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These notes were originally written for a short course held at the Institute of Mathematics and Statistics, University of São Paulo, S.P. Brazil, January 9–20, 2012. The notes are based on the author’s book [17], Lie Sphere Geometry With Applications to Submanifolds, Second Edition, published in 2008, and many passages are taken directly from that book. The notes have been updated from their original version to include some recent developments in the field.

A hypersurface Mn−1 in Euclidean space Rn is proper Dupin if the number of distinct principal curvatures is constant on Mn−1, and each principal curvature function is constant along each leaf of its principal foliation. The main goal of this course is to develop the method for studying proper Dupin hypersurfaces and other submanifolds of Rn within the context of Lie sphere geometry. This method has been particularly effective in obtaining classification theorems of proper Dupin hypersurfaces.


Financial support for Prof. Cecil's visit to University of São Paulo was provided by the Brazilian Government, through Capes (Coordination for the Improvement of Higher Education Personnel). Prof. Martha Patricia Dussan Angulo, University of Sao Paulo, Brazil, hosted the short course.