Sphere-preserving maps in inversive geometry

Abstract

We give an extensive discussion of sphere-preserving maps defined on subdomains of Euclidean<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"><mml:semantics><mml:mi>n</mml:mi><mml:annotation encoding="application/x-tex">n</mml:annotation></mml:semantics></mml:math></inline-formula>-space, and their relationship to Möbius maps and to the preservation of cross-ratios. In the case<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n equals 2"><mml:semantics><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:annotation encoding="application/x-tex">n=2</mml:annotation></mml:semantics></mml:math></inline-formula>(the complex plane) we also relate these ideas to the solutions of certain functional equations.