Affine hypersurfaces with parallel cubic forms

Abstract

In this note, we first investigate the symmetry properties of the tensors VC and F 2 C, where V is the induced affine connection and C is the cubic form of a nondegenerate affine hypersurface M n in R n + 1 . In particular, we study hypersurfaces with parallel cubic form, i.e. PC=0. In the case = 2, this condition is known to characterize a Cayley surface (Nomizu and Pinkall [3]). We obtain a certain class of more general affine surfaces and hypersurfaces.