Removing a ray from a noncompact symplectic manifold

TL;DR

This paper demonstrates a method to remove a ray from a noncompact symplectic manifold, showing it is symplectomorphic to its complement, and applies this to construct Liouville vector fields on cotangent bundles.

Contribution

It introduces an explicit symplectomorphism for removing a ray from Euclidean space and uses this to build Liouville vector fields on cotangent bundles.

Findings

Noncompact symplectic manifolds with a properly embedded ray are symplectomorphic to the complement of the ray.

Constructs explicit symplectomorphisms in Euclidean space.

Provides a method to create Liouville vector fields on cotangent bundles.

Abstract

We prove that any noncompact symplectic manifold which admits a properly embedded ray with a wide neighborhood is symplectomorphic to the complement of the ray by constructing an explicit symplectomorphism in the case of the standard Euclidean space. We use this excision trick to construct a nowhere vanishing Liouville vector fields on every cotangent bundle.