RUI: Differential Geometry of Submanifolds

College Of The Holy Cross, Worcester MA

Investigators

Abstract

Abstract Award: DMS-0405529 Principal Investigator: Thomas E. Cecil The goal of this proposal is to study submanifolds of Euclidean space and the unit sphere which have special curvature properties. Of particular interest are isoparametric hypersurfaces, which have constant principal curvatures, and Dupin hypersurfaces, which have the property that each principal curvature is constant along each of its curvature surfaces. Although these important classes of hypersurfaces have been studied since the nineteenth century, many natural problems remain open at this time. The principal investigator and his collaborators, Quo-Shin Chi and Gary Jensen of Washington University, will employ the method of moving frames on Legendre submanifolds in Lie sphere geometry in their research on these problems. This method is applicable to the study of any submanifold in Euclidean space, not just to those mentioned here, and it has been used successfully by the principal investigator and his collaborators in previous research. Dupin hypersurfaces have been studied extensively since the introduction of the cyclides of Dupin in 1822, and great progress has been made over the past 25 years in their classification. Dupin hypersurfaces have played a major role in various mathematical theories, such as the theory of taut embeddings, the study of Hamiltonian systems of hydrodynamic type, and the theory of higher-dimensional Laplace invariants. The cyclides of Dupin have also apperared in recent papers on computer aided geometric design. The study of isoparametric hypersurfaces in spheres was initiated by the renowned French mathematician, Elie Cartan, in the 1930's, and many mathematicians have made significant contributions to this beautiful theory. Included in the class of isoparametric hypersurfaces and their focal sets are many famous geometric objects, such as the Veronese surface and the Clifford-Stiefel manifolds, which have been studied by research mathematicians from several different points of view. In terms of Research in Undergraduate Institutions (RUI) activities and the broader impact of the proposal, the principal investigator plans to continue his successful program of new course development and mentoring of individual students. Over the past fifteen years, this has resulted in 13 honors theses and a total of 25 students in geometry courses, who have attended or plan to attend graduate school in mathematics or related fields.