the Levi-Civita connection of $g$ and $f:N\to(0,\infty)$ is the conformal factor given by $f\cdot g=g_J$. We refer to the work of Kerman in [@Ker99] for results on the existence of closed magnetic geodesics in this setting.

In the companion paper [@BK19b], we use Corollary \[c:mainsys\] and Proposition \[p:cla\] to establish a systolic-diastolic inequality for magnetic geodesics when $N$ is a surface, namely $m=1$.

Structure of the paper
----------------------

-   In Section \[c:vol\], we introduce the volume of a closed two-form on an odd-dimensional oriented closed manifold and explore its invariance properties.

-   We define odd-symplectic forms and H-forms in Section \[c:odd\]. Under certain conditions, we prove a stability result for H-forms in the set of odd-symplectic forms.

-   We devote Section \[sec:oriented\] to review some basic facts about oriented $S^1$-bundles, which will be crucially used in the following discussion.

-   In Section \[sec:weak\_Zoll\], we define weakly Zoll pairs and Zoll odd-symplectic forms. We also prove Proposition \[p:cla\], which classifies Zoll odd-symplectic forms on oriented three-manifolds.

-   In Section \[s:acfun\], we introduce the action of a closed two-form on an odd-dimensional manifold which is the total space of some oriented $S^1$-bundle. We prove Theorem \[thm:mainaction\], which is just a reformulation of 