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Classification of Möbius Isoparametric Hypersurfaces in \({\Bbb S}^5\)

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Abstract.

Let M n be an immersed umbilic-free hypersurface in the (n + 1)-dimensional unit sphere \({\Bbb S}^{n+1}\), then M n is associated with a so-called Möbius metric g, a Möbius second fundamental form B and a Möbius form Φ which are invariants of M n under the Möbius transformation group of \({\Bbb S}^{n+1}\). A classical theorem of Möbius geometry states that M n (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 hypersurfaces are automatically Möbius isoparametrics, whereas the latter are Dupin hypersurfaces.

In this paper, we determine all Möbius isoparametric hypersurfaces in \({\Bbb S}^5\) by proving the following classification theorem: If \(x:M\rightarrow{\Bbb S}^5\) is a Möbius isoparametric hypersurface, then x is Möbius equivalent to either (i) a hypersurface having parallel Möbius second fundamental form in \({\Bbb S}^5\); or (ii) the pre-image of the stereographic projection of the cone in \({\Bbb R}^5\) over the Cartan isoparametric hypersurface in \({\Bbb S}^4\) with three distinct principal curvatures; or (iii) the Euclidean isoparametric hypersurface with four principal curvatures in \({\Bbb S}^5\). The classification of hypersurfaces in \({\Bbb S}^{n+1} (n\ge2)\) with parallel Möbius second fundamental form has been accomplished in our previous paper [7]. The present result is a counterpart of the classification for Dupin hypersurfaces in \({\Bbb E}^5\) up to Lie equivalence obtained by R. Niebergall, T. Cecil and G. R. Jensen.

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Partially supported by DAAD; TU Berlin; Jiechu grant of Henan, China and SRF for ROCS, SEM.

Partially supported by the Zhongdian grant No. 10531090 of NSFC.

Partially supported by RFDP, 973 Project and Jiechu grant of NSFC.

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Hu, Z., Li, H. & Wang, C. Classification of Möbius Isoparametric Hypersurfaces in \({\Bbb S}^5\) . Mh Math 151, 201–222 (2007). https://doi.org/10.1007/s00605-006-0420-x

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