Reductions of Locally Conformal Symplectic Structures and De Rham Cohomology Tangent to a Foliation

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 of Hamiltonian dynamical systems since the form of the Hamiltonian equations is preserved by homothetic canonical transformations [17]. Two locally conformal symplectic forms Ω1 and Ω2 on M are conformally equivalent if Ω2 = fΩ1 for some smooth positive function f on M . A conformal equivalence class of locally conformal symplectic forms on M is a locally conformal symplectic structure on M ([3]). Let Q be a smooth submanifold of M . Let ι : Q →֒ M denote the standard inclusion. We say that two locally conformal symplectic forms Ω1 and Ω2 on M are conformally equivalent on Q if ιΩ2 = fι Ω1 for some smooth positive function f on Q. Clearly the Lee form of a locally conformal symplectic form is exact if and only if Ω is conformally equivalent to a symplectic form [17]. Then the locally conformal symplectic structure is globally conformal symplectic. Locally conformal symplectic forms were introduced by Lee [11]. They have been intensively studied in [17], [8], [9], [10], [3]. In [18] the symmetry of the Lyapunov spectrum in locally conformal Hamiltonian systems is studied. It was shown that Gaussian isokinetic dynamics, Nosé-Hoovers