Conformally flat affine hypersurfaces with semi-parallel cubic form

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