The Symplectic Geometry of Polygons in Euclidean Space (original) (raw)

The Symplectic Geometry of Polygons in the 3-Sphere

Canadian Journal of Mathematics, 2002

We study the symplectic geometry of the moduli spaces Mr = Mr(S 3) of closed n-gons with fixed side-lengths in the 3-sphere. We prove that these moduli spaces have symplectic structures obtained by reduction of the fusion product of n conjugacy classes in SU(2) by the diagonal conjugation action of SU(2). Here the fusion product of n conjugacy classes is a Hamiltonian quasi-Poisson SU(2)manifold in the sense of [AKSM]. An integrable Hamiltonian system is constructed on Mr in which the Hamiltonian flows are given by bending polygons along a maximal collection of nonintersecting diagonals. Finally, we show the symplectic structure on Mr relates to the symplectic structure obtained from gauge-theoretic description of Mr. The results of this paper are analogues for the 3-sphere of results obtained for Mr(H 3), the moduli space of n-gons with fixed side-lengths in hyperbolic 3-space [KMT], and for Mr(E 3), the moduli space of n-gons with fixed side-lengths in E 3 [KM1].

The Symplectic Geometry of Polygons in Hyperbolic 3-space

Asian Journal of Mathematics

We study the symplectic geometry of the moduli spaces Mr = Mr(H 3 ) of closed n-gons with fixed side-lengths in hyperbolic three-space. We prove that these moduli spaces have almost canonical symplectic structures. They are the symplectic quotients of B n by the dressing action of SU (2) (here B is the subgroup of the Borel subgroup of SL2(C) defined below). We show that the hyperbolic Gauss map sets up a real analytic isomorphism between the spaces Mr and the weighted quotients of (S 2 ) n by P SL2(C) studied by Deligne and Mostow. We construct an integrable Hamiltonian system on Mr by bending polygons along nonintersecting diagonals. We describe angle variables and the momentum polyhedron for this system. The results of this paper are the analogues for hyperbolic space of the results of [KM2] for Mr(E 3 ), the space of n-gons with fixed side-lengths in E 3 . We prove Mr(H 3 ) and Mr(E 3 ) are symplectomorphic.

Integrable Evolutions of Twisted Polygons in Centro-Affine ℝm

International Mathematics Research Notices

We show that discrete WmW_mWm lattices are bi-Hamiltonian, using geometric realizations of discretizations of the Adler–Gel’fand–Dikii flows as local evolutions of arc length-parametrized polygons in centro-affine space. We prove the compatibility of two known Hamiltonian structures defined on the space of geometric invariants by lifting them to a pair of pre-symplectic forms on the space of arc length parametrized polygons. The simplicity of the expressions of the pre-symplectic forms makes the proof of compatibility straightforward. We also study their kernels and possible integrable systems associated to the pair.

On The Moduli Space Of Polygons In The Euclidean Plane

Journal of differential geometry

. We study the topology of moduli spaces of polygons with fixed side lengths in the Euclidean plane. We establish a duality between the spaces of marked Euclidean polygons with fixed side lengths and marked convex Euclidean polygons with prescribed angles. 1. We consider the space P n of all polygons with n distinguished vertices in the Euclidean plane E 2 whose sides have nonnegative length allowing all possible degenerations of the polygons except of the degeneration of the polygon to a point. Two polygons are identified if there exists an orientation preserving similarity of the complex plane C = E 2 which sends vertices of one polygon to vertices of another one. We shall denote the edges of the n-gon P by: e 1 ; :::; e n and vertices by v 1 ; :::; v n so that Gamma! e j = v j+1 Gamma v j+1 . The space P n is canonically isomorphic to the complex projective space P (H) where H ae C n is the hyperplane given by H = f(e 1 ; :::; e n ) 2 C n : e 1 + :::: + e n = 0g Th...

Adapted hyperbolic polygons and symplectic representations for group actions on Riemann surfaces

Journal of Pure and Applied Algebra, 2013

We prove that given a finite group G together with a set of fixed geometric generators, there is a family of special hyperbolic polygons that uniformize the Riemann surfaces admitting the action of G with the given geometric generators. From these special polygons, we obtain geometric information for the action: a basis for the homology group of surfaces, its intersection matrix, and the action of the given generators of G on this basis. We then use the Frobenius algorithm to obtain a symplectic representation G of G corresponding to this action. The fixed point set of G in the Siegel upper half-space corresponds to a component of the singular locus of the moduli space of principally polarized abelian varieties. We also describe an implementation of the algorithm using the open source computer algebra system SAGE.

The Geometry of Polygons in R^5 and Quaternions

2002

We consider the moduli space Mr of polygons with fixed side lengths in five-dimensional Euclidean space. We analyze the local structure of its singularities and exhibit a real-analytic equivalence between Mr and a weighted quotient of n-fold products of the quaternionic projective line HP 1 by the diagonal P SL(2, H)-action. We explore the relation between Mr and the fixed point set of an anti-symplectic involution on a GIT quotient Gr C (2, 4) n /SL(4, C). We generalize the Gel ′ fand-MacPherson correspondence to more general complex Grassmannians and to the quaternionic context, and realize our space Mr as a quotient of a subspace in the quaternionic Grassmannian Gr H (2, n) by the action of the group Sp(1) n . We also give analogues of the Gel ′ fand-Tsetlin coordinates on the space of quaternionic Hermitean matrices and briefly describe generalized action-angle coordinates on Mr.

Multi-Oriented Symplectic Geometry

Symplectic Geometry and Quantum Mechanics, 2006

Multi-oriented symplectic geometry, also called q-symplectic geometry, is a topic which has not been studied as it deserves in the mathematical literature; see however Leray [107], Dazord [28], de Gosson [57, 61]; also [54, 55]. The idea is the following: one begins by observing that since symplectic matrices have determinant 1, the action of Sp(n) on a Lagrangian plane preserves the orientation of that Lagrangian plane. Thus, ordinary symplectic geometry is not only the study of the action Sp(n) × Lag(n) −→ Lag(n) but it is actually the study of the action Sp(n) × Lag 2 (n) −→ Lag 2 (n) where Lag 2 (n) is the double covering of Lag(n). More generally, q-symplectic geometry will be the study of the action Sp q (n) × Lag 2q (n) −→ Lag 2q (n)

Remarks on Symplectic Geometry

arXiv: Symplectic Geometry, 2019

We survey the progresses on the study of symplectic geometry past four decades. We briefly deal with the convexity properties of a moment map, the classification of symplectic actions, the symplectic embedding problems, and the theory of Gromov-Witten invariants.

On the Moduli Space of a Spherical Polygonal Linkage

Canadian Mathematical Bulletin, 1999

We give a \wall-crossing" formula for computing the topology of the moduli space of a closed n-gon linkage on S 2 . We do this by determining the Morse theory of the function n on the moduli space of n-gon linkages which is given by the length of the last side{the length of the last side is allowed to vary, the rst (n ? 1) sidelengths are xed. We obtain a Morse function on the (n ? 2)-torus with level sets moduli spaces of n-gon linkages. The critical points of n are the linkages which are contained in a great circle. We give a formula for the signature of the Hessian of n at such a linkage in terms of the number of back-tracks and the winding number. We use our formula to determine the moduli spaces of all regular pentagonal spherical linkages. the property that 0 < r i < ; 1 i n. In this case P is determined by its vertices u 1 ; : : : ; u n and we may write P = u = (u 1 ; : : : ; u n ).