manifold. Moreover, we will formulate a conjectural systolic-diastolic inequality in this setting which recovers the contact (Theorem \[thm:contact\_sys\]) and the symplectic one (Proposition \[p:hamsys\]) as two extreme cases.

Odd-symplectic systolic geometry
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Let $(\Sigma,\mathfrak o_\Sigma)$ be a connected oriented closed manifold of dimension $2n+1$, and let $C\in H^2_{\mathrm{dR}}(\Sigma)$ be a class in its de Rham cohomology. We write $\Omega^1(\Sigma)$ for the set of one-forms on $\Sigma$ and $\Xi^2_C(\Sigma)$ for the set of closed two-forms on $\Sigma$ representing the class $C$. We pick an auxiliary element $\Omega_0\in \Xi^2_C(\Sigma)$ and get a surjective map $$\label{e:omega1xi2}
\Omega^1(\Sigma)\to \Xi^2_C(\Sigma),\qquad \alpha\mapsto \Omega_\alpha:=\Omega_0+{{\mathrm d}}\alpha,$$ whose kernel is the space of closed one-forms on $\Sigma$. Using this map, we can associate to $\alpha\in\Omega^1(\Sigma)$ a real number ${\mathrm{Vol}}(\alpha)$, which generalises both the contact volume and the Calabi invariant . Namely, we define a functional ${\mathrm{Vol}}:\Omega^1(\Sigma)\to {{\mathbb R}}$ by $${\mathrm{Vol}}(0)=0,\qquad {{\mathrm d}}_{\alpha}{\mathrm{Vol}}\cdot \beta=\int_\Sigma\beta\wedge\Omega_\alpha^{n},\quad \forall\,\beta\in\Omega^1(\Sigma).$$ For instance, when $n=1$, we have $${\mathrm{Vol}}(\alpha)=\frac12\int_\Sigma \alpha\wedge{{\mathrm d}}\alpha+\int_\Sigma\alpha\wedge\Omega_0.$$ We also remark that, when $C=0$ and $\Omega_0=0$, the function ${\mathrm{Vol}}$ recovers the Chern-Simons action for principal $S^1$-bundles over $\Sigma$ (see [@CS74] and Remark \[r:cs\]). For $n=1$, $\Omega_0=0$ and ${{\mathrm d}}\alpha=\iota_F\mu$ for some vector field $F$ on $\Sigma$, whose flow preserves a global volume form $\mu$ 