Consider a symplectic manifold $(M,\omega)$, for a 2 dimensional submanifold $N \subset M$, do we always have $\omega|_{N}$ gives a volume form on $N$?If not, is there any condition we can put on $N$ to make this happen?
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Consider $M = T^4$ with the symplectic form $\omega = dx^1\wedge dx^2 + dx^3\wedge dx^4$ and $N = T^2$ embedded in $M$ via the map $i : N \to M$ given by $i(y^1, y^2) = (y^1, 1, y^2, 1)$. Then $\omega|_N = i^*\omega = 0$ which is not a symplectic form on $N$.
A situation where $\omega|_N$ is necessarily a symplectic form is the following: $M$ is a Kähler manifold and $N$ is a complex submanifold (and hence Kähler).
Michael Albanese
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