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| Classification of the Conformally Flat Centroaffine Hypersurfaces with Vanishing Centroaffine Shape Operator |
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Citation： |
Miaoxin LEI,Ruiwei XU,Peibiao ZHAO.Classification of the Conformally Flat Centroaffine Hypersurfaces with Vanishing Centroaffine Shape Operator[J].Chinese Annals of Mathematics B,2025,46(2):163~180 |
| Page view： 502
Net amount： 249 |
Authors： |
Miaoxin LEI; Ruiwei XU;Peibiao ZHAO |
Foundation： |
the National Natural Science Foundation of China (Nos. 11871197,
12141104, 12271254). |
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| Abstract： |
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+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). |
Keywords： |
Centroaffine hypersurface Centroaffine shape operator Calabi product,
Locally conformally flat Calabi hypersurface |
Classification： |
53A15, 53C24, 53C42 |
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