Li, Tongzhu Homogeneous surfaces in Lie sphere geometry. (English) Zbl 1231.53014 Geom. Dedicata 149, 15-43 (2010). 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. Reviewer: Anton Gfrerrer (Graz) Cited in 1 Document MSC: 53A40 Other special differential geometries Keywords:Lie sphere geometry; canal surface; Lie homogeneous surface; 3-sphere; stereographic projection PDFBibTeX XMLCite \textit{T. Li}, Geom. Dedicata 149, 15--43 (2010; Zbl 1231.53014) Full Text: DOI References: [1] Blaschke W.: Vorlesungen über Differentialgeometrie, vol. 3. Springer, Berlin (1929) · JFM 55.0422.01 [2] Bryant R.: A duality theorem for Willmore surfaces. J. Differ. Geom. 20, 23–53 (1984) · Zbl 0555.53002 [3] Cecil T., Chern S.S.: Tautness and Lie sphere geometry. Math. Ann. 278, 199–381 (1987) · Zbl 0635.53029 [4] Cecil T.: Reducible Dupin submanifolds. Geom. Dedicata 32, 281–300 (1989) · Zbl 0697.53056 [5] Cecil T.: Lie Sphere Geometry, With Applications to Submanifolds. Springer, New York (1992) · Zbl 0752.53003 [6] Cecil, T., Chern, S.S.: Dupin Submanifolds in Lie Sphere Geometry, Lecture Notes in Mathematics 1369, pp. 1–48. Springer, New York (1989) · Zbl 0678.53003 [7] Cecil T., Jensen G.R.: Dupin hypersurfaces with three principal curvatures. Invent. Math. 132, 121–178 (1998) · Zbl 0908.53007 [8] Niebergall R.: Dupin hypersurfaces in R 5, I. Geom. Dedicata 40, 1–22 (1991) · Zbl 0733.53031 [9] Pinkall, U.: W-Kurven in der ebenen Lie-Geometrie. Elemente der Mathematik 39, 28–33 and 67–78 (1984) · Zbl 0552.51008 [10] Pinkall U.: Dupin hypersurfaces. Math. Ann. 270, 427–440 (1985) · Zbl 0538.53004 [11] Pinkall U., Thorbergsson G.: Deformations of Dupin hypersurfaces. Proc. Am. Math. Soc. 107, 1037–1043 (1989) · Zbl 0682.53061 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.