Affine geometry of surfaces and hypersurfaces in R

In the following we will give an overview of our research area. Most of it belongs to the field of affine differential geometry. Geometry, as defined in Felix Klein’s Erlanger Programm, is the theory of invariants with respect to a given transformation group. In this sense affine geometry corresponds to the affine group (general linear transformations and translations) and it’s subgroups acting on a vector space. So far, most often studied are surfaces in R3, but also locally strongly convex hypersurfaces in general, mainly with respect to the unimodular (or equiaffine) group. An overview can be found in [LSZ93] and [NS94], see also [LMSS96]. More recently some important global results were achieved (see e. g. [JL05], [TW05]), and the classification of affine hyperspheres with constant sectional curvature is almost complete (see [Vra00] and the references therein). Most of our research is either about questions concerning surfaces or hypersurfaces in R4, often in the context of centroaffine geometry.