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

Immervoll, Stefan

doi:10.4007/annals.2008.168.1011

Cited by 72 publications

(8 citation statements)

References 14 publications

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“…On the other hand, in S n+1 , there are many more examples (cf. [5,12,18,42,44]). Münzner ([30, 31]) showed that the number r of distinct principal curvatures of an isoparametric hypersurface in S n+1 must be 1, 2, 3, 4 or 6.…”

Section: Introductionmentioning

confidence: 99%

Möbius curvature, Laguerre curvature and Dupin hypersurface

Qing

Wang

2017

Advances in Mathematics

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In this paper we show that a Dupin hypersurface with constant Möbius curvatures isMöbius equivalent to either an isoparametric hypersurface in the sphere or a cone over an isoparametric hypersurface in a sphere. We also show that a Dupin hypersurface with constant Laguerre curvatures is Laguerre equivalent to a flat Laguerre isoparametric hypersurface. These results solve the major issues related to the conjectures of Cecil et al on the classification of Dupin hypersurfaces.

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Section: Introductionmentioning

confidence: 99%

Möbius curvature, Laguerre curvature and Dupin hypersurface

Qing

Wang

2017

Advances in Mathematics

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“…After this, Immervoll [72] gave a different proof of the theorem of Cecil, Chi and Jensen using isoparametric triple systems. The use of triple systems to study isoparametric hypersurfaces was introduced in a series of papers in the 1980's by Dorfmeister and Neher [45,46,47,48,49,50].…”

Section: Multiplicities Of the Principal Curvatures Of Fkm-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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“…It has been proven that every foliation in round spheres by isoparametric hypersurfaces with 4 principal curvatures is either homogeneous or of FKM type, except possibly for a finite number of isolated cases (cf. [10]).…”

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confidence: 99%

Clifford algebras and new singular Riemannian foliations in spheres

Radeschi

2014

Geom. Funct. Anal.

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Using representations of Clifford algebras we construct indecomposable singular Riemannian foliations on round spheres, most of which are nonhomogeneous. This generalises the construction of non-homogeneous isoparametric hypersurfaces due to by Ferus, Karcher and Münzner.A singular Riemannian foliation on a Riemannian manifold M is, roughly speaking, a partition of M into connected complete submanifold, not necessarily of the same dimension, that locally stay at a constant distance from each other. Singular Riemannian foliations on round spheres provide local models of general singular Riemannian foliations around a point.An example of singular Riemannian foliation on round spheres is given by the decomposition into the orbits of an isometric group action, and such a foliation is called homogeneous.A different family of singular Riemannian foliations on spheres is induced by isoparametric hypersurfaces. A hypersurfaces of S n is called isoparametric if it has constant principal curvatures. Isoparametric hypersurfaces were first studied by Cartan who classified those with g ≤ 3 distinct principal curvatures, and a lot of progress has been made (cf. for example the surveys [Cec08,Tho00]), even though the complete classification is still an important open problem. Every isoparametric hypersurface partitions the sphere into parallel hypersurfaces, which are isoparametric as well, and this partition is a special example of a singular Riemannian foliation. For a long time all the known codimension 1 singular Riemannian foliations from isoparametric hypersurfaces appeared to be orbits of some isometric group action on S n , so much so that Cartan asked [Car39] whether every isoparametric hypersurface arised in this way. The question was answered in the negative by Ozeki and Takeuchi [OT75,OT76], who found infinite families of non homogeneous isoparametric foliations with 4 distinct principal curvatures defined in terms of representations Mathematics Subject Classification: 53C12, 57R30

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

Cited by 72 publications

References 14 publications

Möbius curvature, Laguerre curvature and Dupin hypersurface

Möbius curvature, Laguerre curvature and Dupin hypersurface

Isoparametric and Dupin Hypersurfaces

Clifford algebras and new singular Riemannian foliations in spheres

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