Mathematical Sciences: RUI: Lie Sphere Geometry and Dupin Submanifolds

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The Dupin condition can be naturally formulated in Lie sphere geometry. It is known to be invariant under certain Lie transformations. Using this knowledge several theorems were proven which concern hypersurfaces whose principal curvatures are constant along their corresponding curvature surfaces. Recently the principal investigator and a collaborator successfully used these transformations to partially solve the problem of classifying all Dupin submanifolds. The principal investigator will continue his work on this classification. Spheres and tori as they appear in three-dimensional space are examples of what mathematicians call two-dimensional submanifolds. These are Dupin if they satisfy certain curvature requirements. The principal investigator has made substantial progress towards the classification of such submanifolds. He will continue his work on this deep and renowned problem.