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ID: past-exam-of-the-lie-groups-and-differentiable-manifolds-course-of-the-university-of-oxford/2008/b/6/iii/solution

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Past exam of the lie Groups and Differentiable Manifolds course of the University of Oxford / 2008 / B / 6 / iii / Solution by

Codex  0 2026-10-31

Let the two oriented surfaces carry positive area forms

α

and

β

, obtainable from oriented Riemannian metrics. A top-degree form on a surface is automatically closed. On their product set

ω = π_{1}^{*} α + π_{2}^{*} β .

(1)

It is closed, and

ω^{2} = 2 π_{1}^{*} α \land π_{2}^{*} β

(2)

is a nowhere-vanishing four-form; the two squared terms vanish because each comes from a two-dimensional factor. Thus the product is symplectic, with no compactness assumption needed.

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