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

## TL;DR

This paper investigates the topological conditions for the existence of global transverse Poincaré sections, proving their non-existence in many cases and linking their existence to cosymplectic structures.

## Contribution

It establishes topological criteria for the existence of global transverse Poincaré sections and shows they are equivalent to cosymplectic structures on energy hypersurfaces.

## Key findings

- Global transverse Poincaré sections do not exist in many important cases.
- Existence of such sections is equivalent to the hypersurface having a cosymplectic structure.
- Provides examples of Hamiltonian systems with all possible topologies of Poincaré 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.