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:**



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> Is $E$ a trivial bundle over $\Lambda(n)$? If not, what is the maximum number of global independent sections for $(E, \Lambda(n))$? 
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Canonical n plane bundle over Lagrangian Grassmanian
It is not trivial. Its characteristic classes were worked out by Dmitrii Fuchs. In particular, the Maslov class is one of its nontrivial characteristic classes, and was the subject of a famous paper of Vladimir Arnol'd.


