Lie  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 ) 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  dissertation in 1981. In this introduction, we will outline the contents of the book and mention some related results.
KeywordsPrincipal Curvature Isoparametric Hypersurface Contact Transformation Distinct Principal Curvature Dupin Hypersurface
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