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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

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