Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 146 / 1 / e

Table of contents
- Solution e

Solution

Take the torus

M = T^{2 n} = R^{2 n} / Z^{2 n}

with the translation-invariant symplectic form

ω = \sum_{j = 1}^{n} d x_{j} \land d y_{j} .

The coordinate vector fields give a global symplectic frame of

TM

, so

TM

is symplectically trivial. Passing fiberwise to the Lagrangian Grassmannian bundle gives

L G r (TM) ≅ T^{2 n} \times LGr (R^{2 n}),

which is a smooth trivialization over the compact symplectic manifold

T^{2 n}

.

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