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Sphere-preserving maps in inversive geometry


Authors: A. F. Beardon and D. Minda
Journal: Proc. Amer. Math. Soc. 130 (2002), 987-998
MSC (1991): Primary 30C35; Secondary 51F15
DOI: https://doi.org/10.1090/S0002-9939-01-06427-9
Published electronically: November 9, 2001
MathSciNet review: 1873771
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Abstract: We give an extensive discussion of sphere-preserving maps defined on subdomains of Euclidean $n$-space, and their relationship to Möbius maps and to the preservation of cross-ratios. In the case $n=2$ (the complex plane) we also relate these ideas to the solutions of certain functional equations.


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  • [1] Aczél, J. and McKiernan, M.A., On the characterization of plane projective and complex Moebius-transformations, Math. Nach. 33 (1967), 315-337 MR 36:5806
  • [2] Beardon, A.F., The geometry of discrete groups, Springer-Verlag, GTM 91, 1983. MR 85d:22026
  • [3] Busemann, H. and Kelly, P.L., Projective geometry and projective metrics, Academic Press, New York, 1953. MR 14:1008e
  • [4] Carathéodory, C., The most general transformations of plane regions which transform circles into circles, Bull. Amer. Math. Soc. 43 (1937), 573-579.
  • [5] Chubarev, A. and Pinelis, I., Fundamental Theorem of geometry without the 1-to-1 assumption, Proc. Amer. Math. Soc. 127 (1999), 2735-2744. MR 99m:51002
  • [6] Coolidge, J.L., A treatise on the circle and the sphere, Chelsea, 1971 (reprinted from Oxford, 1916). MR 52:10346
  • [7] Coxeter, H.S.M., Similarities and conformal transformations, Annali di Matematica pura ed applicata 53 (1961), 165-172. MR 26:648
  • [8] Coxeter, H.S.M., Introduction to Geometry, Wiley, 1969. MR 49:11369
  • [9] Haruki, H. and Rassias, T.M., A new characteristic of Möbius transformations by use of Apollonius quadilaterals, Proc. Amer. Math. Soc. 126 (1998), 2857-2861. MR 99a:30012
  • [10] Hungerford, T.W., Algebra, Holt, Rinehart and Winston, New York, 1974. MR 50:6693
  • [11] Jeffers, J., Lost theorems of geometry, American Math. Monthly 107 (2000), 800-812. CMP 2001:03
  • [12] McKemie, M.J. and Väisälä, J., Spherical maps of Euclidean spaces, Result. Math. 35 (1999), 145-160. MR 2000a:30044
  • [13] Radford, J.G., Foundations of hyperbolic manifolds, Springer-Verlag, GTM 149, 1994.

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Additional Information

A. F. Beardon
Affiliation: Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, England
Email: A.F.Beardon@dpmms.cam.ac.uk

D. Minda
Affiliation: Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025
Email: David.Minda@math.uc.edu

DOI: https://doi.org/10.1090/S0002-9939-01-06427-9
Received by editor(s): February 29, 2000
Published electronically: November 9, 2001
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2001 American Mathematical Society

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