Totally focal embeddings

Carter, Sheila; West, Alan

doi:10.4310/jdg/1214434490

Cited by 8 publications

(15 citation statements)

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“…Either one can work in U" + 2 and take the cone on M and the radial projection of this cone onto S" + 1 from the origin; or the method can be used with a suitably defined distance function in S" + 1 itself. THEOREM This follows immediately from Theorem (2.5) of [4]. The method there was only applied to a particular component U of JV(M)\F but it works for any component.…”

Section: Totally Focal Hyper Surfaces In S" +mentioning

confidence: 96%

“…Thus there exists e > 0 such that rj(p,9)eW + and r\{p, -6)eW~ i f O < 0 < e . We now state the results we need and which can be obtained immediately by applying the results and methods of [4].…”

Section: Totally Focal Hyper Surfaces In S" +mentioning

confidence: 99%

“…In this way we can apply results from previous papers about totally focal hypersurfaces in Euclidean space. This follows by application of the method of Theorem (2.1) in [4]. Either one can work in U" + 2 and take the cone on M and the radial projection of this cone onto S" + 1 from the origin; or the method can be used with a suitably defined distance function in S" + 1 itself.…”

Section: Totally Focal Hyper Surfaces In S" +mentioning

confidence: 99%

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A Characterisation of Isoparametric Hypersurfaces in Spheres

Carter

West

1982

Journal of the London Mathematical Society

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Section: Totally Focal Hyper Surfaces In S" +mentioning

confidence: 96%

Section: Totally Focal Hyper Surfaces In S" +mentioning

confidence: 99%

Section: Totally Focal Hyper Surfaces In S" +mentioning

confidence: 99%

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A Characterisation of Isoparametric Hypersurfaces in Spheres

Carter

West

1982

Journal of the London Mathematical Society

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“…Carter and West [14,15,16,18], introduced the notion of totally focal submanifolds and studied its relationship to the isoparametric property. A submanifold φ : V → R n is said to…”

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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“…(vi) We have shown that isoparametric manifolds are totally focal. For hypersurfaces in Euclidean space or the sphere we have shown that the converse holds [2,3]. However, there is a non-compact totally focal surface in U A , in fact a Mobius band, which cannot be part of an isoparametric system as its normal bundle is not trivial [2].…”

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confidence: 91%