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<p>This paper is concerned with an open problem regarding classifying <i>n</i>-dimensional locally strongly convex centroaffine hypersurfaces with conformally flat centroaffine metrics for <i>n</i> ≥ 3. First of all, we give a classification of such hypersurfaces by assuming that the Tchebychev operator (i.e., centroaffine shape operator) is Codazzi; this includes those hypersurfaces with vanishing or parallel Tchebychev operators as subclasses. Second, these hypersurfaces are classified under the geometric condition that the Ricci tensor is semi-parallel, as a natural extension of the Einstein condition. New examples are constructed in this paper, and all examples in the classifications above are calculated in detail in order to better illustrate our results.</p>

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On conformally flat centroaffine hypersurfaces

  • 期刊论文

  • 发表日期：

    2026年5月9日

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  • Cheng Xing,
  • Cece Li

摘要

This paper is concerned with an open problem regarding classifying n-dimensional locally strongly convex centroaffine hypersurfaces with conformally flat centroaffine metrics for n ≥ 3. First of all, we give a classification of such hypersurfaces by assuming that the Tchebychev operator (i.e., centroaffine shape operator) is Codazzi; this includes those hypersurfaces with vanishing or parallel Tchebychev operators as subclasses. Second, these hypersurfaces are classified under the geometric condition that the Ricci tensor is semi-parallel, as a natural extension of the Einstein condition. New examples are constructed in this paper, and all examples in the classifications above are calculated in detail in order to better illustrate our results.