Four-dimensional locally strongly convex homogeneous affine hypersurfaces

Abstract

We study four-dimensional locally strongly convex, locally homogeneous, hypersurfaces whose affine shape operator has two distinct principal curvatures. In case that one of the eigenvalues has dimension 1 these hypersurfaces have been previously studied in Dillen and Vrancken (Math Z 212:61–72, 1993, J Math Soc Jpn 46:477–502, 1994) and Hu et al. (Differ Geom Appl 33:46–74, 2014) in which a classification of such submanifolds was obtained in dimension 4 and 5 under the additional assumption that the multiplicity of one of the eigenvalues is 1. In this paper we complete the classification in dimension 4 by considering the case that the multiplicity of both eigenvalues is 2.