Non-existence of global transverse Poincaré sections

Abstract

We study global transverse Poincaré sections and give topological obstructions to their existence. We prove that any energy hypersurface equipped with a global transverse Poincaré section has an induced cosymplectic structure. We give a family of Hamiltonian systems with global Poincaré sections of all possible topologies. Finally, we address the question of when a compact hypersurface of a symplectic manifold possesses an induced cosymplectic structure.