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

Let the two oriented surfaces carry positive area forms $\alpha$ and $\beta$, obtainable from oriented Riemannian metrics. A top-degree form on a surface is automatically closed. On their product setIt is closed, andis 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.