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On characterizations of sphere-preserving maps

Published online by Cambridge University Press:  01 September 2009

BAOKUI LI  and
GUOWU YAO
BAOKUI LI
Affiliation:
Department of Mathematics, Beijing Institute of Technology, Beijing, 100081, China. e-mail: henan_lbk@bit.edu.cn
GUOWU YAO*
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China. e-mail: gwyao@math.tsinghua.edu.cn
*
†Corresponding author.
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Abstract

Recently, the first author and Y. Wang proved that (n ≥ 2) is a Möbius transformation if and only if f is a non-degenerate circle-preserving map. In this paper, we will further the result to show that f is a Möbius transformation if and only if f is a non-degenerate r–dimensional sphere-preserving map. The versions for the Euclidean and hyperbolic cases are also obtained. These results make no surjectivity or injectivity or even continuity assumptions on f. Moreover, certain degenerate sphere-preserving maps are given, which completes the characterizations of sphere-preserving maps.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 147 , Issue 2 , September 2009 , pp. 439 - 446
Copyright
Copyright © Cambridge Philosophical Society 2009

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References

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