Is every $n$-dimensional linear subspace of $\mathbb{R}^{2n}$ a Lagrangian submanifold?

79 Views Asked by At

Let $\mathbb{R}^{2n}$ have the standard symplectic form $\omega = dx_1\wedge dy_1 + \cdots + dx_n\wedge dx_n$. If we consider an $n$-dimensional subvector space of $\mathbb{R}^{2n}$ as a submanifold, is this submanifold Lagrangian? My main motivation for this question is to see if $Gr_\mathbb{R}(2n,n)$ is a parameter space for Lagrangian submanifolds.