A group-theoretic property of the Euclidean metric
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- by Robert Williamson and Ludvik Janos PDF
- Proc. Amer. Math. Soc. 98 (1986), 150-152 Request permission
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
- F. S. Beckman and D. A. Quarles Jr., On isometries of Euclidean spaces, Proc. Amer. Math. Soc. 4 (1953), 810–815. MR 58193, DOI 10.1090/S0002-9939-1953-0058193-5
- Leonard M. Blumenthal, Theory and applications of distance geometry, Oxford, at the Clarendon Press, 1953. MR 0054981
- Ludvík Janoš, On maximal groups of isometries, Proc. Amer. Math. Soc. 28 (1971), 584–586. MR 275403, DOI 10.1090/S0002-9939-1971-0275403-0
- Daryl Tingley, Metric transformations of the real line, Rocky Mountain J. Math. 15 (1985), no. 1, 199–206. MR 779264, DOI 10.1216/RMJ-1985-15-1-199
Additional Information
- © Copyright 1986 American Mathematical Society
- 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