Classification of Calabi Hypersurfaces in Rn+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 Cal-abi products in Calabi geometry and prove decomposition theorems in terms of their Calabi invariants. As the main result, we obtain a complete classification of Calabihypersurfaces in Rn+1 with parallel Fubini-Pick tensor with respect to the Levi-Civitaconnection of the Calabi metric. This result is a counterpart in Calabi geometry of theclassification 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).