Affine Hypersurfaces with Parallel Shape Operator

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 κ(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa (t)$$\end{document} and C(vi;t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(v_i;t)$$\end{document}, the latter being solutions of certain Monge–Ampère 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 R4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb R^4$$\end{document}, this construction is determined by only three functions of a single variable.