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On Möbius form and Möbius isoparametric hypersurfaces

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Abstract

An umbilic-free hypersurface in the unit sphere is called Möbius isoparametric if it satisfies two conditions, namely, it has vanishing Möbius form and has constant Möbius principal curvatures. In this paper, under the condition of having constant Möbius principal curvatures, we show that the hypersurface is of vanishing Möbius form if and only if its Möbius form is parallel with respect to the Levi-Civita connection of its Möbius metric. Moreover, typical examples are constructed to show that the condition of having constant Möbius principal curvatures and that of having vanishing Möbius form are independent of each other.

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Authors and Affiliations

  1. Department of Mathematics, Zhengzhou University, Zhengzhou, 450052, P. R. China

    Ze Jun Hu

  2. Department of Humanities and Social Sciences, He’nan Vocational and Technical College of Communications, Zhengzhou, 450007, P. R. China

    Xiao Li Tian

Authors
  1. Ze Jun Hu
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  2. Xiao Li Tian
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Correspondence to Ze Jun Hu.

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Supported by National Natural Science Foundation of China (Grant No. 10671181)

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Hu, Z.J., Tian, X.L. On Möbius form and Möbius isoparametric hypersurfaces. Acta. Math. Sin.-English Ser. 25, 2077–2092 (2009). https://doi.org/10.1007/s10114-009-7682-x

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  • DOI: https://doi.org/10.1007/s10114-009-7682-x

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