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Ali Taghavi
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Canonical n plane bundle on 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 $(E, \Lambda(n))$?

Ali Taghavi
  • 356
  • 8
  • 31
  • 123