The classification of 4-dimensional non-degenerate affine hypersurfaces with parallel cubic form

In this paper, we complete the classification of 4-dimensional non-degenerate affine hypersurfaces with parallel cubic form with respect to the Levi-Civita connection of the affine Berwald–Blaschke metric.

► We complete the classification for 4-dimensional affine hypersurfaces with parallel cubic form. ► We construct several affine invariant functions and study the relative extremal point in different cases. ► An algebra lemma for a self-adjoint operator is employed in other cases. ► Finally all information about affine invariants are obtained.