|
Sphere-preserving maps in inversive geometry
Author(s):
A.
F.
Beardon;
D.
Minda
Journal:
Proc. Amer. Math. Soc.
130
(2002),
987-998.
MSC (1991):
Primary 30C35;
Secondary 51F15
Posted:
November 9, 2001
Retrieve article in:
PDF DVI PostScript
This article is available free of charge
Abstract |
References |
Similar articles |
Additional information
Abstract:
We give an extensive discussion of sphere-preserving maps defined on subdomains of Euclidean  -space, and their relationship to Möbius maps and to the preservation of cross-ratios. In the case  (the complex plane) we also relate these ideas to the solutions of certain functional equations.
References:
- [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.
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical Society
with MSC
(1991):
30C35,
51F15
Retrieve articles in all Journals with MSC
(1991):
30C35,
51F15
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:
10.1090/S0002-9939-01-06427-9
PII:
S 0002-9939(01)06427-9
Received by editor(s):
February 29, 2000
Posted:
November 9, 2001
Communicated by:
Juha M. Heinonen
Copyright of article:
Copyright
2001,
American Mathematical Society
|
Comments: webmaster@ams.org