Lie  introduced his 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 ) and in Volume III of Blaschke’s  Vorlesungen über Differentialgeometrie, published in 1929. In recent years, sphere geometry has become a valuable tool in the study of Dupin submanifolds in Euclidean space ℝn, beginning with Pinkall’s  dissertation in 1981. In this introduction, we will outline the contents of the book and mention some related results.
KeywordsPrincipal Curvature Sphere Geometry Isoparametric Hypersurface Contact Transformation Distinct Principal Curvature
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