Characterizations of the Calabi Product of Hyperbolic Affine Hyperspheres

There exists a well known construction which allows to associate with two hyperbolic affine hyperspheres \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_{i} : M^{n_{i}}_{i} \rightarrow {\mathbb{R}}^{n_{i}+1}$$\end{document} a new hyperbolic affine hypersphere immersion of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I \times M_{1} \times M_{2}$$\end{document} into \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}}^{n_{1}+n_{2}+2}$$\end{document}. In this paper we deal with the inverse problem: how to determine from properties of the difference tensor whether a given hyperbolic affine hypersphere immersion of a manifold \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M^{n} \rightarrow R^{n+1}$$\end{document} can be decomposed in such a way.