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Translation Of: Dupin’Sche Hyperflächen, Doctoral Dissertation, Universität Freiburg (1981) By Ulrich Pinkall, Thomas E. Cecil

Mathematics Department Faculty Scholarship

This is an unofficial translation of the original dissertation which was written in German. A few minor typographical errors have been corrected by the translator. All references should be made to the original dissertation. The classification of Dupin hypersurfaces in E⁴ contained in this dissertation is also contained in the journal article by Ulrich Pinkall, Dupin’sche Hyperflächen in E⁴, Manuscr. Math. 51 (1985), 89–119.

Using Lie Sphere Geometry To Study Dupin Hypersurfaces In R^N, Thomas E. Cecil

Mathematics Department Faculty Scholarship

A hypersurface M in Rⁿ or Sⁿ is said to be Dupin if along each curvature surface, the corresponding principal curvature is constant. A Dupin hypersurface is said to be proper Dupin if each principal curvature has constant multiplicity on M, i.e., the number of distinct principal curvatures is constant on M. The notions of Dupin and proper Dupin hypersurfaces in Rⁿ or Sⁿ can be generalized to the setting of Lie sphere geometry, and these properties are easily seen to be invariant under Lie sphere transformations. This makes Lie sphere geometry an effective …

Lie Sphere Geometry And Dupin Hypersurfaces, Thomas E. Cecil

Mathematics Department Faculty Scholarship

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 M^{n−1 …}