Cliffordalgebren und neue isoparametrische Hyperfl�chen

Ferus, Dirk; Karcher, Hermann; Münzner, Hans

doi:10.1007/bf01219082

Cited by 267 publications

(270 citation statements)

References 6 publications

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“…Next, Ferus, Karcher and Münzner [13] used representations of Clifford algebras to construct for any positive integer m 1 an infinite series of isoparametric hypersurfaces with four principal curvatures having multiplicities (m 1 , m 2 ), where m 2 is nondecreasing and unbounded in each series. In fact, m 2 = kδ(m 1 ) − m 1 − 1, where δ(m 1 ) is the positive integer such that the Clifford algebra C m1−1 has an irreducible representation on R δ(m1) (see [2]), and k is any positive integer for which m 2 is positive.…”

Section: Introductionmentioning

confidence: 99%

Isoparametric hypersurfaces with four principal curvatures

2007

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Let M be an isoparametric hypersurface in the sphere S n with four distinct principal curvatures. Münzner showed that the four principal curvatures can have at most two distinct multiplicities m 1 , m 2 , and Stolz showed that the pair (m 1 , m 2 ) must either be (2, 2), (4, 5), or be equal to the multiplicities of an isoparametric hypersurface of FKM-type, constructed by Ferus, Karcher and Münzner from orthogonal representations of Clifford algebras. In this paper, we prove that if the multiplicities satisfy m 2 ≥ 2m 1 − 1, then the isoparametric hypersurface M must be of FKM-type. Together with known results of Takagi for the case m 1 = 1, and Ozeki and Takeuchi for m 1 = 2, this handles all possible pairs of multiplicities except for four cases, for which the classification problem remains open.

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Section: Introductionmentioning

confidence: 99%

Isoparametric hypersurfaces with four principal curvatures

2007

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“…Molino, not being aware of the existence of non-homogeneous isoparametric foliations (cf. [13]), has conjectured that all singular Riemannian foliations in Euclidean spheres are homogeneous. However, there is in fact a vast class of non-homogeneous examples (cf.…”

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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“…To see this, notice that the poly- If we rewrite the Pj in terms of the E. (see [FKM,3.3]) and substitute (u,v ) for x we see that these equations are equivalent to…”

Section: Clifford 2-framesmentioning

confidence: 99%

Deformations of Dupin hypersurfaces

Pinkall¹,

Thorbergsson²

1989

Proc. Amer. Math. Soc.

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Abstract.Examples are given of compact proper-Dupin hypersurfaces that are not Lie equivalent to an isoparametric hypersurface. DUPIN HYPERSURFACESA question raised by many authors is whether a compact proper-Dupin hypersurface is equivalent to an isoparametric hypersurface under a Lie transformation. Here we say that a hypersurface or more generally a submanifold of S"4 is proper-Dupin if the principal curvatures have globally constant multiplicities and are constant along the corresponding curvature leaves which lie in the unit normal bundle if the codimension is greater than one. An evidence for a positive answer to the question is that the geometry and topology of proper-Dupin hypersurfaces and isoparametric hypersurfaces are very similar. E.g., for both classes the number g of principal curvatures can only be 1, 2, 3, 4 or 6 and the dimension of the Z-,-cohomology ring is 2g, see [Mii] and [Th]. See also [GH] for further similarities in the topology of these two classes of hypersurfaces.The question has been answered positively for the cases g = 1 (trivial), g -2 [CR], see also [Pi], and g = 3 [Mi 1]. It will be shown in the present paper that the answer is negative for g = 4. More precisely, we will find a oneparameter deformation M(t) of any isoparametric hypersurface M of Clifford

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Cliffordalgebren und neue isoparametrische Hyperfl�chen

Cited by 267 publications

References 6 publications

Isoparametric hypersurfaces with four principal curvatures

Isoparametric hypersurfaces with four principal curvatures

Algebraic nature of singular Riemannian foliations in spheres

Deformations of Dupin hypersurfaces

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