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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.

Information

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

REFERENCES

[1]Aczél, J. and McKiernan, M. A.On the characterization of hyperplane projective and complex Möbius-transformation. Math. Nachr. 33 (1967), 315–337.Google Scholar
[2]Beardon, A. F. and Minda, D.Sphere-preserving maps in inversive geometry. Proc. Amer. Math. Soc. 130 (2001), 987–998.CrossRefGoogle Scholar
[3]Chubarev, A. and Pinelis, I.Fundamental theorem of geometry without the 1-to-1 assumption. Proc. Amer. Math. Soc. 127 (1999), 2735–2744.Google Scholar
[4]Jeffers, J.Lost theorems of geometry. Amer. Math. Monthly. 107 (2000), 800–812.Google Scholar
[5]Li, B. and Wang, Y.Transformations and non-degenerate maps. Sci. China Ser. A, Mathematics. 48 Supp. (2005), 195–205.Google Scholar
[6]Nehari, Z.Conformal Mapping. (McGraw-Hill, 1952).Google Scholar
[7]Yao, G. W.On existence of degenerate circle-preserving maps. J. Math. Anal. Appl. 334 (2007), 950–953.Google Scholar
[8]Yao, G. W. Fundamental theorem of hyperbolic geometry without the injectivity assumption, to appear in Math. Nachr. available at http://arXiv.org-arXiv:0810.1580[math.CV].Google Scholar
[9]Yao, G. W. Transformations of spheres without the injectivity assumption. Preprint, 2008.Google Scholar