Applications of Lie Sphere Geometry to Submanifold Theory

College Of The Holy Cross, Worcester MA

Investigators

Abstract

Abstract Award: DMS-0071390 Principal Investigator: Thomas E. Cecil The principal investigator and his collaborators, Quo-Shin Chi and Gary Jensen, will study submanifolds of Euclidean space and the sphere within the context of Lie sphere geometry. Of particular interest are submanifolds with special curvature properties. These include isoparametric hypersurfaces, which have constant principal curvatures, and Dupin hypersurfaces, which have the property that each principal curvature is constant along each of its corresponding curvature surfaces. The main problems to be studied are the classification of isoparametric hypersurfaces of the sphere with four principal curvatures, and the classification of locally irreducible Dupin hypersurfaces with four or six principal curvatures. This research is primarily local in nature, using the method of moving frames in Lie sphere geometry. This project focuses on an important class of surfaces, Dupin surfaces, which have very special curvature properties. Examples of Dupin surfaces are planes, spheres, circular cylinders and the cyclides of Dupin, which have been useful in recent years in advanced computer-aided design. Dupin surfaces have higher dimensional analogues which were first studied by the great French mathematician Elie Cartan in the 1930's and which have been researched extensively by many mathematicians over the past thirty years. The goal of the proposed research is to classify these higher dimensional Dupin surfaces. Another important aspect of the proposal is the principal investigator's mentoring work with undergraduate students. Over the period of the grant, three undergraduate students will be supported by the grant for a summer of directed independent study in an area related to the principal investigator's own research. Each of these students will then write an honors thesis based on this study. In the past ten years, most of the students who have written honors theses under the principal investigator's supervision have then pursued graduate study in mathematics. In this way, the principal investigator's previous grants have played a significant role in the education of some of the next generation of scientists.