How to find lagrangian submanifolds. I am quite confused on the definition of a lagrangian submanifold $L$ of a symplectic manifold $(M,\omega)$. In particular, I read that $L \subset M$ is lagrangian iff the symplectic form field $\omega(x)$ evaluated on every point $p\in L$ gives zero. How is it possible that $\omega$ assumes the value zero on a subset $L$ of $M$, but still it is not degenerate (it is never zero) over the whole $M$?! Furthermore, consider $M=\mathbb{R}^2$ with the standard symplectic form $\omega=dq\wedge dp$. It appears to me that there are no lagrangian submanifolds at all, since the symplectic form is everywhere constant and never zero. However, I am reading that for example all the submanifolds of the tipe $q=const$ are lagrangian, in this case. Clearly, I am not understanding well the definition of Lagrangian submanifold. Where am I going wrong?

The condition of being a lagrangian submanifold $L\subset M$ means that for every point $p\in L$ and every pair of tangent vectors $X,Y\in T_pL$, $$\omega_p(X,Y)=0\,.$$ This does not violate the non-degeneracy of $\omega$, since $T_pL$ is merely a subspace of $T_pM$. However, non-degeneracy implies that any submanifold satisfying this condition is of dimension at most $\frac12\dim M$. Lagrangian submanifolds are submanifolds satisfying the condition above of maximal possible dimension, i.e. $\dim L=\frac12\dim M$.

Another was of saying the same thing is that the pullback $i^*\omega$ of $\omega$ under the inclusion (smooth map) $i:L\hookrightarrow M$ of the $2$-form $\omega$ is zero: $$i^*\omega=0\,.$$