2008/09/08 by Thomas E. Cecil · 3 citations
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Algebraic Geometry and Number Theory #Geometric Analysis and Curvature Flows #math.DG
paper · pdf · doi:10.3842/sigma.2008.062
published as SIGMA 4 (2008), 062, 28 pages · This is a contribution to the Special Issue "Elie Cartan and Differential Geometry", published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA
arxiv created 2008/09/08 · openalex publication_date 2008/09/08 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 as lie 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.