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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A group-theoretic property of the Euclidean metric


Authors: Robert Williamson and Ludvik Janos
Journal: Proc. Amer. Math. Soc. 98 (1986), 150-152
MSC: Primary 54E35; Secondary 54E40, 54H15
DOI: https://doi.org/10.1090/S0002-9939-1986-0848893-0
MathSciNet review: 848893
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Abstract: Let $ d$ denote a metric on $ {{\mathbf{R}}^n}(n \in {\mathbf{N}})$ compatible with its Euclidean toplogy, and let $ I(d)$ be the group of isometries on $ {{\mathbf{R}}^n}$ relative to this metric. We show that whenever $ I(d)$ includes the group of motions then $ I(d)$ is identical with it.


References [Enhancements On Off] (What's this?)

  • [1] T. S. Beckman and D. A. Quarles, Jr., On isometries of Euclidean spaces, Proc. Amer. Math. Soc. 4 (1953), 810-815. MR 0058193 (15:335a)
  • [2] L. M. Blumenthal, Theory and applications of distance geometry, Oxford University Press, 1953. MR 0054981 (14:1009a)
  • [3] Ludvik Janos, On maximal groups of isometries, Proc. Amer. Math. Soc. 28 (1971), 584-586. MR 0275403 (43:1160)
  • [4] Daryl Tingley, Metric transformations on the real line, Rocky Mountain J. Math 15 (1985), 199-206. MR 779264 (86f:51022)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1986-0848893-0
Article copyright: © Copyright 1986 American Mathematical Society

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