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arXiv · 0912.2002

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

Edward Crane·Ian Short
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Abstract

We answer two questions of Beardon and Minda that 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 Möbius 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 by the absolute cross-ratios of 4-tuples of its elements. The proofs use the hyperboloid model of hyperbolic $(N+1)$-space.

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Edward Crane, Ian Short. 2009-12-10. Rigidity of configurations of balls and points in the $N$-sphere. https://arxiv.org/abs/0912.2002

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