Lagrangian submanifold

Remarks

Example: in $\R^4$ with $\omega_0 = dx_1 \wedge dy_1 + dx_2 \wedge dy_2$, the plane $L = \set{y_1 = y_2 = 0}$. It has dimension $2$, and on it $dy_1 = dy_2 = 0$, so $\iota^* \omega_0 = 0$.

Non-example: the plane $P = \set{x_2 = y_2 = 0}$ in the same space has dimension $2$, but $\omega_0$ restricts to $dx_1 \wedge dy_1 \ne 0$ on it, so clause 2 fails. Such a plane is called symplectic.

Clause 2 alone forces $\dim L \le n$, so Lagrangian submanifolds are the isotropic submanifolds of largest dimension. In a cotangent bundle $T^*Q$ with its canonical form, the graph of a $1$-form $\alpha$ on $Q$ is Lagrangian exactly when $\alpha$ is closed. A diffeomorphism $\phi \colon M \to M$ is a symplectomorphism exactly when its graph is Lagrangian in $M \times M$ with the form $\mathrm{pr}_1^* \omega - \mathrm{pr}_2^* \omega$.