the set ${{\mathcal Z}}_C(\Sigma)$ is connected.

The classification of Zoll odd-symplectic forms up to diffeomorphism on a three-manifold is equivalent to the classification of bundles in the set $\{\mathfrak p_\Omega\ |\ \Omega\in{{\mathcal Z}}(\Sigma)\}$ up to isomorphism. Analogously, the connected components of ${{\mathcal Z}}_C(\Sigma)$ on a three-manifold are in bijection with the connected components of $\{\mathfrak p_\Omega\ |\ \Omega\in{{\mathcal Z}}_C(\Sigma)\}$. This is due to the fact that the map ${{\mathcal Z}}_C(\Sigma)\to\mathfrak P(\Sigma)$, given by $\Omega\mapsto\mathfrak p_\Omega$, has contractible fibres.

Let us assume that ${{\mathcal Z}}_C(\Sigma)$ is not empty and take $\Omega_*\in{{\mathcal Z}}_C(\Sigma)$ with associated $S^1$-bundle $$\mathfrak p_{\Omega_*}:\Sigma\to M_*.$$ This implies that there exists a positive symplectic form ${\mathrm{\omega}}_*$ on $M_*$ such that $\Omega_*=\mathfrak p_{\Omega_*}^*{\mathrm{\omega}}_*$. We set $c_*:=[{\mathrm{\omega}}_*]\in H^2_{{\mathrm{dR}}}(M_*)$. We write $\mathfrak h\in[S^1,\Sigma]$ for the free-homotopy class of the oriented $\mathfrak p_{\Omega_*}$-fibres and $e_*\in H^2_{{\mathrm{dR}}}(M_*)$ for minus the real Euler class of $\mathfrak p_{\Omega_*}$. Let $\mathfrak P^0(\Sigma)$ be the connected component of $\mathfrak p_{\Omega_*}$ inside $\mathfrak P(\Sigma)$. As we did for the volume, the action will be computed with respect to some reference object, which we now define.

A **weakly Zoll pair** is a couple $(\mathfrak p,c)$, where $\mathfrak p:\Sigma\to M$ is an element in $\mathfrak P(\Sigma)$ and $c\in H^2_{{\mathrm{dR}}}(M)$ is a 