Rigidity of configurations of balls and points in the N-sphere

ET Crane, I Short

Research output: Contribution to journalArticle (Academic Journal)

Abstract

We answer two questions of Beardon and Minda which arose from their study of the conformal symmetries of circular regions in the complex plane. We show that a configuration of closed balls in the N-sphere is determined up to Mbius transformations by the signed inversive distances between pairs of its elements, except when the boundaries of the balls have a point in common, and that a configuration of points in the N-sphere is determined up to Mbius transformations by the absolute cross-ratios of 4-tuples of its elements. The proofs use the hyperboloid model of hyperbolic (N 1)-space.
Original languageEnglish
Pages (from-to)351 - 362
Number of pages12
JournalQuarterly Journal of Mathematics
Volume62
Issue number2
Early online date29 Jan 2010
DOIs
Publication statusPublished - Jun 2011

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Bibliographical note

Publisher: Oxford University Press

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