Conformal symplectic geometry of cotangent bundles

Abstract

We prove a version of the Arnol'd conjecture for Lagrangian submanifolds of conformal symplectic manifolds: a Lagrangian L which has nonzero Morse-Novikov homology for the restriction of the Lee form β cannot be disjoined from itself by a C 0 -small Hamiltonian isotopy. Furthermore for generic such isotopies the number of intersection points equals at least the sum of the free Betti numbers of the Morse-Novikov homology of β. We also give a short exposition of conformal symplectic geometry, aimed at readers who are familiar with (standard) symplectic or contact geometry. * (X 0 , X 0 \ Q × B) ≥ 2r: Consider the Mayer-Vietoris sequence:

Locally conformally symplectic and Kähler geometry

“…A result on neighbourhoods of Lagrangian submanifolds in locally conformally symplectic manifolds was obtained in [109], analogous to the known result of Weinstein in the symplectic case, [134]. The problem of displacing a Lagrangian submanifold in a locally conformally symplectic manifold is tackled in [30]. The paper also contains some interesting observations on the issues that appear when one tries to apply Floer's machinery or results such as Gromov compactness to the locally conformally symplectic situation.…”

“…5 Vaisman uses locally conformal symplectic, while I stick with the terminology locally conformally symplectic in this note. Some recent papers use conformal symplectic, see [30,37]. The 1-form ϑ was baptized the Lee form by Vaisman.…”

On time-dependent Hamiltonian realizations of planar and nonplanar systems

“…Locally, the inverse of this assertion is also true. That is, if a vector field is conformal, then there exist a function H and the conformal vector field can be written in the form of X a H satisfying (7). More general, the set of conformal vector fields on Q is given by {X H + aZ} where Z being the Liouville vector field satisfying i Z Ω = −θ.…”

Structure of locally conformally symplectic Lie algebras and solvmanifolds

“…Notice that a lcs structure of the first kind is automatically exact. As shown in [23] (see also [19,Theorem 2.15]), exact lcs structures exist on every closed manifold M with H 1 pM ; Rq ‰ 0 endowed with an almost symplectic form.…”