On the hyperbolic symplectic geometry

The Symplectic Geometry of Polygons in Hyperbolic 3-space∗

We study the symplectic geometry of the moduli spaces Mr = Mr(H) 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 Bn 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 set...

Notes on Symplectic Geometry

1. Week 1 1 1.1. The cotangent bundle 1 1.2. Geodesic flow as Hamiltonian flow 4 2. Week 2 7 2.1. Darboux’s theorem 7 3. Week 3 10 3.1. Submanifolds of symplectic manifolds 10 3.2. Contact manifolds 12 4. Week 4 15 4.1. Symplectic linear group and linear complex structures 15 4.2. Symplectic vector bundles 18 5. Week 5 21 5.1. Almost complex manifolds 21 5.2. Kähler manifolds 24 6. Week 6 26 6....

Lectures on Symplectic Geometry

Symplectic geometry on symplectic knot spaces

Symplectic knot spaces are the spaces of symplectic subspaces in a symplectic manifold M . We introduce a symplectic structure and show that the structure can be also obtained by the symplectic quotient method. We explain the correspondence between coisotropic submanifolds in M and Lagrangians in the symplectic knot space. We also define an almost complex structure on the symplectic knot space,...

On the hyperbolic unitary geometry

Hans Cuypers (Preprint) describes a characterisation of the geometry on singular points and hyperbolic lines of a finite unitary space—the hyperbolic unitary geometry—using information about the planes. In the present article we describe an alternative local characterisation based on Cuypers’ work and on a local recognition of the graph of hyperbolic lines with perpendicularity as adjacency. Th...