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Centroaffine T-Umbilical Hypersurfaces and Pseudo-Parallel Cubic Form

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Abstract

In this article, we introduce the notion of locally strongly convex centroaffine T-umbilical hypersurfaces, which may be the “simplest” centroaffine hypersurfaces next to the hyperquadrics centered at the origin. These hypersurfaces are proved to be of a conformally flat, quasi-Einstein centroaffine metric and the pseudo-parallel cubic form relative to the Levi-Civita connection of the centroaffine metric. As the main results, we find a geometric characterization of those hypersurfaces as not being of constant sectional curvature, and further classify these hypersurfaces. Meanwhile, for locally strongly convex centroaffine hypersurfaces with pseudo-parallel cubic form, we completely determine the 3-dimensional case, and all dimensions for the conformally flat case.

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Funding

This work was supported by National Natural Science Foundation of China, Grant Numbers 11401173, 12101194.

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

    Cece Li & Huiyang Xu

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  1. Cece Li
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Correspondence to Cece Li.

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Li, C., Xu, H. Centroaffine T-Umbilical Hypersurfaces and Pseudo-Parallel Cubic Form. Results Math 79, 14 (2024). https://doi.org/10.1007/s00025-023-02045-8

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