geodesics {#the-example-of-magnetic-geodesics .unnumbered}

Our main motivation to study the systolic-diastolic inequality for odd-symplectic forms comes from twisted cotangent bundles, where the magnetic form is symplectic. Let $(N,\sigma)$ be a connected closed symplectic manifold of dimension $2m$. Let $\mathfrak p_{{{\mathrm T}}^*N}:{{\mathrm T}}^*N\to N$ be the cotangent bundle map and consider the twisted symplectic form $${{\mathrm d}}\lambda_\mathrm{can}+\mathfrak p_{{{\mathrm T}}^*N}^*\sigma\in\Omega^2({{\mathrm T}}^*N),$$ where $\lambda_\mathrm{can}$ is the canonical one-form on ${{\mathrm T}}^*N$. We fix a $\sigma$-compatible almost complex structure $J$ on $N$ with associated metric $g_J$. The structure $J$ turns $\mathfrak p_{{{\mathrm T}}^*N}$ into a complex vector bundle and we denote by $\mathfrak p_J:\mathbb P_{{\mathbb C}}({{\mathrm T}}^*N)\to N$ its projectivisation. For all Riemannian metrics $g$ on $N$ in the same conformal class of $g_J$, let $S_g^*N$ be the unit co-sphere bundle of $g$, and write $\mathfrak p_g:S^*_gN\to N$ and $\mathfrak i_g:S_g^*N\hookrightarrow{{\mathrm T}}^*N$ for the natural projection and inclusion. The one-parameter group $t\mapsto e^{tJ}$ acts fibrewise on ${{\mathrm T}}^*N$ and yields a free $S^1$-action on $S^*_gN$, since $J$ is $g$-orthogonal. The quotient is naturally identified with $\mathbb P_{{\mathbb C}}({{\mathrm T}}^*N)$, so that we have an oriented $S^1$-bundle $\mathfrak p$ making the following diagram commute $$\xymatrix{
        {{\mathrm T}}^*N\ar[dr]_{\mathfrak p_{{{\mathrm T}}^*N}}&S^*_gN\ar[r]^-{\mathfrak p}\ar[d]^{\mathfrak p_g}\ar[l]_{\ \, \mathfrak i_g}&\mathbb P_{{\mathbb C}}({{\mathrm 