two new characterizations of Calabi composition

Provided a direct proof method for the second characterization

Enhanced understanding of hyperbolic affine hypersphere structures

Abstract

In this paper, we mainly prove a theorem with a corollary establishing two characterizations of the Calabi composition of hyperbolic hyperspheres, where the second characterization (i.e., the corollary) has been given via a dual correspondence theorem earlier but now we would like to use a very direct method. Note that Z.J. Hu, H.Z. Li and L. Vrancken also gave a characterization of the $2$-factor Calabi composition in a different manner.