等仿射超曲面是仿射微分几何主要的研究对象，其研究无论从整体还是局部都取得了巨大的成果，但也遗留了一些重要的问题。一方面，与欧氏几何相比，齐性仿射超曲面的研究较少，其分类远远没有完成；另一方面，一些典型仿射超曲面的分类还没有完成。. 在此背景下，本项目拟在已有的工作基础上，主要对以下内容展开研究：1. 研究具有下述性质之一的齐性仿射超曲面的分类问题：（1）具有常截面曲率和零Pick不变量的仿射超球面；（2）局部强凸的Einstein仿射超球面；（3）局部强凸的拟脐仿射超曲面；（4）局部强凸的常截面曲率仿射超曲面。在齐性情形研究Vrancken关于等参仿射超曲面主曲率个数的猜想。2. 首次引入信息几何中的仿射alpha联络，研究关于此联络平行的差异张量或cubic形式的仿射超曲面的分类问题，以及典型齐性仿射超曲面的性质和几何特征，丰富和发展仿射超曲面理论。

Equiaffine hypersurface is the main object of affine differential geometry, there are great results on both whole and local geometry, however some important questions are also left. On the one hand, compared with Euclidean geometry there are few results on homogeneous affine hypersurfaces, whose classification is far from complete; On the other hand, the classifications of some canonical affine hypersurfaces are not yet completed. . Against this background, based on our experience of previous work we manily study the following subjects: 1. Study the classification of homogeneous affine hypersurfaces with one of the following properties: (1) affine hyperspheres with constant sectional curvature and zero Pick invariant; (2) locally strongly convex Einstein affine hyperspheres; (3) locally strongly convex quasi-umblic affine hypersurfaces; (4) locally strongly convex affine hypersurfaces with constant sectional curvature. And study Vrancken's conjecture on the number of principle curvature of isoparametric affine hypersurfaces under the homogeneous assumption. 2. Introduce the affine alpha connection of information geometry for the first time, we study the classification of affine hypersurfaces with parallel difference tensor or cubic form relative to such connection, and the properties and geometrical characterization of canonical homogeneous affine hypersurfaces, whose results will enrich and develop the theory of affine hypersurface.