Conformally flat affine hypersurfaces with semi-parallel cubic form

Abstract

In this paper, we study locally strongly convex affine hypersurfaces with the vanishing Weyl curvature tensor and semi-parallel cubic form relative to the Levi-Civita connection of the affine metric. As a main result, we classify these hypersurfaces as not being of a flat affine metric. In particular, 2 and 3-dimensional locally strongly convex affine hypersurfaces with semi-parallel cubic forms are completely determined.