Removing parametrized rays symplectically

TL;DR

The paper proves that removing parametrized rays from a symplectic manifold under certain conditions results in a manifold symplectomorphic to the original, extending to higher dimensions with an extra condition.

Contribution

It establishes conditions under which parametrized rays can be removed from a symplectic manifold without changing its symplectic type, including higher-dimensional cases.

Findings

Removing parametrized rays preserves symplectic structure.

Higher-dimensional parametrized rays can be removed under additional conditions.

The result applies to injective, proper, immersed maps satisfying a specific symplectic orthogonality condition.

Abstract

Extracting isolated rays from a symplectic manifold result in a manifold symplectomorphic to the initial one. The same holds for higher dimensional parametrized rays under an additional condition. More precisely, let $(M,\omega)$ be a symplectic manifold. Let $[0,\infty)\times Q\subset\mathbb{R}\times Q$ be considered as parametrized rays $[0,\infty)$ and let $\varphi:[-1,\infty)\times Q\to M$ be an injective, proper, continuous map immersive on $(-1,\infty)\times Q$. If for the standard vector field $\frac{\partial}{\partial t}$ on $\mathbb{R}$ and any further vector field $\nu$ tangent to $(-1,\infty)\times Q$ the equation $\varphi^*\omega(\frac{\partial}{\partial t},\nu)=0$ holds then $M$ and $M\setminus \varphi([0,\infty)\times Q)$ are symplectomorphic.