We consider Riemannian metrics compatible with the natural 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. The con...

A symplectic groupoid G. := (G1 ⇉ G0) determines a Poisson structure on G0. In this case, we call G. a symplectic groupoid of the Poisson manifold G0. However, not every Poisson manifold M has such a symplectic groupoid. This keeps us away from some desirable goals: for example, establishing Morita equivalence in the category of all Poisson manifolds. In this paper, we construct symplectic Wein...

An interesting question in symplectic geometry concerns whether or not a closed symplectic manifold can have a free symplectic circle action with orbits contractible in the manifold. Here we present a c-symplectic example, thus showing that the problem is truly geometric as opposed to topological. Furthermore, we see that our example is the only known example of a c-symplectic manifold having n...

We classify all homogeneous symplectic manifolds with a torsion free connection of special symplectic holonomy, i.e. a connection whose holonomy is an absolutely irreducible proper subgroup of the full symplectic group. Thereby, we obtain many new explicit descriptions of manifolds with special symplectic holonomies. We also show that manifolds with such a connection are homogeneous iff they co...

the osteology of the suspensorial and opercular series in representatives of 49 genera and 41 families of eurypterygian fishes were studied. the suspensorium consists of the palatine, ectopterygoid, endopterygoid, metapterygoid, quadrate, symplectic, and hyomandibular bones. the hyomandibular foramen is present at the base of the anterior head. the opercular series consists of the preopercle, s...

where ω is a closed 1-form. ω is uniquely determined by Ω and is called the Lee form of Ω. (M,Ω, ω) is called a locally conformal symplectic manifold. If Ω satisfies (1) then ω|Ua = d(ln fa) for all a ∈ A. If fa is constant for all a ∈ A then Ω is a symplectic form on M . The Lee form of the symplectic form is obviously zero. Locally conformal symplectic manifolds are generalized phase spaces o...

In symplectic geometry, it is often useful to consider the so-called Poisson bracket on the algebra of functions on a C ∞ symplectic manifold M. The bracket determines, and is determined by, the symplectic form; however, many of the features of symplectic geometry are more conveniently described in terms of the Poisson bracket. When one turns to the study of symplectic manifolds in the holomorp...

We characterize general symplectic manifolds and their structure groups through a family of isotropic or symplectic submanifolds and their diffeomorphic invariance. In this way we obtain a complete geometric characterization of symplectic diffeomorphisms and a reinterpretation of symplectomorphisms as diffeomorphisms acting purely on isotropic or symplectic submanifolds. DOI: 10.1134/S008154380...

We study neighborhoods of configurations of symplectic surfaces in symplectic 4–manifolds. We show that suitably “positive” configurations have neighborhoods with concave boundaries and we explicitly describe open book decompositions of the boundaries supporting the associated negative contact structures. This is used to prove symplectic nonfillability for certain contact 3–manifolds and thus n...

Let X be the plumbing of copies of the cotangent bundle of a 2−sphere as prescribed by an ADE Dynkin diagram. We prove that the only exact Lagrangian submanifolds in X are spheres. Our approach involves studying X as an ALE hyperkähler manifold and observing that the symplectic cohomology of X will vanish if we deform the exact symplectic form to a generic non-exact one. We will construct the s...