RUI: Differential Geometry of Submanifolds

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AbstractAward: DMS-0405529Principal Investigator: Thomas E. CecilThe goal of this proposal is to study submanifolds of Euclideanspace and the unit sphere which have special curvatureproperties. Of particular interest are isoparametrichypersurfaces, which have constant principal curvatures, andDupin hypersurfaces, which have the property that each principalcurvature is constant along each of its curvature surfaces.Although these important classes of hypersurfaces have beenstudied since the nineteenth century, many natural problemsremain open at this time. The principal investigator and hiscollaborators, Quo-Shin Chi and Gary Jensen of WashingtonUniversity, will employ the method of moving frames on Legendresubmanifolds in Lie sphere geometry in their research on theseproblems. This method is applicable to the study of anysubmanifold in Euclidean space, not just to those mentioned here,and it has been used successfully by the principal investigatorand his collaborators in previous research.Dupin hypersurfaces have been studied extensively since theintroduction of the cyclides of Dupin in 1822, and great progresshas been made over the past 25 years in their classification.Dupin hypersurfaces have played a major role in variousmathematical theories, such as the theory of taut embeddings, thestudy of Hamiltonian systems of hydrodynamic type, and the theoryof higher-dimensional Laplace invariants. The cyclides of Dupinhave also apperared in recent papers on computer aided geometricdesign. The study of isoparametric hypersurfaces in spheres wasinitiated by the renowned French mathematician, Elie Cartan, inthe 1930's, and many mathematicians have made significantcontributions to this beautiful theory. Included in the class ofisoparametric hypersurfaces and their focal sets are many famousgeometric objects, such as the Veronese surface and theClifford-Stiefel manifolds, which have been studied by researchmathematicians from several different points of view. In termsof Research in Undergraduate Institutions (RUI) activities andthe broader impact of the proposal, the principal investigatorplans to continue his successful program of new coursedevelopment and mentoring of individual students. Over the pastfifteen years, this has resulted in 13 honors theses and a totalof 25 students in geometry courses, who have attended or plan toattend graduate school in mathematics or related fields.