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Locally Strongly Convex Affine Hypersurfaces with Semi-parallel Cubic Form

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Abstract

In this paper, we investigate the locally strongly convex affine hypersurfaces with semi-parallel cubic form relative to the Levi-Civita connection of affine metric. We obtain two results on such hypersurfaces which admit at most one affine principal curvature of multiplicity one: (1) classify these being not affine hyperspheres; (2) classify these affine hyperspheres with constant scalar curvature. For the latter, by proving the parallelism of their cubic forms we translate the classification into that of affine hypersurfaces with parallel cubic form, which has been completed by Hu-Li-Vrancken (J Differ Geom 87:239–307, 2011).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, 471023, People’s Republic of China

    Cece Li & Huiyang Xu

  2. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, People’s Republic of China

    Cheng Xing

  3. School of Mathematical Sciences, Nankai University, Tianjin, 300071, People’s Republic of China

    Cheng Xing

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  1. Cece Li
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  2. Cheng Xing
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Li was supported by NNSF of China, Grant Number 11401173; Xing was supported by NNSF of China, Grant Number 12171437; Xu was supported by NNSF of China, Grant Number 12101194.

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Li, C., Xing, C. & Xu, H. Locally Strongly Convex Affine Hypersurfaces with Semi-parallel Cubic Form. J Geom Anal 33, 81 (2023). https://doi.org/10.1007/s12220-022-01133-5

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