# Canonical n plane bundle over Lagrangian Grassmanian

Recall that the Lagrangian Grassmanian, which is denoted by $\Lambda(n)$, is the subset of the standard Grassmanian $G(n,2n)$, which consists lagrangian sub vector spaces of $\mathbb{R}^{2n}$. Lets restrict the canonical n plane bundle over $G(n,2n)$ to $\Lambda(n)$. We denote this restricted bundle by $E$.

Question:

Is $E$ a trivial bundle over $\Lambda(n)$? If not, what is the maximum number of global independent sections for $(E, \Lambda(n))$?

No, the bundle is not trivial in general. If we consider the case $n=1$, then the associated Lagrangian Grassmannian is actually the ordinary Grassmannian, namely $\mathbb{R}\mathbb{P}^1$. Also, the tautological bundle over $\mathbb{R}\mathbb{P}^1$ is non-trivial.