A Survey on Isoparametric Hypersurfaces and Their Generalizations

Thorbergsson, Gudlaugur

doi:10.1016/s1874-5741(00)80013-8

Cited by 147 publications

(112 citation statements)

References 87 publications

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“…For a survey on these and related objects, see [Th2]. In order to generalize results known for isoparametric and equifocal submanifolds, we pursue a completely different approach than in the respective theories.…”

Section: Introductionmentioning

confidence: 99%

Parallel focal structure and singular Riemannian foliations

Töben

2005

Trans. Amer. Math. Soc.

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Abstract. We give a necessary and sufficient condition for a submanifold with parallel focal structure to give rise to a global foliation of the ambient space by parallel and focal manifolds. We show that this is a singular Riemannian foliation with complete orthogonal transversals. For this object we construct an action on the transversals that generalizes the Weyl group action for polar actions.

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Section: Introductionmentioning

confidence: 99%

Parallel focal structure and singular Riemannian foliations

Töben

2005

Trans. Amer. Math. Soc.

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“…They then showed that a submanifold with constant principal curvatures is either isoparametric or a focal submanifold of an isoparametric submanifold. The excellent survey article of Thorbergsson [157] gives a more detailed account of the theory of isoparametric submanifolds of codimension greater than one in S n .…”

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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“…For further information on this topic, see [2], [5], [13], [17], or [6], [7]. The theory of isoparametric triple systems was introduced in Dorfmeister's and Neher's paper [4].…”

Section: Isoparametric Triple Systemsmentioning

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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A Survey on Isoparametric Hypersurfaces and Their Generalizations

Cited by 147 publications

References 87 publications

Parallel focal structure and singular Riemannian foliations

Parallel focal structure and singular Riemannian foliations

Isoparametric and Dupin Hypersurfaces

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

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