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...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 λ_(1)=2λ_(2)),complementing the type I Calabi product(in caseλ_(1)≠2λ_(2)),and achieve a classification of the locally strongly convex centroaffine hypersurfaces in R^(n+1) with 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).展开更多
基金supported by the National Natural Science Foundation of China(Nos.11871197,12141104,12271254).
文摘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 λ_(1)=2λ_(2)),complementing the type I Calabi product(in caseλ_(1)≠2λ_(2)),and achieve a classification of the locally strongly convex centroaffine hypersurfaces in R^(n+1) with 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).
