Applications of Lie Sphere Geometry to Submanifold Theory

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AbstractAward: DMS-0071390Principal Investigator: Thomas E. CecilThe principal investigator and his collaborators, Quo-Shin Chiand Gary Jensen, will study submanifolds of Euclidean space andthe sphere within the context of Lie sphere geometry. Ofparticular interest are submanifolds with special curvatureproperties. These include isoparametric hypersurfaces, whichhave constant principal curvatures, and Dupin hypersurfaces,which have the property that each principal curvature is constantalong each of its corresponding curvature surfaces. The mainproblems to be studied are the classification of isoparametrichypersurfaces of the sphere with four principal curvatures, andthe classification of locally irreducible Dupin hypersurfaceswith four or six principal curvatures. This research isprimarily local in nature, using the method of moving frames inLie sphere geometry.This project focuses on an important class of surfaces, Dupinsurfaces, which have very special curvature properties. Examplesof Dupin surfaces are planes, spheres, circular cylinders and thecyclides of Dupin, which have been useful in recent years inadvanced computer-aided design. Dupin surfaces have higherdimensional analogues which were first studied by the greatFrench mathematician Elie Cartan in the 1930's and which havebeen researched extensively by many mathematicians over the pastthirty years. The goal of the proposed research is to classifythese higher dimensional Dupin surfaces. Another importantaspect of the proposal is the principal investigator's mentoringwork with undergraduate students. Over the period of the grant,three undergraduate students will be supported by the grant for asummer of directed independent study in an area related to theprincipal investigator's own research. Each of these studentswill then write an honors thesis based on this study. In thepast ten years, most of the students who have written honorstheses under the principal investigator's supervision have thenpursued graduate study in mathematics. In this way, theprincipal investigator's previous grants have played asignificant role in the education of some of the next generationof scientists.