TITLE: compact symplectic manifolds
QUESTION [5 upvotes]: Why there is no compact symplectic submanifold with dimension greater than 2 in $\mathbb{R}^{2n}$ ?

REPLY [6 votes]: The symplectic form on $R^{2n}$ is exact, however the symplectic form on a compact symplectic manifold cannot be exact. Hence no symplectic submanifolds (whether of dim.2 or higher).