Non-existence of Global Transverse Poincar\'{e} Sections

Abstract

We study global transverse Poincar\'{e} sections and give topological conditions for their existence, showing they never exist in many important cases. We prove that an energy hypersurface possessing global transverse Poincar\'{e} section is equivalent to the hypersuface having a cosymplectic structure. We give a family of Hamiltonian systems with global Poincar\'{e} sections of all possible topologies. Finally, we address the question of when a compact hypersurface of a symplectic manifold possesses an induced cosymplectic structure.