The symplectic geometry of polygons in Euclidean space. (English) Zbl 0889.58017

Kapovich, Michael; Millson, John J.

J. Differ. Geom. 44, No. 3, 479-513 (1996).

Let \(M_r\) be the moduli space of \(n\)-gons with fixed side lengths \((r_1, \dots, r_n)\) in Euclidean space. The space \(M_r\) is naturally homeomorphic to \(\widetilde M_r/SO (3) \), where \(\widetilde M_r =\{u\in (S^2)^n: \sum r_ju_j =0\}\).
The authors give \(M_r\) a symplectic and a complex analytic structure by the following philosophy: whenever a compact Lie group \(G\) and its complexification \(G^\mathbb{C}\) operate on a complex manifold \(M\) with a given \(G\)-invariant Kähler form, one obtains a \(G\)-equivariant moment map \(\mu\) and \( \mu^{-1} (0)/G \approx G^\mathbb{C} \mu^{-1} (0)/G^\mathbb{C}\); the left hand side is a “symplectic space” and the right hand side a “complex analytic space”. By beautiful and natural geometric constructions, the authors produce functions which Poisson-commute and make a dense open subset \(M_r'\) of \(M_r\) into a toric variety. They also identify functions which give action-angle coordinates on a dense open subset \(M_0\) of \(M_r'\).
There is some overlap with the work of A. Klyachko [Aspects Math. E 25, 67-84 (1992; Zbl 0820.51016)].

Reviewer: H.Azad (Dharan)

MSC:

58D27

Moduli problems for differential geometric structures

Keywords:

moduli spaces; symplectic structures; moment map