Classification of Calabi Hypersurfaces in $\bbr^{n+1}$ with parallel Fubini-Pick form

Abstract

The classifications of locally strongly convex equiaffine hypersurfaces (resp. centroaffine hypersurfaces) with parallel Fubini-Pick form with respect to the Levi-Civita connection of the Blaschke-Berwald affine metric (resp. centroaffine metric) have been completed by several geometers in the last decades, see \cite{HLV} and \cite{CHM}. In this paper we define a generalized Calabi product 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 $\bbr^{n+1}$ with parallel Fubini-Pick form 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 \cite{HLV} and centroaffine situation \cite{CHM}.