Hostname: page-component-848d4c4894-75dct Total loading time: 0 Render date: 2024-05-18T07:50:54.462Z Has data issue: false hasContentIssue false
  • English
  • Français

Surfaces in Möbius geometry

Published online by Cambridge University Press:  22 January 2016

Changping Wang
Changping Wang*
Affiliation:
Nankai Institute of Mathematics Nankai University, Tianjin, 300071, P.R., China and Technische Universität Berlin Fachbereich Mathematik, MA 8-3, Straße des 17 Juni 136 1000, Berlin 12
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Our purpose in this paper is to give a basic theory of Möbius differential geometay. In such geometry we study the properties of hypersurfaces in unit sphere Sn which are invariant under the Möbius transformation group on Sn.

Since any Möbius transformation takes oriented spheres in Sn to oriented spheres, we can regard the Möbius transformation group Gn as a subgroup MGn of the Lie transformation group on the unit tangent bundle USn of Sn. Furthermore, we can represent the immersed hypersurfaces in Sn by a class of Lie geometry hypersurfaces (cf. [9]) called Möbius hypersurfaces. Thus we can use the concepts and the techniques in Lie sphere geometry developed by U. Pinkall ([8], [9]), T. Cecil and S. S. Chern [2] to study the Möbius differential geometry.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 125 , March 1992 , pp. 53 - 72
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1992

References

[ 1 ] Bryant, R., A duality theorem for Willmore surfaces, J. Differential Geom., 20 (1984), 23–53.Google Scholar
[ 2 ] Cecil, T. and Chern, S. S., Tautness and Lie sphere geometry, Math. Ann., 278 (1987), 381–399.CrossRefGoogle Scholar
[ 3 ] Cecil, T. and Chern, S. S., Dupin submanifolds in Lie sphere geometry, Lecture Notes in Mathematics, 1369, Springer-Verlag, 1–48.Google Scholar
[ 4 ] Cecil, T. and Ryan, P., Tight and Taut immersions of manifolds, Res. Notes Math., 107, Pitman, London, 1985.Google Scholar
[ 5 ] Eisenhart, L., A treatise on the differential geometry of curves and surfaces, Ginn Boston, 1909.Google Scholar
[ 6 ] Miyaoka, R., Compact Dupin hypersurfaces with three principal curvatures, Math. Z., 187 (1984), 433–452.Google Scholar
[ 7 ] Palmer, B., The conformal Gauss map and the stability of Willmore surfaces, Preprint TU Berlin, No. 267/1990.Google Scholar
[8] Pinkall, U., Dupin’sche Hyperflachen, Manuscr. Math., 51 (1985), 89–119.Google Scholar
[9] Pinkall, U., Dupin hypersurfaces, Math. Ann., 270 (1985), 427–440.Google Scholar