Lie [104] introduced the geometry of oriented spheres in his dissertation, published as a paper in Mathematische Annalen in 1872. Sphere geometry was also prominent in his study of contact transformations (Lie–Scheffers [105]) and in Volume III of Blaschke’s book on differential geometry published in 1929. In recent years, Lie sphere geometry has become a valuable tool in the study of Dupin submanifolds, beginning with Pinkall’s [146] dissertation in 1981. In this introduction, we will outline the contents of the book and mention some related results.


Principal Curvature Isoparametric Hypersurface Contact Transformation Distinct Principal Curvature Dupin Hypersurface 
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© Springer Science+Business Media, LLC 2008

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