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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
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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

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