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Classification of Möbius Isoparametric Hypersurfaces in 4

Published online by Cambridge University Press:  11 January 2016

Zejun Hu  and
Haizhong Li
Zejun Hu
Affiliation:
Department of Mathematics Zhengzhou University, 450052 Zhengzhou People’s Republic of [email protected]
Haizhong Li
Affiliation:
Department of Mathematical Sciences, Tsinghua University, 100084 Beijing People’s Republic of [email protected]
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Abstract

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Let Mn be an immersed umbilic-free hypersurface in the (n + 1)-dimensional unit sphere n+1, then Mn is associated with a so-called Möbius metric g, a Möbius second fundamental form B and a Möbius form Φ which are invariants of Mn under the Möbius transformation group of n+1. A classical theorem of Möbius geometry states that Mn (n ≥ 3) is in fact characterized by g and B up to Möbius equivalence. A Möbius isoparametric hypersurface is defined by satisfying two conditions: (1) Φ ≡ 0; (2) All the eigenvalues of B with respect to g are constants. Note that Euclidean isoparametric hyper-surfaces are automatically Möbius isoparametric, whereas the latter are Dupin hypersurfaces.

In this paper, we prove that a Möbius isoparametric hypersurface in 4 is either of parallel Möbius second fundamental form or Möbius equivalent to a tube of constant radius over a standard Veronese embedding of ℝP2 into 4. The classification of hypersurfaces in n+1 (n ≥ 2) with parallel Möbius second fundamental form has been accomplished in our previous paper [6]. The present result is a counterpart of Pinkall’s classification for Dupin hypersurfaces in 4 up to Lie equivalence.

Keywords

53A3053B25
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 179 , 2005 , pp. 147 - 162
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2005

References

[1] Akivis, M. A. and Goldberg, V. V., Conformal differential geometry and its generalizations, Wiley, New York, 1996.Google Scholar
[2] Akivis, M. A. and Goldberg, V. V., A conformal differential invariant and the conformal rigidity of hypersurfaces, Proc. Amer. Math. Soc., 125 (1997), 2415–2424.CrossRefGoogle Scholar
[3] Cartan, E., Sur des familles remarquables d’hypersurfaces isoparametriques dans les espace spheriques, Math. Z., 45 (1939), 335–367.CrossRefGoogle Scholar
[4] Guo, Z., Li, H. and Wang, C. P., The second variation formula for Willmore submanifolds in n , Results in Math., 40 (2001), 205–225.CrossRefGoogle Scholar
[5] Hu, Z. J. and Li, H., Submanifolds with constant Möbius scalar curvature in n , Manuscripta Math., 111 (2003), 287–302.CrossRefGoogle Scholar
[6] Hu, Z. J. and Li, H., Classification of hypersurfaces with parallel Möbius second fundamental form in n+1 , Science in China Ser. A, Mathematics, 47 (2004), 417–430.CrossRefGoogle Scholar
[7] Hu, Z. J. and Li, H., A rigidity theorem for hypersurfaces with positive Möbius Ricci curvature in n+1 , to appear, Tsukuba J. Math., 29 (2005), no. 1.CrossRefGoogle Scholar
[8] Li, H., Liu, H., Wang, C. P. and Zhao, G. S., Möbius isoparametric hypersurfaces in n+1 with two distinct principal curvatures, Acta Math. Sinica, English Series, 18 (2002), 437–446.CrossRefGoogle Scholar
[9] Li, H. and Wang, C. P., Surfaces with vanishing Möbius form in n , Acta Math. Sinica, English Series, 19 (2003), 671–678.Google Scholar
[10] Li, H. and Wang, C. P., Möbius geometry of hypersurfaces with constant mean curvature and constant scalar curvature, Manuscripta Math., 112 (2003), 1–13.CrossRefGoogle Scholar
[11] Li, H., Wang, C. P. and Wu, F., A Möbius characterization of Veronese surfaces in n , Math. Ann., 319 (2001), 707–714.CrossRefGoogle Scholar
[12] Liu, H. L., Wang, C. P. and Zhao, G. S., Möbius isotropic submanifolds in n , Tohoku Math. J., 53 (2001), 553–569.CrossRefGoogle Scholar
[13] Pinkall, U., Dupinsche hyperflächen in 4 , Manuscripta Math., 51 (1985), 89–119.CrossRefGoogle Scholar
[14] Thorbergsson, G., Dupin hypersurfaces, Bull. London Math. Soc., 15 (1983), 493–498.CrossRefGoogle Scholar
[15] Wang, C. P., Möbius geometry of submanifolds in n , Manuscripta Math., 96 (1998), 517–534.CrossRefGoogle Scholar
[16] Wang, C. P., Möbius geometry for hypersurfaces in 4 , Nagoya Math. J., 139 (1995), 1–20.CrossRefGoogle Scholar