Introduction

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.