A characterisation of the Calabi product of hyperbolic affine spheres

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.