The symplectic geometry of polygons in hyperbolic 3-space
Symplectic Geometry
2007-05-23 v2 Differential Geometry
Abstract
We study the symplectic geometry of the moduli space of closed n-gons with fixed side-lengths in hyperbolic 3-space. We prove that these moduli spaces have a symplectic structure coming from Poisson Lie theory. We construct completely integrable systems on these moduli spaces by bending n-gons along their diagonals. The results of this paper are the analogues of the results of the first two authors for n-gon linkages in Euclidean 3-space in JDG 44. We conclude by proving by a deformation argument that the moduli spaces of n-gon linkages in hyperbolic 3-space and Euclidean 3-space with the same set of side-lengths are symplectomorphic.
Cite
@article{arxiv.math/9907143,
title = {The symplectic geometry of polygons in hyperbolic 3-space},
author = {Michael Kapovich and John J. Millson and Thomas Treloar},
journal= {arXiv preprint arXiv:math/9907143},
year = {2007}
}