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

In a coordinate chart, define the exterior derivative byThe chain rule makes this independent of coordinates. It is real linear, increases form degree by one, agrees with the differential on functions, satisfies $d^2=0$, and obeys the graded Leibniz ruleIt also commutes with pullback by a smooth map. These properties characterize it locally and hence globally.

de Rham cohomology is the quotient of closed forms by exact forms. If $d\alpha =d\beta =0$, Leibniz gives $d(\alpha \wedge \beta )=0$. Changing $\alpha$ to $\alpha +d\eta$ changes the product by $d(\eta \wedge \beta )$; changing $\beta$ to $\beta +d\xi$ changes it by $(-1{)}^{r}d(\alpha \wedge \xi )$. Thusis independent of representatives. This gives the graded-commutative de Rham cohomology ring.

The nonvanishing top-degree form ${\omega }^m$ is a volume form. Declare an ordered tangent basis positive when its value under ${\omega }^m$ is positive. Smooth nonvanishing ensures these choices agree on overlapping oriented charts and vary continuously. Hence a symplectic manifold is orientable; this is its symplectic orientation.

Use the standard convention that the compact manifold is nonempty and has no boundary. Orient it by ${\omega }^m$. Since $d\omega =0$, every power ${\omega }^k$ is closed. If ${\omega }^k=d\eta$ for some $1\le k\le m$, thenagain by Leibniz and closedness. Stokes theorem would give ${\int }_M{\omega }^m=0$. But this integral is strictly positive in the symplectic orientation, a contradiction. ThereforeThe boundaryless hypothesis matters: a compact disk has a symplectic area form but no nonzero ordinary second cohomology class.

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.