Symplectic normal bundle

The symplectic normal bundle of $S$ is the symplectic orthogonal complement $(TS{)}^{\omega }\subset TM{|}_S$. It is a symplectic vector bundle and is naturally isomorphic to the ordinary normal bundle.

If a symplectomorphism between compact symplectic submanifolds lifts to an isomorphism of their symplectic normal bundles, then it extends to a symplectomorphism between neighborhoods. Thus the germ of a symplectic neighborhood is determined by the restricted form and the symplectic normal bundle.