Affine hypersurfaces with parallel difference tensor relative to affine α-connection

Li and Zhang (2014) studied affine hypersurfaces of Rn+1 with parallel difference tensor relative to the affine alpha-connection del((alpha)), and characterized the generalized Cayley hypersurfaces by Kn-1 not equal 0 and del K-(alpha) = 0 for some nonzero constant a, where the affine alpha-connection del((alpha)) of information geometry was introduced on affine hypersurface. In this paper, by a slightly different method we continue to study affine hypersurfaces with del K-(alpha) = 0, if alpha = Owe further assume that the Pick invariant vanishes and affine metric is of constant sectional curvature. It is proved that they are either hyperquadrics or improper affine hypersphere with flat indefinite affine metric, the latter can be locally given as a graph of a polynomial of at most degree n + 1 with constant Hessian determinant. In particular, if the affine metric is definite, Lorentzian, or its negative index is 2, we complete the classification of such hypersurfaces. (C) 2014 Elsevier B.V. All rights reserved.