Example of symplectic structure

Question

Let $S$ be a Riemann surface and $\Omega^1(S)$ algebra of $1$-forms on $S$.

How to prove that $\omega(a,b)=\int_S a\wedge b$ defines symplectic structure on $\Omega^1(S)$?

Any help is welcome. Thanks in advance.

Answer 1

Clearly, $\omega$ is antisymmetric.

It remains to show that $\omega$ is nondegenerate, but as $\Omega^1(S)$ is infinite dimensional, one has to specify the meaning of 'nondegenerate'. I will show here that $\omega$ is weakly nondegenerate, which means that for any nonzero $\alpha \in \Omega^1(S)$, there exists $\beta \in \Omega^1(S)$ such that $\omega(\alpha, \beta) \neq 0$.

Let $\alpha \in \Omega^1(S)$ be nonzero. Hence there is $p \in S$ such that $\alpha(p) \neq 0$. By continuity of $\alpha$, $\alpha$ is nonzero on a open neighbourhood $p \in U \subseteq S$. Choose coordinates $(x,y)$ on an open neighbourhood $p \in V \subseteq U$ such that $p$ corresponds to $(0,0)$. Without lost of generality, possibly after reducing $V$ and performing a linear combination, we can assume that $\alpha(p) = dx$. Again reducing $V$ if necessary, $\alpha(x,y) = f(x,y)dx + g(x,y)dy$ where $f$ and $g$ are smooth functions such that $f(x,y) > 0$ on $V$ (and $f(0,0) =1$, $g(0,0) = 0$, but these won't matter).

Choose a nonzero positive smooth bump function $\rho$ supported in $V$, so that the function $f \rho > 0$ on some open set $W \subseteq V$. Set $\beta(x,y) = \rho(x,y)dy$ and extend $\beta$ by zero outside $V$. Hence $\beta \in \Omega^1(S)$. Now, we compute

$$\omega(\alpha, \beta) = \int_V \alpha \wedge \beta = \int_V f(x,y)\rho(x,y) dx \wedge dy \ge \int_W f(x,y)\rho(x,y) dx \wedge dy > 0. $$

Remark: There is also strong nondegeneracy which would mean that the map $\omega^{\flat} : \Omega^1(S) \to (\Omega^1(S))^* : \alpha \mapsto \omega(\alpha, -)$ is not only injective (which is a reformulation of weak nondegeneracy), but bijective. This is a much more subtle question, as the topology chosen on $\Omega^1(S)$ influences what is the set $(\Omega^1(S))^*$. For the most common topology on the former, the latter contains singular distributions (such as 'Dirac delta forms'), which makes it doubtful that the above map could be a bijection.

Edit: It is quite possible that one would require $\omega : \Omega^1(S) \times \Omega^1(S) \to \mathbb{R}$ to be continuous, in which case the topology on $\Omega^1(S)$ should be specified. For instance, when $S$ is compact and endowed with a metric $g$, one can set $\langle a, b \rangle = \int_S g^*(a(x), b(x)) dV_g(x)$ for $a,b \in \Omega^1(S)$ (where $g^*$ is the induced metric on the cotangent bundle); the Cauchy-Schwarz inequality then implies $| \omega(a,b) | \le \|a\|\|b\|$, establishing the continuity of $\omega$.