Möbius Isoparametric Hypersurfaces in S n +1 with Two Distinct Principal Curvatures

Li, Hai Zhong; Liu, Huili; Wang, Chang Ping; Zhao, Guo Song

doi:10.1007/s10114-002-0173-y

Cited by 77 publications

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“…Therefore it is remarkable to find that the classification theorem above can be regarded as a counterpart of Pinkall's classification theorem [13] for Dupin hypersurfaces in E 4 under equivalence of Lie sphere transformation. [8], done under the condition γ = 2 for all dimensions, covers for n = 3 the first three cases of the above classification theorem. We note that (1) and (5) are Euclidean isoparametric hypersurfaces in S 4 , whereas (2) and (3) are euclidean isoparametric hypersurfaces in R 4 and H 4 (−1), respectively.…”

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confidence: 99%

Classification of Möbius Isoparametric Hypersurfaces in ⁴

Hu¹,

Li²

2005

Nagoya Mathematical Journal

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Abstract. Let M n be an immersed umbilic-free hypersurface in the (n + 1)-dimensional unit sphere 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 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 isoparametric, whereas the latter are Dupin hypersurfaces.In this paper, we prove that a Möbius isoparametric hypersurface in 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 RP 2 into S 4 . The classification of hypersurfaces in S 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 E 4 up to Lie equivalence. §1. Introduction Let x : M n → S n+1 be a hypersurface in the (n + 1)-dimensional unit sphere S n+1 without umbilic point and {e i } be a local orthonormal basis with respect to the induced metric I = dx · dx with dual basis {θ i }. Let II = i,j h ij θ i ⊗ θ j be the second fundamental form with length square II 2 = i,j (h ij ) 2 and H = 1 n i h ii the mean curvature of x, respectively. Define ρ 2 = n/(n − 1) · ( II 2 − nH 2 ), then the positive definite form g = ρ 2 dx · dx is a Möbius invariant and is called the Möbius metric of x : M n → S n+1 . The Möbius second fundamental form B, another basic Möbius invariant of x, together with g determine completely a hypersurface of S n+1 up to Möbius equivalence, see Theorem 2.2 below. vanishes and all eigenvalues w.r.t. g of the Möbius shape operatorare constant, here S denotes the shape operator of x : M n → S n+1 . This definition of Möbius isoparametric hypersurfaces is meaningful when we compared it with that of (Euclidean) isoparametric hypersurfaces in S n+1 , as we see that the images of all hypersurfaces of the sphere with constant mean curvature and constant scalar curvature under Möbius transformation satisfy Φ ≡ 0 and that the Möbius invariant operator S play the same role in Möbius geometry as S does in the Euclidean situation (see with γ = 2. In this paper, by relaxing the restriction of γ = 2, we will consider Möbius isoparametric hypersurfaces of S 4 and finally determine all of them in explicit form. To state the result, we first recall that for the n-dimensional hyperbolic space of constant sectional curvature −c < 0and the hemisphere S n + = (x 1 , . . . , x n+1 ) ∈ S n i x 2 i = 1, x 1 > 0 , we can define, respectively, the conformal diffeomorphism σ : R n → S n \ (1) the torus S k (a) × S 3−k (b) with k = 1, 2 and a 2 + b 2 = 1;(2) the image of σ of the standard cylinder S k...

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confidence: 99%

Classification of Möbius Isoparametric Hypersurfaces in ⁴

Hu¹,

Li²

2005

Nagoya Mathematical Journal

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“…According to [Li et al 2002], a Möbius isoparametric hypersurface of ‫ޓ‬ n+1 is an umbilic-free hypersurface of ‫ޓ‬ n+1 such whose Möbius-invariant 1-form = −ρ −1 i e i (H ) + j (h i j − H δ i j )e j (log ρ) θ i vanishes and whose Möbius principal curvatures are all constant. These curvatures are the eigenvalues of the Möbius shape operator := ρ −1 (S − H id) with respect to g, where S denotes the shape operator of x : M n → ‫ޓ‬ n+1 .…”

Section: Introductionmentioning

confidence: 99%

“…But there are other examples which cannot be obtained by this way; for example, one occurs in our classification for hypersurfaces of ‫ޓ‬ n+1 with parallel Möbius second fundamental form, that is, those whose Möbius second fundamental form is parallel with respect to the Levi-Civita connection of the Möbius metric g; see [Hu and Li 2004;Li et al 2002] for details. On the other hand, it was proved in [Li et al 2002] that any Möbius isoparametric hypersurface is in particular a Dupin hypersurface, which implies from [Thorbergsson 1983] that for a compact Möbius isoparametric hypersurface embedded in ‫ޓ‬ n+1 , the number γ of distinct principal curvatures can only take the values γ = 2, 3, 4, 6. A characterization of Möbius isoparametric hypersurfaces in terms of Dupin hypersurfaces was given in [Li et al 2002] and was obtained very recently also by L. A.…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, it was proved in [Li et al 2002] that any Möbius isoparametric hypersurface is in particular a Dupin hypersurface, which implies from [Thorbergsson 1983] that for a compact Möbius isoparametric hypersurface embedded in ‫ޓ‬ n+1 , the number γ of distinct principal curvatures can only take the values γ = 2, 3, 4, 6. A characterization of Möbius isoparametric hypersurfaces in terms of Dupin hypersurfaces was given in [Li et al 2002] and was obtained very recently also by L. A. Rodriques and K. Tenenblat [2009]; this characterization states that a Möbius isoparametric hypersurface is either a cyclide of Dupin or a Dupin hypersurface whose Möbius curvatures are constant.…”

Section: Introductionmentioning

confidence: 99%

“…In [Li et al 2002], the authors classified locally all Möbius isoparametric hypersurfaces of ‫ޓ‬ n+1 with γ = 2. By relaxing the restriction that γ = 2, local classifications for all Möbius isoparametric hypersurfaces in ‫ޓ‬ 4 , ‫ޓ‬ 5 and ‫ޓ‬ 6 were established in [Hu and Li 2005], and [Hu and Zhai 2008], respectively.…”

Section: Introductionmentioning

confidence: 99%

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Möbius isoparametric hypersurfaces with three distinct principal curvatures, II

Hu¹,

Zhai²

2011

Pacific J. Math.

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Using the method of moving frames and the algebraic techniques of T. E. Cecil and G. R. Jensen that were developed while they classified the Dupin hypersurfaces with three principal curvatures, we extend Hu and Li's main theorem in Pacific J. Math. 232:2 (2007), 289-311 by giving a complete classification for all Möbius isoparametric hypersurfaces in ‫ޓ‬ n+1 with three distinct principal curvatures.

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Isoparametric Hypersurfaces

Cecil¹,

Ryan²

2015

Springer Monographs in Mathematics

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The classification of isoparametric hypersurfaces with four principal curvatures in spheres in [2] hinges on a crucial characterization, in terms of four sets of equations of the 2nd fundamental form tensors of a focal submanifold, of an isoparametric hypersurface of the type constructed by Ferus, Karcher and Münzner. The proof of the characterization in [2] is an extremely long calculation by exterior derivatives with remarkable cancellations, which is motivated by the idea that an isoparametric hypersurface is defined by an over-determined system of partial differential equations. Therefore, exterior differentiating sufficiently many times should gather us enough information for the conclusion. In spite of its elementary nature, the magnitude of the calculation and the surprisingly pleasant cancellations make it desirable to understand the underlying geometric principles.In this paper, we give a conceptual, and considerably shorter, proof of the characterization based on Ozeki and Takeuchi's expansion formula for the Cartan-Münzner polynomial. Along the way the geometric meaning of these four sets of equations also becomes clear.1991 Mathematics Subject Classification. Primary 53C40.

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Möbius Isoparametric Hypersurfaces in S n +1 with Two Distinct Principal Curvatures

Cited by 77 publications

References 8 publications

Classification of Möbius Isoparametric Hypersurfaces in ⁴

Classification of Möbius Isoparametric Hypersurfaces in ⁴

Möbius isoparametric hypersurfaces with three distinct principal curvatures, II

Isoparametric Hypersurfaces

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Möbius Isoparametric Hypersurfaces in S n +1 with Two Distinct Principal Curvatures

Cited by 77 publications

References 8 publications

Classification of Möbius Isoparametric Hypersurfaces in 4

Classification of Möbius Isoparametric Hypersurfaces in 4

Möbius isoparametric hypersurfaces with three distinct principal curvatures, II

Isoparametric Hypersurfaces

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Classification of Möbius Isoparametric Hypersurfaces in ⁴

Classification of Möbius Isoparametric Hypersurfaces in ⁴