Let V be a non-degenerate symplectic space of dimension 2n over the field F and for a natural number l < n denote by Cl(V ) the incidence geometry whose points are the totally isotropic l-dimensional subspaces of V . Two points U, W of Cl (V ) will be collinear when W ⊂ U⊥ and dim(U ∩ W ) = l − 1 and then the line on U and W will consist of all the l-dimensional subspaces of U + W which contain...

This paper gives methods for understanding invariants of symplectic quotients. The symplectic quotients that we consider are compact symplectic manifolds (or more generally orbifolds), which arise as the symplectic quotients of a symplectic manifold by a compact torus. A companion paper [23] examines symplectic quotients by a nonabelian group, showing how to reduce to the maximal torus. Through...

In this note we apply a 4-fold sum operation to develop an associativity rule for the pairwise symplectic sum. This allows us to show that certain diffeomorphic symplectic 4-manifolds made out of elliptic surfaces are in fact symplectically deformation equivalent. We also show that blow-up points can be traded from one side of a symplectic sum to another without changing the symplectic deformat...

We give an elementary construction of symplectic connections through reduction. This provides an elegant description of a class of symmetric spaces and gives examples of symplectic connections with Ricci type curvature, which are not locally symmetric; the existence of such symplectic connections was unknown. Key-words: Marsden-Weinstein reduction, symplectic connections, symmetric spaces MSC 2...

We consider Riemannian metrics compatible with the symplectic structure on T 2 ×M , where T 2 is a symplectic 2-Torus and M is a closed symplectic manifold. To each such metric we attach the corresponding Laplacian and consider its first positive eigenvalue λ1. We show that λ1 can be made arbitrarily large by deforming the metric structure, keeping the symplectic structure fixed. This extends a...

Let X be a 2n−dimensional smooth manifold. A 2−form ω on X is said to be non-degenerate if, for each q ∈ X and for each nonzero vector v in the tangent space TqX, there is a tangent vector v ∈ TqX such that ω(u, v) 6= 0. A symplectic structure onX is a non-degenerate closed 2−form. The fundamental example of a symplectic structure is ω0 = ∑ i dxi ∧ dyi on R = {(x1, y1, ..., xn, yn)}. In fact, b...

Hofer’s metric is a very interesting way of measuring distances between compactly supported Hamiltonian symplectic maps. Unfortunately, it is not known yet how to compute it in general, for example for symplectic maps far away from each other. It is known that Hofer’s metric is locally flat, and it can be computed by the so-called oscillation norm of the difference between the Poincare generati...

In this paper we consider  (extended) metaplectic representation of the  semidirect product  $G_{mathbb{J}}=mathbb{R}^{2d}timesmathbb{J}$  where $mathbb{J}$ is a closed subgroup of $Sp(d,mathbb{R})$, the symplectic group. We will investigate continuous representation frame on $G_{mathbb{J}}$. We also discuss the existence of duals for such frames and give several characterization for them. Fina...