Affine Hypersurfaces With Parallel Difference Tensor

The Pick-Berwald Theorem is one of the most attractive results in the classical affine differential geometry. One of its natural generalizations is the work on hypersurfaces whose difference tensor K is parallel with respect to the induced affine connection▽. In [5], F. Dillen and L. Vrancken start with this work, and they obtained the classifications for dimensions up to n≤4. However, up to now, the problem of classifying affine hypersurfaces with▽K= 0 for all dimensions is still open.In this paper, we will give a complete classification of affine hypersurfaces in Rn+1 satisfying the conditions▽K=0, Kn-3≠0 and Kn-2=0. From this, we immediately obtain a complete classification of the hypersurfaces with▽K= 0 for dimension n=5.