Isoparametric Hypersurfaces with Four Principal Curvatures

By Thomas E. Cecil, Quo-Shin Chi, G. Jensen. 2004

212 citations recorded by the source corpus.

Let M be an isoparametric hypersurface in the sphere S n with four distinct principal curvatures. Munzner showed that the four principal curvatures can have at most two distinct multiplicities m1 ,m 2, and Stolz showed that the pair (m1 ,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 Munzner from orthogonal representations of Clifford algebras. In this paper, we prove that if the multiplicities satisfy m2 ≥ 2m1 −1, then the isoparametric hypersurface M must be of FKM-type. Together with known results of Takagi for the case m1 = 1, and Ozeki and Takeuchi for m1 = 2, this handles all possible pairs of multiplicities except for four cases, for which the classification problem remains open.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2ee3620978b355c58edf87cd976042ebf9c440a2

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In Applied Mathematics

1. Zizhou Tang
2. G. Thorbergsson
3. Stefan Immervoll
4. Quo-Shin Chi
5. Jianquan Ge
6. R. Miyaoka

In Geometry and Topology

1. Zizhou Tang
2. G. Thorbergsson
3. Stefan Immervoll
4. Quo-Shin Chi
5. Jianquan Ge
6. R. Miyaoka

In Mathematical Physics

1. G. Thorbergsson
2. Stefan Immervoll
3. Zizhou Tang
4. Jianquan Ge

Citations in the corpus

• Isoparametric hypersurfaces with four principal curvatures, II (2010, background): “The classification theorem in [6] (see also [7]), where it is assumed that m2 ≥ 2m1 − 1, eventually comes down to an estimate on the dimension of certain singular varieties associated with the complexified second fundamental form of the focal manifold of codimension m1 +1, restricted to the direct s”
• On the classification of isoparametric hypersurfaces with four distinct curvatures in spheres (2008, methodology): “In this paper we present a new proof for the following classification result in [3].”
• Singular Riemannian Foliations and Isoparametric Submanifolds (2010, methodology): “A new proof of a part of the results in [ 17 ] can be found in [18].”
• Isoparametric functions and exotic spheres (2010, background): “rfaces (cf. [CR85]). From then on, besides geometrical and topological viewpoints, people also put extensive attention on algebraical approach to this area (see [Th00] for an excellent survey and see [CCJ07], [Imm08], [GX10] for recent progresses and applications). To study such submanifold geometry”
• Geometry of isoparametric hypersurfaces in Riemannian manifolds (2010, methodology): “number g of distinct principal curvatures of isoparametric hypersurfaces in spheres (For the theory of isoparametric hypersurfaces in the model space forms, see [Th00] for an excellent survey and see [CCJ07], [Imm08], [GX10] for recent progresses and applications). 2. The shapeoperatorof Tubes In th”