四维仿射空间的标准曲面

Canonical Centroaffine Surfaces in E^4

在仿射微分几何中,研究的重心是将有常截面曲率仿射度量的曲面(超曲面)进行分类。如果一中心仿射曲面的中心仿射度量为平坦的,并且其Fubini-Pick形式关于中心仿射度量是平行的,则称此曲面为标准超曲面,因此讨论并构造E4空间中的标准超曲面是重要的。

An important problem in affine differential geometry is to classify all the affine hypersurfaces with constant sectional curvature affine metric.A centroaffine hypersurface is said to be canonical if its centroaffine metric is flat and its Fubini-Pick form is parallel with respect to its centroaffine metric. Calabi＇s composition can make one regular centroaffine hypersurfaces into two regular centroaffine hypersurfaces.In this paper,we will generalize and apply Calabi＇s composition to centroaffine hypersurfaces.As a result,we give the canonical centroaffine surfaces in E4.