Compact Symplectic Manifold Research Articles

A class of non-polarizable symplectic manifolds

A class of compact 4-dimensional symplectic manifolds which admit no polarizations whatsoever is presented. These spaces also provide examples of nonparallelizable manifolds which are symplectic but have no complex, and hence no Kahler, structures.

A lifting theorem for compact symplectic manifolds

It is shown that the Lie algebra of globally Hamiltonian vector fields on a compact symplectic manifold can be lifted to a Lie algebra of smooth functions on the manifold under Poisson bracket. This implies that any algebra of symmetries of a classical mechanical system described by such a manifold may be realised as an algebra of observables (smooth functions). Parallels between lifting problems in classical and quantum mechanics are explored.

Homogeneous contact compact manifolds and homogeneous symplectic manifolds

A homogeneous contact compact manifold can be considered as the total space of a principal circle bundle over a simply connected homogeneous symplectic manifold whose fundamental form determines an integral cohomology class. A similar result but assuming the contact manifold to be simply connected was given by Boothby and Wang. The proof we give here is independent of that of Boothby and Wang. We also prove that if a Lie group acts transitivelly on a simplectic manifold of integral class by diffeomorphism of the symplectic structure, there is a Lie group acting transitivelly on the total space of the bundle obtained from the symplectic form (Kobayashi's method) by diffeomorphisms of the contact structure. (The contact form is the correspondent connection form).

On compact homogeneous symplectic manifolds

On compact homogeneous symplectic manifolds

Constants of Motion and Lie Group Actions

A classical Hamiltonian dynamical system with 2N-dimensional phase space is studied in the case when a Lie algebra of constants of the motion exists which contains 2N −- 1 functionally independent elements and when each constant of motion generates a complete integral curve. It is proved that a connected (global) Lie group G acts on the phase space and acts transitively on each connected component of each surface of constant energy. When G is compact, each component of the space of time orbits corresponding to a fixed energy is shown to be a (2N − 2)-dimensional compact symplectic manifold diffeomorphic to an orbit of G in the dual of the adjoint representation. It is shown that a (global) Lie group does act in the case of the harmonic oscillator, but does not act in the case of the negative energy Kepler problem.

Symplectic quotients of unstable Morse strata for normsquares of moment maps

Let K be a compact Lie group and fix an invariant inner product on its Lie algebra. Given a Hamiltonian action of K on a compact symplectic manifold X, the normsquare of the moment map defines a Morse stratification of X by locally closed symplectic submanifolds such that the stratum to which any x in X belongs is determined by the limiting behaviour of its downwards trajectory under the gradient flow with respect to a suitably compatible Riemannian metric on X. The open stratum indexed by 0 retracts K-equivariantly via this gradient flow to the minimum which is the zero-locus of the moment map (if this is not empty). The usual 'symplectic quotient' for the action of K on any other stratum is empty. Nonetheless, motivated by recent results in non-reductive geometric invariant theory, we find that the symplectic quotient construction can be modified to provide natural 'symplectic quotients' for the unstable strata labelled by nonzero indices. There is an analogous infinite-dimensional picture for the Yang--Mills functional over a Riemann surface with strata determined by Harder-Narasimhan type.

The symplectic displacement energy

We define the symplectic displacement energy of a non-empty subset of a compact symplectic manifold as the infimum of the Hofer-like norm [5] of symplectic diffeomorphisms that displace the set. We show that this energy (like the usual displacement energy defined using Hamiltonian diffeomorphisms) is a strictly positive number on sets with non-empty interior. As a consequence we prove a result justifying the introduction of the notion of strong symplectic homeomorphisms [4].