Nagoya Mathematical Journal

Classification of Möbius isoparametric hypersurfaces in $\mathbb{S}^{4}$

Zejun Hu and Haizhong Li

Full-text: Open access

Abstract

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

In this paper, we prove that a Möbius isoparametric hypersurface in $\mathbb{S}^{4}$ is either of parallel Möbius second fundamental form or Möbius equivalent to a tube of constant radius over a standard Veronese embedding of $\mathbb{R}P^{2}$ into $\mathbb{S}^{4}$. The classification of hypersurfaces in $\mathbb{S}^{n+1}$ $(n\ge2)$ with parallel Möbius second fundamental form has been accomplished in our previous paper \cite{6}. The present result is a counterpart of Pinkall's classification for Dupin hypersurfaces in $\mathbb{E}^{4}$ up to Lie equivalence.

Article information

Source
Nagoya Math. J., Volume 179 (2005), 147-162.

Dates
First available in Project Euclid: 5 October 2005

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1128518459

Mathematical Reviews number (MathSciNet)
MR2164403

Zentralblatt MATH identifier
1110.53010

Subjects
Primary: 53A30: Conformal differential geometry
Secondary: 53B25: Local submanifolds [See also 53C40]

Citation

Hu, Zejun; Li, Haizhong. Classification of Möbius isoparametric hypersurfaces in $\mathbb{S}^{4}$. Nagoya Math. J. 179 (2005), 147--162. https://projecteuclid.org/euclid.nmj/1128518459


Export citation

References

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