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A characterization of Möbius transformations

Author: Roland Höfer
Journal: Proc. Amer. Math. Soc. 128 (2000), 1197-1201
MSC (1991): Primary 51B10; Secondary 51M04, 51M09
Published electronically: August 3, 1999
MathSciNet review: 1646191
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Abstract: Let $n\ge 2$ be an integer and let $\mathcal{D}$ be a domain of $\mathbb{R}^n$. Let $f:\mathcal{D}\to\mathbb{R}^n$ be an injective mapping which takes hyperspheres whose interior is contained in $\mathcal{D}$ to hyperspheres in $\mathbb{R}^n$. Then $f$ is the restriction of a Möbius transformation.

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  • 1. A. D. Alexandrov. Seminar report. Uspekhi Mat. Nauk, 37(3):187, 1950.
  • 2. A. D. Alexandrov. On the axioms of relativity theory. Vestnik Leningrad Univ. Math., 19:5-28, 1976.
  • 3. W. Benz. Characterizations of geometrical mappings under mild hypotheses: Über ein modernes Forschungsgebiet der Geometrie. Hamb. Beitr. Wiss.gesch., 15:393-409, 1994.
  • 4. W. Benz. Real Geometries. BI Wissenschaftsverlag, Mannheim, Leipzig, Wien, Zürich, 1994. MR 95k:51019
  • 5. C. Carathéodory. The most general transformations of plane regions which transform circles into circles. Bull. Am. Math. Soc., 43:573-579, 1937.
  • 6. T. E. Cecil. Lie Sphere Geometry. Springer-Verlag, New York Berlin Heidelberg, 1992. MR 94m:53076
  • 7. J. A. Lester. A physical characterization of conformal transformations of Minkowski spacetime. Ann. Discrete Math., 18:567-574, 1983. MR 84g:83004b
  • 8. J. A. Lester. Distance preserving transformations. In F. Buekenhout, editor, Handbook of Incidence Geometry, pages 921-944, Amsterdam, 1995. Elsevier. MR 96j:51019
  • 9. I. Popovici and D. C. Radulescu. Characterizing the conformality in a Minkowski space. Ann. Inst. H. Poincaré. Phys. Théor., 35:131-148, 1981. MR 83b:53011

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Additional Information

Roland Höfer
Affiliation: Mathematisches Seminar, Universität Hamburg, Bundesstr. 55, 20146 Hamburg, Germany

Keywords: M{\"o}bius transformation, Lie transformation, mappings preserving hyperspheres, Alexandrov's theorem for domains
Received by editor(s): June 4, 1998
Published electronically: August 3, 1999
Communicated by: Christopher Croke
Article copyright: © Copyright 2000 American Mathematical Society

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