Removing parametrized rays symplectically

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 $\phi :[-1,\infty )\times Q\to M$ 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 ${\phi }^{\ast }\omega (\frac{\partial }{\partial t},\nu )=0$ holds then $M$ and $M\setminus \phi ([0,\infty )\times Q)$ are symplectomorphic.

7 pages