On Liouville’s Theorem for Conformal Maps

A theorem of Liouville asserts that the simplest angle-preserving transformations on Euclidean space-translations, dilations, reflections, and inversions-generate all angle-preserving transformations when the dimension is at least 3. This note gives a proof which uses only elementary multivariable c...

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Bibliographic Details
Published in:The American mathematical monthly 2024-08, Vol.131 (7), p.619-623
Main Authors: Kushelman, Mathew, McGrath, Peter
Format: Article
Language:English
Subjects:
Calculus
Conformal mapping
Differential calculus
Euclidean geometry
Euclidean space
Inversions
Liouville theorem
Mathematical functions
Proof theory
Theorems
Translations
Citations: Items that this one cites
Online Access:Get full text
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Summary:A theorem of Liouville asserts that the simplest angle-preserving transformations on Euclidean space-translations, dilations, reflections, and inversions-generate all angle-preserving transformations when the dimension is at least 3. This note gives a proof which uses only elementary multivariable calculus and simplifies a differential-geometric argument of Flanders.
ISSN:0002-9890
1930-0972
DOI:10.1080/00029890.2024.2344409