Affine Hypersurfaces with Parallel Shape Operator

Abstract

We prove the following: a relative hypersurface with parallel shape operator is either a relative hypersphere, or it is affinely equivalent to an example constructed by Th. Binder. Furthermore, based on Binder’s example, we give another simple and more explicit example; this way we improve the classification and show that it is completely determined by functions \(\kappa (t)\) and \(C(v_i;t)\), the latter being solutions of certain Monge–Ampere equations. Our example geometrically is constructed from a plane curve and a family of relative hyperspheres. In case of an affine sphere with Blaschke geometry we show that our classification can be considered as a construction coming from a plane curve together with a family of improper affine hyperspheres. Especially in \(\mathbb R^4\), this construction is determined by only three functions of a single variable.