'systolic\_bib.bib'
title: 'On a local systolic inequality for odd-symplectic forms'
---

Introduction
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In this article we formulate a local systolic-diastolic inequality for odd-symplectic forms and establish it in some situations. Before discussing the precise set-up, we recall two inequalities which motivate us to study the local systolic-diastolic inequality for odd-symplectic forms and which arise as extreme cases of this inequality. Throughout the paper, we normalise the circle $S^1$ to have length one, namely $S^1:={{\mathbb R}}/{{\mathbb Z}}$.

Two known inequalities
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### Contact systolic-diastolic inequality {#contact-systolic-diastolic-inequality .unnumbered}

Let $\Sigma$ be a connected closed manifold of dimension $2n+1$. Let $\alpha$ be a contact form on $\Sigma$, namely $\alpha$ is a one-form such that $\alpha\wedge({{\mathrm d}}\alpha)^n$ is nowhere vanishing. Orienting $\Sigma$ so that $\alpha\wedge({{\mathrm d}}\alpha)^n$ is positive, we define the total volume of $\Sigma$ with respect to $\alpha$ by $$\label{eq:contact_volume}
\mathrm{Volume}(\alpha):=\int_{\Sigma}\alpha\wedge({{\mathrm d}}\alpha)^n>0.$$ We consider the Reeb vector field $R_\alpha$ on $\Sigma$ characterised by the relations ${{\mathrm d}}\alpha(R_\alpha,\cdot)=0$ and $\alpha(R_\alpha)=1$ and denote by $\Phi_{\alpha}$ the generated flow.

A contact form $\alpha_*$ on $\Sigma$ is called *Zoll* if all orbits of $\Phi_{\alpha_*}$ are periodic with the same prime period $T(\alpha_*)$. This is equivalent to saying that $\Phi_{\alpha_*}$ induces a free circle action on $\Sigma$ 