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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.

Information

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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