Isoparametrische Hyperflächen in Sphären

Münzner, Hans

doi:10.1007/bf01420281

Cited by 276 publications

(174 citation statements)

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“…By the work of Münzner [14], [15], in order for λ to be Lie equivalent to an isoparametric hypersurface with four principal curvatures, it is necessary that the multiplicities of the four curvature spheres satisfy m 1 = m 2 and m 3 = m 4 and that the Lie curvature r = −1. In this section, we assume these necessary conditions and then find sufficient conditions in Theorem 7 for Lie equivalence to an isoparametric hypersurface.…”

Section: A Sufficient Condition To Be Isoparametricmentioning

confidence: 99%

“…Thorbergsson [21] showed that the number g of distinct principal curvatures of a compact proper Dupin hypersurface M immersed in S n must be 1, 2, 3, 4 or 6, the same as Münzner's [14,15] restriction on the number of distinct principal curvatures of an isoparametric (constant principal curvatures) hypersurface in S n . In the cases g = 1, 2, 3, compact proper Dupin hypersurfaces in S n have been completely classified.…”

Section: Introductionmentioning

confidence: 96%

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Dupin hypersurfaces with four principal Curvatures, II

2007

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Abstract. If M is an isoparametric hypersurface in a sphere S n with four distrinct principal curvatures, then the principal curvatures κ 1 , . . . , κ 4 can be ordered so that their multiplicities satisfy m 1 = m 2 and m 3 = m 4 , and the cross-ratio r of the principal curvatures (the Lie curvature) equals −1. In this paper, we prove that if M is an irreducible connected proper Dupin hypersurface in R n ( or S n ) with four distinct principal curvatures with multiplicities m 1 = m 2 ≥ 1 and m 3 = m 4 = 1, and constant Lie curvature r = −1, then M is equivalent by Lie sphere transformation to an isoparametric hypersurface in a sphere. This result remains true if the assumption of irreducibility is replaced by compactness and r is merely assumed to be constant.

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Section: A Sufficient Condition To Be Isoparametricmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

Dupin hypersurfaces with four principal Curvatures, II

2007

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“…Such hypersurfaces are natural and very interesting generalizations of (orbits of) isometric cohomogeneity one actions on spheres. A major step towards the understanding of isoparametric hypersurfaces has been done by Münzner in [19], [20]. He proved a finiteness result controlling the topology of the hypersurfaces and an algebraicity result building a bridge between geometry and algebra: any isoparametric hypersurface is given as the zero set of a polynomial equation.…”

Section: Introductionmentioning

confidence: 99%

Algebraic nature of singular Riemannian foliations in spheres

Lytchak

Radeschi

2016

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…By [12,Satz 1], the distinct principal curvatures have at most two different multiplicities m 1 , m 2 . In the following we assume that M has four distinct principal curvatures.…”

Section: Introductionmentioning

confidence: 99%

On the classification of isoparametric hypersurfaces with four distinct curvatures in spheres

Immervoll

2008

Ann. Math.

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In this paper we give a new proof for the classification result in [3]. We show that isoparametric hypersurfaces with four distinct principal curvatures in spheres are of Clifford type provided that the multiplicities m 1 , m 2 of the principal curvatures satisfy m 2 ≥ 2m 1 − 1. This inequality is satisfied for all but five possible pairs (m 1 , m 2 ) with m 1 ≤ m 2 . Our proof implies that for (m 1 , m 2 ) = (1, 1) the Clifford system may be chosen in such a way that the associated quadratic forms vanish on the higher-dimensional of the two focal manifolds. For the remaining five possible pairs (m 1 , m 2 ) with m 1 ≤ m 2 (see [13], [1], and [15]) this stronger form of our result is incorrect: for the three pairs (3, 4), (6, 9), and (7, 8) there are examples of Clifford type such that the associated quadratic forms necessarily vanish on the lower-dimensional of the two focal manifolds, and for the two pairs (2, 2) and (4, 5) there exist homogeneous examples that are not of Clifford type; cf. [5, 4.3, 4.4].

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Isoparametrische Hyperflächen in Sphären

Cited by 276 publications

References 8 publications

Dupin hypersurfaces with four principal Curvatures, II

Dupin hypersurfaces with four principal Curvatures, II

Algebraic nature of singular Riemannian foliations in spheres

On the classification of isoparametric hypersurfaces with four distinct curvatures in spheres

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