Classification of Calabi Hypersurfaces in \({{\mathbb {R}}}^{n+1}\) with Parallel Fubini-Pick Tensor

The classifications of locally strongly convex equiaffine hypersurfaces (resp. centroaffine hypersurfaces) with parallel Fubini–Pick tensor with respect to the Levi-Civita connection of the Blaschke–Berwald affine metric (resp. centroaffine metric) have been completed in the last decades. In this paper we define generalized Calabi products in Calabi geometry and prove decomposition theorems in terms of their Calabi invariants. As the main result, we obtain a complete classification of Calabi hypersurfaces in \({{\mathbb {R}}}^{n+1}\) ${\mathbb{R}}^{n+1}$ with parallel Fubini–Pick tensor with respect to the Levi-Civita connection of the Calabi metric. This result is a counterpart in Calabi geometry of the classification theorems in equiaffine situation Hu et al. (J Diff Geom 87:239-307, 2011) and centroaffine situation Cheng et al. (Results Math 72:419-469, 2017).