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

TL;DR

This paper provides a complete classification of Calabi hypersurfaces in Euclidean space with parallel Fubini-Pick form, extending previous affine and centroaffine classification results to Calabi geometry.

Contribution

It introduces a generalized Calabi product and proves decomposition theorems, leading to a full classification of Calabi hypersurfaces with parallel Fubini-Pick form.

Findings

Complete classification of Calabi hypersurfaces with parallel Fubini-Pick form.

Introduction of generalized Calabi product in Calabi geometry.

Decomposition theorems based on Calabi invariants.

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}.