Classification of the Conformally Flat Centroaffine Hypersurfaces with Vanishing Centroaffine Shape Operator

Cheng-Hu-Moruz（2017） completely classified the locally strongly convex centroaffine hypersurfaces with parallel cubic form based on the Calabi product（called the type I Calabi product for short） proposed by Li-Wang（1991）.In the present paper, the authors introduce the type II Calabi product（in case λ1 =2λ2）, complementing the type I Calabi product（in case λ1 =≠ 2λ2）, and achieve a classification of the locally strongly convex centroaffine hypersurfaces in Rn+1with vanishing centroaffine shape operator and Weyl curvature tensor by virtue of the types I and II Calabi product.As a corollary, 3-dimensional complete locally strongly convex centroaffine hypersurfaces with vanishing centroaffine shape operator are completely classified, which positively answers the centroaffine Bernstein problems III and V by Li-Li-Simon（2004）.