Conformally flat affine hypersurfaces with semi-parallel cubic form

TL;DR

This paper classifies certain convex affine hypersurfaces with semi-parallel cubic form and vanishing Weyl curvature, providing a complete description in 2 and 3 dimensions.

Contribution

It offers a classification of non-flat, locally strongly convex affine hypersurfaces with semi-parallel cubic form and vanishing Weyl curvature tensor.

Findings

Complete classification of 2 and 3-dimensional cases

Identification of hypersurfaces with semi-parallel cubic form

Results applicable to non-flat affine metrics

Abstract

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.