 Lagrangian Grassmannian bundle

For a rank-$2n$ symplectic vector bundle $E\to B$, the Lagrangian Grassmannian bundle has fiber $\mathrm{LGr}(E_b)$ over $b$. A symplectic trivialization of $E$ induces a trivialization $\mathcal{L}Gr(E)\cong B\times \mathrm{LGr}({\mathbb{R}}^{2n})$.