On widetilde{J}-tangent Affine Hyperspheres

Abstract

In this paper we study widetilde{J}-tangent affine hyperspheres, where widetilde{J} is the canonical para-complex structure on mathbb {R}^{2n+2}. The main purpose of this paper is to give a classification of widetilde{J}-tangent affine hyperspheres of an arbitrary dimension with an involutive distribution mathcal {D}. In particular, we classify all such hyperspheres in the 3-dimensional case. We also show that there is a direct relation between widetilde{J}-tangent affine hyperspheres and Calabi products. As an application we obtain certain classification results. In particular, we show that, with one exception, all odd dimensional proper flat affine hyperspheres are, after a suitable affine transformation, widetilde{J}-tangent. Some examples of widetilde{J}-tangent affine hyperspheres are also given.