Decomposable affine hypersurfaces

Abstract

In affine differential geometry, Calabi discovered how to associate a new hyperbolic affine hypersphere with two hyperbolic affine hyperspheres. This was later generalized by Dillen and Vrancken in order to obtain a large class of examples of equiaffine homogeneous affine hypersurfaces. Note that the constructions defined above remain valid if one of the affine hyperspheres is a point. In this paper we consider the converse question: how can we determine, given properties of the difference tensor K and the affine shape operator S, whether a given hypersurface can be decomposed as a generalized Calabi product of an affine sphere and a point? © 2014 Faculty of Mathematics, Kyushu University.