Acta mathematica scientia,Series B ›› 2023, Vol. 43 ›› Issue (6): 2413-2429.doi: 10.1007/s10473-023-0606-5

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CONFORMALLY FLAT AFFINE HYPERSURFACES WITH SEMI-PARALLEL CUBIC FORM*

Huiyang XU, Cece LI   

  1. School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471023, China
  • Received:2022-06-10 Revised:2023-05-20 Published:2023-12-08
  • Contact: †Cece LI, E-mail: ceceli@haust.edu.cn
  • About author:Huiyang XU, E-mail: xuhuiyang@haust.edu.cn
  • Supported by:
    This work was supported by the NNSF of China (12101194, 11401173).

Abstract: In this paper, we study locally strongly convex affine hypersurfaces with the vanishing Weyl curvature tensor and semi-parallel cubic form relative to the Levi-Civita connection of the affine metric. As a main result, we classify these hypersurfaces as not being of a flat affine metric. In particular, 2 and 3-dimensional locally strongly convex affine hypersurfaces with semi-parallel cubic forms are completely determined.

Key words: affine hypersurface, semi-parallel cubic form, Levi-Civita connection, conformally flat, warped product

CLC Number: 

  • 53A15
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