Generic Affine Hypersurfaces with Self Congruent Center Map

In this paper, we study n-dimensional locally strongly convex hypersurfaces in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${\mathbb{R}}^{n+1}$$ \end{document} with the property that its center map is congruent to the original immersion. Our aim is to study such hypersurfaces in the generic case, i.e., when the affine shape operator has n different non zero eigenvalues. We will show that for such hypersurfaces the centroaffine metric is flat and the centroaffine difference tensor is parallel with respect to the Levi Civita connection of the centroaffine metric. In particular this means that they are canonical hypersurfaces in the sense of [3]. Extending slightly the classification of such hypersurfaces in [3], we obtain a complete classification.