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

Published electronically:
November 9, 2001

MathSciNet review:
1873771

Full-text PDF Free Access

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.

**[1]**J. Aczél and M. A. McKiernan,*On the characterization of plane projective and complex Moebius-transformations*, Math. Nachr.**33**(1967), 315–337. MR**0222756****[2]**Alan F. Beardon,*The geometry of discrete groups*, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR**698777****[3]**Herbert Busemann and Paul J. Kelly,*Projective geometry and projective metrics*, Academic Press Inc., New York, N. Y., 1953. MR**0054980****[4]**Carathéodory, C., The most general transformations of plane regions which transform circles into circles,*Bull. Amer. Math. Soc.*43 (1937), 573-579.**[5]**Alexander Chubarev and Iosif Pinelis,*Fundamental theorem of geometry without the 1-to-1 assumption*, Proc. Amer. Math. Soc.**127**(1999), no. 9, 2735–2744. MR**1657778**, 10.1090/S0002-9939-99-05280-6**[6]**Julian Lowell Coolidge,*A treatise on the circle and the sphere*, Chelsea Publishing Co., Bronx, N.Y., 1971. Reprint of the 1916 edition. MR**0389515****[7]**H. S. M. Coxeter,*Similarities and conformal transformations*, Ann. Mat. Pura Appl. (4)**53**(1961), 165–172. MR**0143083****[8]**H. S. M. Coxeter,*Introduction to geometry*, 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1969. MR**0346644****[9]**Hiroshi Haruki and Themistocles M. Rassias,*A new characteristic of Möbius transformations by use of Apollonius quadrilaterals*, Proc. Amer. Math. Soc.**126**(1998), no. 10, 2857–2861. MR**1485479**, 10.1090/S0002-9939-98-04736-4**[10]**Thomas W. Hungerford,*Algebra*, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1974. MR**0354211****[11]**Jeffers, J., Lost theorems of geometry,*American Math. Monthly*107 (2000), 800-812. CMP**2001:03****[12]**M. Jean McKemie and Jussi Väisälä,*Spherical maps of Euclidean spaces*, Results Math.**35**(1999), no. 1-2, 145–160. MR**1678056**, 10.1007/BF03322029**[13]**Radford, J.G.,*Foundations of hyperbolic manifolds*, Springer-Verlag, GTM 149, 1994.

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:
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