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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 $(E, \Lambda(n))$?

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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.

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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.

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  • $\begingroup$ thank you. But lets think to the problem for non obvious and general cases (i.e n >1) $\endgroup$ – Ali Taghavi Mar 2 '14 at 19:57

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