On (J)over-tilde-tangent Affine Hyperspheres

摘要

In this paper we study (J) over tilde -tangent affine hyperspheres, where (J) over tilde is the canonical para-complex structure on R2n+2. The main purpose of this paper is to give a classification of (J) over tilde -tangent affine hyperspheres of an arbitrary dimension with an involutive distribution D. In particular, we classify all such hyperspheres in the 3-dimensional case. We also show that there is a direct relation between (J) over tilde -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, (J) over tilde -tangent. Some examples of (J) over tilde -tangent affine hyperspheres are also given.