Three-dimensional quasi-umbilical locally strongly convex homogeneous affine hypersurfaces

Abstract

The study of strongly convex, locally homogeneous hypersurfaces in the affine space ℝn+1 has a rich history. In dimension 2, these hypersurfaces were first examined in Guggenheimer's 1977 book “Differential Geometry” [4], and the classification was completed by Nomizu and Sasaki in their 1991 paper “A new model of unimodular-affinely homogeneous surfaces” [6]. Apart from the work of Magid and Vrancken in their 1995 paper [5] and Ooguri’s 2013 paper [9], most studies have focused on locally strongly convex affine hypersurfaces. These are characterized by an induced affine metric that is positive definite, and have been extensively studied in Sasaki’s 1980 paper “Hyperbolic affine hyperspheres” [10]. When the hypersurface is an affine sphere, where the shape operator has all eigenvalues equal to zero, Dillen and Vrancken provided a complete classification in a series of papers. In dimension 3, their 1993 paper “The classification of 3-dimensional locally strongly convex homogeneous affine hypersurfaces” [3] offers a comprehensive classification for any dimensions of eigenvalues of the shape operator. In dimension 4, the quasi-umbilical case was explored by Dillen and Vrancken in their 1994 paper “Quasi-umbilical, locally strongly convex homogeneous affine hypersurfaces” [2]. The case where the affine shape operator has two distinct eigenvalues, each with a multiplicity of two, was investigated by Chikh-Salah and Vrancken in their 2017 paper “Four-dimensional locally strongly convex homogeneous affine hypersurfaces” [1]. In this paper, we provide a different proof of the final theorem in Dillen and Vrancken’s [3], utilizing Mathematica software to calculate all Christoffel symbol functions.