TY - JOUR
T1 - Rigidity of configurations of balls and points in the N-sphere
AU - Crane, ET
AU - Short, I
N1 - Publisher: Oxford University Press
PY - 2011/6
Y1 - 2011/6
N2 - 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.
AB - 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.
U2 - 10.1093/qmath/hap044
DO - 10.1093/qmath/hap044
M3 - Article
VL - 62
SP - 351
EP - 362
JO - Quarterly Journal of Mathematics
JF - Quarterly Journal of Mathematics
SN - 0033-5606
IS - 2
ER -