Virtual Reeb Flows and Odd-Symplectic Surgery

We consider odd-symplectic manifolds admitting a cover by a contact manifold of bounded geometry. The characteristic foliation of these manifolds defines a dynamic that is the projection of the Reeb dynamic of the covering manifold. We prove the existence of closed contractible characteristics in several cases. Furthermore we define a surgery construction along isotropic spheres in an odd-symplectic manifold that leads to a symplectic cobordism.