Homogeneous surfaces in Lie sphere geometry
This paper deals with certain classes of surfaces in the 3-sphere \(S^3 \subset {\mathbb R}^4\). It is shown that any flat canal surface in \(S^3\) is Lie equivalent to the stereographic projection of either a generic cylinder or a generic cone or a surface of revolution in \({\mathbb R}^3\). The non-flat canal surfaces in \(S^3\) are classified up to Lie equivalence. Moreover, a classification of Lie homogeneous surfaces is given. If \(US^3 := \{(x, \xi) \in S^3\times S^3\mid x\cdot \xi = 0\}\), then an immersion \(\lambda = (x, \xi): M \longrightarrow US^3\) is called a Lie geometric surface if \(dx\cdot \xi = 0\) holds. The Lie geometric surfaces which allow a Lie sphere transformation for any prescribed preimage-image pair of points chosen on them are called homogeneous Lie surfaces. The author gives a complete classification of such surfaces with respect to the Lie transformation group of \(S^3\). It is shown that there exist 19 types of such surfaces.
scientific article; zbMATH DE number 5818355