on $\Sigma$, ${\mathrm{Vol}}(\alpha)$ recovers the helicity of $F$, see [@Pat09].

If $C^{n}=0$, then we can use the map to push forward ${\mathrm{Vol}}$ to a **volume functional** $${{\mathfrak{Vol}}}:\Xi^2_C(\Sigma)\to{{\mathbb R}}.$$ If $C^{n}\neq 0$, this procedure does not work, since there exists $\tau\in H^1_{{\mathrm{dR}}}(\Sigma)$ such that $\tau\cup C^n\neq 0$. In this case, we say that $\alpha\in\Omega^1(\Sigma)$ is **normalised**, if ${\mathrm{Vol}}(\alpha)=0$. For every $\Omega\in\Xi^2_C(\Sigma)$, there exists $\alpha\in\Omega^1(\Sigma)$ normalised such that $\Omega=\Omega_\alpha$. Therefore, in this situation we can just work with normalised forms and declare ${{\mathfrak{Vol}}}:\Xi^2_C(\Sigma)\to{{\mathbb R}}$ to be identically zero. In both cases, the volume functional is invariant under diffeomorphisms $\Psi:\Sigma\to\Sigma$ isotopic to the identity (see Proposition \[prp:vol\]): $${{\mathfrak{Vol}}}(\Psi^*\Omega)={{\mathfrak{Vol}}}(\Omega).$$

Having identified what the volume should be, we want to introduce the set $\mathcal X(\Omega)$ of closed characteristics of $\Omega$ generalising periodic Reeb and Hamiltonian orbits. To this purpose, we consider the possibly singular distribution $\ker\Omega\to \Sigma$ called the **characteristic distribution** and define $$\mathcal X(\Omega):=\Big\{\gamma:S^1\hookrightarrow \Sigma\ \Big|\ \dot\gamma\in \ker\Omega\Big\}\Big/\!\sim\,,$$ where $\gamma_1\sim\gamma_2$ if and only if $\gamma_1$ and $\gamma_2$ coincide up to an orientation-preserving reparametrisation of $S^1$. The distribution $\ker\Omega$ is co-oriented by $\Omega^n$ and we orient it using the given orientation $\mathfrak o_\Sigma$ on $\Sigma$.

We single out the forms in $\Xi^2_C(\Sigma)$, whose characteristic distribution 