Special classes of three dimensional affine hyperspheres characterized by properties of their cubic form

Abstract

It is well known that locally strongly convex ane hyperspheres can be determined as solutions of dierential equations of Monge-Ampere type. The global properties of those solutions are well understood. However, due to the nature of the Monge-Ampere equation, not much isknown about local solutions, particularly if the dimension of the hypersurface is greater then 2. By the fundamental theorem, ane hyperspheres are completely determined by their metric h and their dierence tensor K which together build the symmetric cubic form C . Following an idea of Bryant [1],we want toinvestigate ane hyperspheres for which at each point there exist isometries with respect to h preserving this cubic form. The rst non-trivial case is the case that M is 3-dimensional which is also the case which is investigated further in this paper.