Tohoku Mathematical Journal
2008
December
SECOND SERIES VOL. 60, NO. 4

Tohoku Math. J.
60 (2008), 499-526

Title CLASSIFICATION OF MÖBIUS ISOPARAMETRIC HYPERSURFACES IN THE UNIT SIX-SPHERE

Dedicated to Professors Udo Simon and Seiki Nishikawa on the occasion of their seventieth and sixtieth birthday
Author Zejun Hu and Shujie Zhai

(Received October 11, 2007, revised April 28, 2008)
Abstract. An immersed umbilic-free hypersurface in the unit sphere is equipped with three Möbius invariants, namely, the Möbius metric, the Möbius second fundamental form and the Möbius form. The fundamental theorem of Möbius submanifolds geometry states that a hypersurface of dimension not less than three is uniquely determined by the Möbius metric and the Möbius second fundamental form. A Möbius isoparametric hypersurface is defined by two conditions that it has vanishing Möbius form and has constant Möbius principal curvatures. It is well-known that all Euclidean isoparametric hypersurfaces are Möbius isoparametrics, whereas the latter are Dupin hypersurfaces. In this paper, combining with previous results, a complete classification for all Möbius isoparametric hypersurfaces in the unit six-sphere is established.

2000 Mathematics Subject Classification. Primary 53A30; Secondary 53B25.
Key words and phrases. Möbius isoparametric hypersurface, Möbius equivalence, Möbius second fundamental form, Möbius metric, Möbius form.

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