Classification of Calabi Hypersurfaces with parallel Fubini-Pick form

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

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-P…

作为 Fubini-Pick 不变量消失情形的两个自变量的三阶仿射不变 PDE（方程组）

Third-order affine-invariant (systems of) PDEs in two independent variables as vanishing of the Fubini-Pick invariant

本文研究两个自变量、一个未知函数的三阶 Aff(3) 不变 PDE（方程组），证明其与 Fubini-Pick 不变量的关系，并用齐性流形上构造不变 PDE 的一般方法加以导出。

A classification theorem of nondegenerate equiaffine symmetric hypersurfaces

Motivated by the ideas and methods used by Naitoh in the consideration of parallel totally real submanifolds in complex space forms, the author of the present paper successfully makes use of the so called Jordan triple and (restricted) structure Lie algebra associated with a given Jordan algebra to establish a one-to-one correspondence between the set of equivalence classes of connected, simply connected and nondegenerate equiaffine symmetric hypersurfaces with a given nonzero affine mean curvature and that of the equivalence classes of semi-simple real Jordan algebras. Then, via the existing …

Einstein Metrics via Intrinsic or Parallel Torsion

The classification of Riemannian manifolds by the holonomy group of their Levi-Civita connection picks out many interesting classes of structures, several of which are solutions to the Einstein equations. The classification has two parts. The first consists of isolated examples: the Riemannian symmetric spaces. The second consists of geometries that can occur in continuous families: these include the Calabi-Yau structures and Joyce manifolds of string theory. One may ask how one can weaken the definitions and still obtain similar classifications. We present two closely related suggestions. The…