Möbius isoparametric hypersurfaces with three distinct principal curvatures

Hu, Zejun; Zhai, Shujie

doi:10.2140/pjm.2007.232.289

Cited by 22 publications

(19 citation statements)

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“…From the above discussions (I)-(IV), it follows that, if x does not have four distinct Möbius principal curvatures then it must have three distinct Möbius principal curvatures with one of which is simple. But in this case x can have at most two distinct Blaschke eigenvalues (see [10]), contradicting our main assumption. This completes the proof of Lemma 4.6.…”

Section: Discussionmentioning

confidence: 74%

“…To the authors' knowledge, many interesting and important results in this area have been obtained. Among these results are some nice classification theorems of submanifolds with particular Möbius invariants, for example, the classification of all immersed hypersurfaces in S m+1 with parallel Möbius second fundamental form [8], and classification theorems of so called Möbius isoparametric hypersurfaces [9][10][11][12][13].…”

Section: Introductionmentioning

confidence: 99%

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Classification of the Blaschke Isoparametric Hypersurfaces with Three Distinct Blaschke Eigenvalues

Peng

2010

Results. Math.

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An immersed umbilic-free submanifold in the unit sphere is called Blaschke isoparametric if its Möbius form vanishes identically and all of its Blaschke eigenvalues are constant. Then the classification of Blaschke isoparametric hypersurfaces is natural and interesting in the Möbius geometry of submanifolds. In this paper, we give a classification of the Blaschke isoparametric hypersurfaces with three distinct Blaschke eigenvalues one of which is simple.Mathematics Subject Classification (2010). Primary 53A30; Secondary 53B25.

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Section: Discussionmentioning

confidence: 74%

Section: Introductionmentioning

confidence: 99%

Classification of the Blaschke Isoparametric Hypersurfaces with Three Distinct Blaschke Eigenvalues

Peng

2010

Results. Math.

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“…(3.41) [7,8]. For any natural number p, q, p + q < n and real number r ∈ (0, 1), consider the immersed hypersurface u : (1) without umbilical points and with vanishing Möbius form, it is denoted by CSS(p, q, r).…”

Section: Propositions and Typical Examplesmentioning

confidence: 99%

“…For any natural number p, q, p + q < n and real number r ∈ (0, 1), consider the immersed hypersurface u : (1) without umbilical points and with vanishing Möbius form, it is denoted by CSS(p, q, r). From [7] and [8], by a direct calculation, we know that CSS(p, q, r) has three distinct Möbius principal curvatures. In particular, if p = q and r = 1 √ 2 then CSS(p, q, r) has exactly three distinct Blaschke eigenvalues.…”

Section: Propositions and Typical Examplesmentioning

confidence: 99%

PARA-BLASCHKE ISOPARAMETRIC HYPERSURFACES IN A UNIT SPHERE S^{n + 1}(1)

Шу

2012

Glasgow Math. J.

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Abstract. Let A = ρ 2 i,j A ij θ i ⊗ θ j and B = ρ 2 i,j B ij θ i ⊗ θ j be the Blaschke tensor and the Möbius second fundamental form of the immersion x. Let D = A + λB be the para-Blaschke tensor of x, where λ is a constant. If x : M n → S n+1 (1) is an n-dimensional para-Blaschke isoparametric hypersurface in a unit sphere S n+1 (1) and x has three distinct Blaschke eigenvalues one of which is simple or has three distinct Möbius principal curvatures one of which is simple, we obtain the full classification theorems of the hypersurface.2001 Mathematics Subject Classification. 53A30, 53B25.

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“…Then Hu and D. Li [69] studied Möbius isoparametric hypersurfaces with three distinct principal curvatures in S n and found a complete classification of such hypersurfaces in S 6 .…”

Section: Dupin Hypersurfacesmentioning

confidence: 99%

Isoparametric and Dupin Hypersurfaces

Cecil¹

2008

SIGMA

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Abstract. A hypersurface M n−1 in a real space-form R n , S n or H n is isoparametric if it has constant principal curvatures. For R n and H n , the classification of isoparametric hypersurfaces is complete and relatively simple, but asÉlie Cartan showed in a series of four papers in 1938-1940, the subject is much deeper and more complex for hypersurfaces in the sphere S n . A hypersurface M n−1 in a real space-form is proper Dupin if the number g of distinct principal curvatures is constant on M n−1 , and each principal curvature function is constant along each leaf of its corresponding principal foliation. This is an important generalization of the isoparametric property that has its roots in nineteenth century differential geometry and has been studied effectively in the context of Lie sphere geometry. This paper is a survey of the known results in these fields with emphasis on results that have been obtained in more recent years and discussion of important open problems in the field.

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Möbius isoparametric hypersurfaces with three distinct principal curvatures

Abstract: We investigate Möbius isoparametric hypersurfaces in the (n+1)-Euclidean unit sphere ‫ޓ‬ n+1 with three distinct Möbius principal curvatures. As direct consequence of our main result, we establish the complete classification for all such hypersurfaces in ‫ޓ‬ 6 .

Cited by 22 publications

References 25 publications

Classification of the Blaschke Isoparametric Hypersurfaces with Three Distinct Blaschke Eigenvalues

Classification of the Blaschke Isoparametric Hypersurfaces with Three Distinct Blaschke Eigenvalues

PARA-BLASCHKE ISOPARAMETRIC HYPERSURFACES IN A UNIT SPHERE S^{n + 1}(1)

Isoparametric and Dupin Hypersurfaces

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Möbius isoparametric hypersurfaces with three distinct principal curvatures

Abstract: We investigate Möbius isoparametric hypersurfaces in the (n+1)-Euclidean unit sphere ‫ޓ‬ n+1 with three distinct Möbius principal curvatures. As direct consequence of our main result, we establish the complete classification for all such hypersurfaces in ‫ޓ‬ 6 .

Cited by 22 publications

References 25 publications

Classification of the Blaschke Isoparametric Hypersurfaces with Three Distinct Blaschke Eigenvalues

Classification of the Blaschke Isoparametric Hypersurfaces with Three Distinct Blaschke Eigenvalues

PARA-BLASCHKE ISOPARAMETRIC HYPERSURFACES IN A UNIT SPHERE Sn + 1(1)

Isoparametric and Dupin Hypersurfaces

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PARA-BLASCHKE ISOPARAMETRIC HYPERSURFACES IN A UNIT SPHERE S^{n + 1}(1)