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

Miaoxin Lei; Ruiwei Xu

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

Differential Geometry 2021-12-06 v1

Authors: Miaoxin Lei , Ruiwei Xu

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}.

Keywords

calabi hypersurfaces parallel fubini-pick form calabi metric levi-civita connection classification calabi invariants decomposition theorems

Cite

@article{arxiv.2112.01947,
  title  = {Classification of Calabi Hypersurfaces in $\bbr^{n+1}$ with parallel Fubini-Pick form},
  author = {Miaoxin Lei and Ruiwei Xu},
  journal= {arXiv preprint arXiv:2112.01947},
  year   = {2021}
}

Comments

For covinent readers' convenience, we keep many detailed calculations and proofs of conclusions. Most of them are the same in [10] and [6]. Comments are welcome

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

Abstract

Keywords

Cite

Comments

Related papers