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Why there is no compact symplectic submanifold with dimension greater than 2 in $\mathbb{R}^{2n}$ ?

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up vote 7 down vote accepted

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).

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