Affine hypersurfaces with parallel difference tensor relative to affineα-connection

Abstract

Li and Zhang (2014) studied affine hypersurfaces of R n + 1 with parallel difference tensor relative to the affine α -connection ∇ ( α ) , and characterized the generalized Cayley hypersurfaces by K n − 1 ≠ 0 and ∇ ( α ) K = 0 for some nonzero constant α , where the affine α -connection ∇ ( α ) of information geometry was introduced on affine hypersurface. In this paper, by a slightly different method we continue to study affine hypersurfaces with ∇ ( α ) K = 0 , if α = 0 we 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.