On locally strongly convex affine hypersurfaces with parallel cubic form. Part I

In this paper, we study locally strongly convex affine hypersurfaces of Rn+1Rn+1 that have parallel cubic form with respect to the Levi-Civita connection of the affine Berwald–Blaschke metric; it is known that they are affine spheres. In dimension n⩽7n⩽7 we give a complete classification of such hypersurfaces; in particular, we present new examples of affine spheres.