Removing parametrized rays symplectically

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 $\varphi^{\ast} \omega \left(\frac{\partial}{\partial t}, \nu \right) = 0$ holds then $M$ and $M \setminus \varphi ([0,\infty) \times Q)$ are symplectomorphic.

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The research for this work was partially supported by the “SFB/Transregio 191 Symplektische Strukturen in Geometrie, Algebra und Dynamik”.

Received 12 October 2020

Accepted 9 March 2021

Published 23 December 2022