Dupin Hypersurfaces

Cecil, Thomas E.; Ryan, Patrick J.

doi:10.1007/978-1-4939-3246-7_5

Cited by 1 publication

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“…As noted above, all of these hypersurfaces are shown to be counterexamples to Conjecture 1 by proving that they do not have constant Lie curvatures. This led to a revision of Conjecture 1 by Cecil, Chi, and Jensen [53] (p. 52) in 2007 that contains the additional assumption of constant Lie curvatures. This revised conjecture is still an open problem, although it has been shown to be true in some cases, which we describe after stating the conjecture.…”

Section: Classifications Of Dupin Hypersurfacesmentioning

confidence: 99%

Using Lie Sphere Geometry to Study Dupin Hypersurfaces in Rn

Cecil

2024

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A hypersurface M in Rn or Sn is said to be Dupin if along each curvature surface, the corresponding principal curvature is constant. A Dupin hypersurface is said to be proper Dupin if each principal curvature has constant multiplicity on M, i.e., the number of distinct principal curvatures is constant on M. The notions of Dupin and proper Dupin hypersurfaces in Rn or Sn can be generalized to the setting of Lie sphere geometry, and these properties are easily seen to be invariant under Lie sphere transformations. This makes Lie sphere geometry an effective setting for the study of Dupin hypersurfaces, and many classifications of proper Dupin hypersurfaces have been obtained up to Lie sphere transformations. In these notes, we give a detailed introduction to this method for studying Dupin hypersurfaces in Rn or Sn, including proofs of several fundamental results. We also give a survey of the results in the field that have been obtained using this approach.

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Section: Classifications Of Dupin Hypersurfacesmentioning

confidence: 99%

Using Lie Sphere Geometry to Study Dupin Hypersurfaces in Rn

Cecil

2024

Axioms

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Dupin Hypersurfaces

Cited by 1 publication

References 115 publications

Using Lie Sphere Geometry to Study Dupin Hypersurfaces in Rn

Using Lie Sphere Geometry to Study Dupin Hypersurfaces in Rn

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