1 Introduction↩︎

1.1 Odd-symplectic forms↩︎

Given a connected symplectic manifold \((Q,\omega)\), a smooth real-valued function \(H \in C^{\infty}(Q)\) is called an autonomous Hamiltonian on \(W\), and it defines a Hamiltonian vector field \(X_H\) on \(Q\). This Hamiltonian vector field is defined to be the unique solution to the following differential equation: \[\begin{align} \iota_{X_H}\omega =dH \end{align}\] When the dimension of \(Q\) is at least four, upon choosing a regular value \(r \in \mathbb{R}\) of \(H\), the restriction \(\varphi^*\omega\) is a closed and maximally non-degenerate \(2\)-form on the hypersurface \(\varphi:H^{-1}(r) \hookrightarrow W\), i.e, it is an instance of an odd-symplectic form. A precise definition of an odd-symplectic form is the following:

Every odd-symplectic manifold \((\Sigma,\Omega)\) canonically determines a distribution \(\ker\Omega \subset T\Sigma\) called its characteristic distribution. Being one-dimensional, the characteristic distribution of an odd-symplectic form is always involutive and determines a foliation of \(\Sigma\) by embedded \(1\)-dimensional submanifolds. This foliation is called the characteristic foliation of \(\Omega\).

If \(\Sigma\) is orientable, a choice of orientation on \(\Sigma\) and the \(2n-2\) form \(\Omega^{n-1}\) determine an orientation on the characteristic distribution of \(\Omega\) and on the leaves of its characteristic foliation. In such a setting, given an oriented odd-symplectic manifold \((\Sigma,\Omega)\), a \(1\)-form \(\alpha \in \Omega^1(\Sigma)\) is said to be positively transverse to \(\Omega\) if \(\ker\alpha \pitchfork \ker\Omega\) and \(\alpha\) evaluates positively on any vector field spanning the distribution \(\ker\Omega\) with the correct orientation. The set of \(1\)-forms positively transverse to an odd-symplectic manifold \((\Sigma,\Omega)\) determine a convex cone in \(\Omega^1(\Sigma)\) and will be denoted by \(\mathcal{C}(\Sigma,\Omega)\). Given an \(\alpha \in \mathcal{C}(\Sigma,\Omega)\), the pair \((\alpha,\Omega)\) is usually called a framed Hamiltonian pair on the manifold \(\Sigma\) (see [1],[2]). The choice of a framed Hamiltonian pair \((\alpha,\Omega)\) can be used to show that odd-symplectic forms always arise as restrictions of symplectic forms to regular energy levels of autonomous Hamiltonians. Indeed, observe that there exists \(\epsilon>0\) such that on the manifold \([-\epsilon,\epsilon] \times \Sigma\), the following \(2\)-form is a symplectic form: \[\begin{align} \tilde{\omega} := \pi^*_{\Sigma}\Omega + d(a\alpha) \end{align}\] where \(a \in [-\epsilon,\epsilon]\) denotes the coordinate on the first factor and \(\pi_{\Sigma}\) is the projection to \(\Sigma\). This observation motivates the definition of the Hamiltonian vector field of the framed Hamiltonian pair \((\alpha,\Omega)\), which is defined to be the unique vector field \(R\) on \(\Sigma\) satisfying the following equations: \[\begin{align} \alpha(R) \equiv 1 \qquad \iota_{R}\Omega \equiv 0 \end{align}\] The main objects of study in this work are Zoll odd-symplectic forms, which are defined in the following way:

It follows from the definition of a Zoll odd-symplectic form that any oriented Zoll odd-symplectic manifold \((\Sigma,\Omega_0)\) determines an isomorphism class of oriented \(S^1\)-bundles. We will call any representative of this isomorphism class an \(S^1\)-bundle associated with \(\Omega_0\). Examples of Zoll odd-symplectic forms are constructed in Section 3.

1.2 Local systolic inequalities for odd-symplectic forms↩︎

The main result of this work is the following. Its proof can be found in Section 2. Below, we will make use of the following notational convention: given a smooth manifold \(\Sigma\) and a cohomology class \(B \in H^k(\Sigma,\mathbb{R})\), we will denote by \(\Xi^k_{B}(\Sigma) \subset \Omega^k(\Sigma)\) the set of representatives of this class.

In Theorem 1 above, \(\mathcal{A}_{\Omega}\) denotes the Hamiltonian action of the odd-symplectic form \(\Omega\) (it is introduced in Section 1.4.1 below), and \(P\) is a polynomial called the Zoll polynomial; it is monotonic close to zero. This polynomial can be thought of as being motivated by the polynomial expression appearing in Weyl’s tube formula [3]; we introduce it in Section 1.4.3 below. \(Vol_{\Omega_0}(\Omega)\) is the Chern-Simons functional for \(S^1\)-bundles [4], which we refer to as the volume function; this volume function is introduced in Section 1.4.2 below. The hypothesis \(C^{n-1}_0 =0\) in Theorem 1 is equivalent to the non-vanishing of the Euler class of the isomorphism class of \(S^1\)-bundles associated with the Zoll odd-symplectic manifold \((\Sigma,\Omega_0)\) [4]; a precise definition of the Euler class can be found in Section 1.3. The statement of Theorem 1 when the Euler class vanishes is not new and was proven in [4]. Applications of Theorem 1 to the study of autonomous Hamiltonian systems are considered in Section 1.5.

It was shown in [4], that the inequalities 1 and 2 reduce to the \(C^3\)-local systolic optimality of Zoll contact forms, in the contact case; a known result proven in [5]. Recall that an oriented odd-symplectic manifold \((\Sigma,\Omega)\) satisfies the contact type condition if \(\Omega\) is exact that is, if \(\Omega = d\alpha\) for some \(\alpha \in \Omega^1(\Sigma)\) and if the \(2n-1\)-form \(\alpha \wedge d\alpha^{n-1}\) is everywhere positive on \(\Sigma\) [6]. In such a situation, the pair \((\Sigma,\alpha)\) is called a strict contract manifold [7]. The Reeb vector field of \(\alpha\) is the unique vector field \(R_{\alpha}\) on \(\Sigma\) satisfying the following conditions: \[\begin{align} \label{Reeb95odd-symp95eqn} \alpha(R_{\alpha}) \equiv 1 \;\;\; \text{and} \;\;\; \iota_{R_{\alpha}} d\alpha \equiv 0 \end{align}\tag{3}\] A flow line of \(R_{\alpha}\) on \(\Sigma\) is called a Reeb orbit of the contact form \(\alpha\). An odd-symplectic manifold \((\Sigma,d\alpha)\) of contact type is Zoll if and only if its Reeb flow induces a free \(S^1\)-action on \(\Sigma\).

Given a strict contact manifold \((\Sigma,\alpha)\), the associated contact systolic ratio is the following ( [8], [9]): \[\begin{align} \label{contat95sys95ratio95dim953} \rho_{\text{sys}}(\alpha):= \frac{T_{\text{min}}(\alpha)^{n}}{\text{Vol} (\alpha)} \end{align}\tag{4}\] where \(T_{\text{min}}(\alpha)\) is the minimal period of a periodic orbit of the Reeb vector field associated of \(\alpha\) and \(\mathrm{Vol}(\alpha): = \int_{\Sigma} \alpha \wedge d\alpha^{n-1}\) is the contact volume. If a contact form \(\alpha \in \Omega^1(\Sigma)\) does not admit any closed Reeb orbits we set \(\rho_{\text{sys}}(\alpha) = + \infty\).

The motivation for the definition of the contact systolic ratio comes from the \(\pi_1\)-systolic ratio in metric geometry, defined in the following way: given a closed, connected smooth manifold \(W\) of dimension \(n\). If \(g\) is a smooth Riemannian metric on \(W\), the associated \(\pi_1\)-systolic ratio is \[\begin{align} \label{metric95sys95ratio} \rho_{\text{sys}}(g):= \frac{\text{sys}^n(g)}{\text{Vol}(g)} \end{align}\tag{5}\] where \(\text{sys}(g)\) is the \(g\)-length of the shortest non-constant closed \(g\)-geodesic and \(\text{Vol}(g)\) is the Riemannian volume of \(W\). The key question in metric systolic geometry is to find an upper bound on the systolic ratio of metrics \(g\) on \(W\) that depends only on \(W\) and not on the specific metric \(g\). Such a uniform bound was first established on non-simply connected surfaces [10], [11], [12], and [13]. And in [14], it was shown that the ratio 5 was bounded from above on \(S^2\). Interestingly, however, the round metric is not the maximizer of the ratio, contradicting what would be expected from what is known about the \(2\)-torus and \(\mathbb{RP}^2\). It was then shown in [15] that the round metric was a local maximizer in an appropriate topology. It was later conjectured (see [16]) that the local maximizers of the ratio 5 on \(S^2\) in the \(C^3\)-topology could be characterized as Zoll metrics, i.e., those metrics all of whose geodesics are closed with the same prime period.

Up to multiplication by a universal dimensional constant proportional to the volume of an \(n\)-dimensional Euclidean ball, the ratio 5 agrees with the contact systolic ratio of the strict contact manifold \((S^*_gW,\lambda)\), where \(S^*_gW\) is the unit cotangent bundle of \((W,g)\) and \(\lambda\) is the restriction of the Liouville \(1\)-form. Since the dynamics of a tight contact form on the lens space \(S_g^*S^2\) is uniquely determined by its lift to the double cover \(S^3\) (for any smooth Riemannian metric \(g\) on \(S^2\)), motivated by the above-mentioned conjecture, Abbondandolo, Bramham, Hryniewicz, and Salomão showed in [8] that Zoll contact forms on \(S^3\) were the \(C^3\)-local maximizes of the contact systolic ratio on the set of contact forms defining the standard contact structure on \(S^3\). The same authors were also able to construct counterexamples to this statement in the \(C^0_{\text{loc}}\)-topology. In [17] Benedetti and Kang showed that Zoll contact forms maximized the contact systolic ratio in dimension 3. Abbondandolo and Benedetti proved the higher-dimensional analog of this result in [5]. Finally, in [18], Abbondandolo, Benedetti, and Edtmair showed that Zoll contact forms were the \(C^2\)-local maximizes of the contact systolic ratio on any contact manifold. The same authors also construct counterexamples to the \(C^1\)-local systolic optimality of Zoll contact forms.

1.3 Main technical results↩︎

In this section, we present the main technical contributions of this work.

We preface the statement of these results with a brief description of the geometric setup: given a connected Zoll odd-symplectic manifold \((\Sigma,\Omega_0)\) and an \(S^1\)-bundle \(\Sigma \xrightarrow{p} M\) associated with \(\Omega_0\), a \(1\)-form \(\alpha_0 \in \Omega^1(\Sigma)\) is called a connection \(1\)-form if the following holds for some vector field \(R\) generating the relevant \(S^1\)-action on \(\Sigma\): \[\begin{align} \mathcal{L}_R\alpha_0 \equiv 0 \qquad \alpha_0(R)\equiv 1 \end{align}\] In particular, fixing a connection \(1\)-form for an \(S^1\)-bundle is equivalent to fixing a vector field generating the relevant \(S^1\)-action on the total space. Upon fixing a connection \(1\)-form \(\alpha_0\) for the \(S^1\)-bundle \(p: \Sigma \rightarrow M\), the real Euler class is defined to be the real cohomology class represented by a closed \(2\)-form \(\nu \in \Omega^2(M)\) such that the following hold: \[\begin{align} p^*\nu = d\alpha_0 \end{align}\] We will denote by \(e_0 \in H^2(M,\mathbb{R})\), \((-1)\) times the real Euler class of the chosen \(S^1\)-bundle.

In the remainder of this work, when working with a Zoll odd-symplectic manifold \((\Sigma,\Omega_0)\), we will always assume that the cohomology class \(C_0 \in H^2(\Sigma,\mathbb{R})\) represented by \(\Omega_0\) satisfies the condition \(C^{n-1}_0=0\).

1.3.1 Normal form theorem↩︎

The following normal form theorem was motivated by [5], its proof can be found in Section 4.

The following is an immediate corollary of the above normal form theorem and is a qualitative refinement of [19] in the Zoll case; its proof can be found in Section 5. In the sequel, unless otherwise specified, when talking about a cohomology functor, we will always mean the singular cohomology functor computed with the ring of real numbers as coefficients.

1.3.2 A formula for the volume↩︎

The following theorem furnishes a convenient expression for the Chern-Simons functional close to Zoll odd-symplectic forms. Its proof can be found in Section 6.

1.4 Preliminaries↩︎

Throughout this section, we will fix a connected Zoll odd-symplectic manifold \((\Sigma,\Omega_0)\) and an \(S^1\)-bundle \(p: \Sigma \rightarrow M\) associated with the Zoll odd-symplectic form \(\Omega_0\); we will also denote by \(C_0 \in H^2(\Sigma,\mathbb{R})\) the cohomology class represented by \(\Omega_0\).

1.4.1 The Action↩︎

In this section, we outline a variational principle that allows us to look for closed leaves of the characteristic foliation of an odd-symplectic form \(\Omega \in \Xi^2_{C_0}(\Sigma)\) that is \(C^1\)-close to \(\Omega_0\). By Lemma 1 and [5], we expect to find such leaves in a small neighborhood of the fibers of the \(S^1\)-bundle \(\Sigma \xrightarrow{p} M\) associated with \(\Omega_0\) in the loop space \(\Lambda(\Sigma)\) i.e, the set of smooth (\(C^{\infty}\)), embedded loops in \(\Sigma\). This fact motivates the following definition: we say that an open cover \(\{\tilde{B}_n\}_{n \in \mathbb{N}}\) of \(M\) is a good cover if pair wise intersections of the form \(\tilde{B}_m \cap \tilde{B}_k\) are contractible for all \(m,k \in \mathbb{N}\). An open cover \(\{B_n\}_{n \in \mathbb{N}}\) of \(\Sigma\) is called a good cover if there exists a corresponding good cover \(\{\tilde{B}_n\}_{n \in \mathbb{N}}\) of \(M\) such that the following holds: \[\begin{align} B_n \subset p^{-1}(\tilde{B}_n) \qquad \forall\; n\in \mathbb{N} \end{align}\] Given a loop \(\gamma_x \subset \Sigma\) passing through a point \(x \in \Sigma\), we say that it is a short loop if there exists a good neighborhood \(B_x\) of the fiber \(p_x\) through \(x\) such that \(\gamma_x \subset B_x\). Upon denoting by \(\mathfrak{h} \in \pi_1(\Sigma)\) the homotopy class represented by the fibers of the fixed \(S^1\)-bundle associated with \(\Omega_0\), we will denote by \(\Lambda_{\mathfrak{h}}(\Sigma)\) the set of short loops in \(\Sigma\). By the inverse function theorem, we know this is an open subset of \(\Lambda(\Sigma)\) [20]. In the sequel, we will always suppress the base point \(x \in \Sigma\) of the loop \(\gamma_x\) in the notation.

Upon fixing an arbitary representative \(\Omega_* \in \Xi^2_{C_0}(\Sigma)\), for any odd-symplectic form \(\Omega \in \Xi^2_{C_0}(\Sigma)\) there exists an \(\alpha \in \Omega^1(\Sigma)\) such that \(\Omega = \Omega_*+d\alpha\), we define the action \(\mathcal{A}_{\Omega}\) of \(\Omega\) in the following way: \[\begin{align} \label{action95functional95eq} \mathcal{A}_{\Omega}(\gamma):\;&{\Lambda}_{\mathfrak{h}}(\Sigma) \rightarrow \mathbb{R}\\\nonumber \gamma \mapsto \int_{\gamma}&\alpha + \int_{\Gamma_{\gamma}} \Omega_*+d\alpha \end{align}\tag{11}\] where \(\Gamma_{\gamma}\) is an embedded cylinder such that \(\Gamma_{\gamma} \subset \mathcal{B}_x\) that is obtained parameterizing a homotopy from \(\gamma\) to an \(S^1\)-fiber. We will call such cylinders short cylinders.

In [4], it is shown that the above function is well defined and that it is invariant when pulled back.

The relevance of the action functional defined in 11 is that a loop \(\gamma \in {\Lambda}_{\mathfrak{h}}(\Sigma)\) is a critical point of \(\mathcal{A}_{\Omega}\) if and only if it is a leaf of the characteristic foliation of \(\Omega\). This was proven in [4]. In Corollary 1, we will use this functional to show that for any odd-symplectic form \(\Omega \in \Xi^2_{C_0}(\Sigma)\) that is sufficiently close to \(\Omega_0\), \(\mathcal{A}_{\Omega}\) has a strictly positive number of critical points in \(\Lambda_{\mathfrak{h}}(\Sigma)\).

We denote by \(\chi(\Omega)\) the set of closed leaves of the characteristic foliation of an odd-symplectic form \(\Omega \in \Xi^2_{C_0}(\Sigma)\) and make the following definitions: \[\begin{align} \inf \mathcal{A}_{\Omega}:= \inf_{\gamma \in \chi(\Omega) \cap \Lambda_{\mathfrak{h}}(\Sigma)} \mathcal{A}_{\Omega}(\gamma) \\\nonumber \sup \mathcal{A}_{\Omega}:= \sup_{\gamma \in \chi(\Omega)\cap \Lambda_{\mathfrak{h}}(\Sigma)} \mathcal{A}_{\Omega}(\gamma) \end{align}\] \(\inf \mathcal{A}_{\Omega}\) and \(\sup \mathcal{A}_{\Omega}\) are finite and close to zero when \(\Omega\) is \(C^1\)-close to \(\Omega_0\), see [4] for a proof.

1.4.2 The Volume↩︎

Here, we introduce an approach of Benedetti and Kang in [4] to computing a volume-like average of closed \(2\)-form \(\Omega \in \Xi^2_{C_0}(\Sigma)\) on \(\Sigma\) with respect to the fixed reference form \(\Omega_* \in \Xi^2_{C_0}(\Sigma)\). This average reduces to the contact volume when the \(2\)-form being considered is an odd-symplectic form of contact type. In addition, as in the contact case, this quantity coincides with the helicity when \(\Omega\) is odd-symplectic and exact and \(\Sigma\) is of dimension three [4].

We define the volume functional to be: \[\begin{align} \label{Vol95function95def95eqn} Vol_{\Omega_*}: \; &\Xi^2_{C_0}(\Sigma) \rightarrow \mathbb{R}\\\nonumber &\Omega \mapsto \int^1_0 \int_{\Sigma} \alpha \wedge (\Omega_*+rd\alpha)^{n-1} dr \end{align}\tag{12}\] In [4], it is shown that the above functional is well defined and invariant under diffeomorphisms isotopic to the identity.

The relevance of the volume functional here is that, as shown in [4], if \(\Omega\) is a Zoll odd-symplectic form, the functional \(Vol_{\Omega_*}(\Omega)\) defined in 12 is a polynomial. More precisely, since \(\Omega_0\) is a Zoll odd-symplectic form, there exists a symplectic form \(\omega_0 \in \Omega^2(M)\) such that \(p^*\omega_0 = \Omega_0\). Upon denoting by \(c_0 \in H^2(M,\mathbb{R})\) the cohomology class represented by \(\omega_0\), the following holds: \[\begin{align} p^*c_0 = C_0 \end{align}\] In this setting, it is shown in [4] that the following holds when \(\Omega \in \Xi^2_{C_0}(\Sigma)\) is a Zoll odd-symplectic form: \[\begin{align} Vol_{\Omega_*}(\Omega) = \int^{\mathcal{A}_{\Omega}}_0 \langle (c_0+te_0)^{n-2},[M] \rangle \; dt \;\;\; \text{for} \;\;\; \text{where } {\mathcal{A}_{\Omega}} \in \mathbb{R}\;\;\; \text{is the action of \Omega} \end{align}\] In addition, the above polynomial is monotonic close to zero because the class \(c_0\) can be represented by a symplectic form on \(M\). Indeed, the following is true: \[\begin{align} \frac{ Vol_{\Omega_0}(\Omega)}{d A}\Big|_{A=0} = \langle c_0^{n-2}, [M] \rangle >0 \end{align}\] This observation motivates the definition of the Zoll polynomial defined in the following section.

1.4.3 The Zoll polynomial↩︎

The Zoll polynomial is defined in the following way: \[\begin{align} \label{Zoll95poly95def95eqn} P(A) = \int^A_0 \langle (c_0+te_0)^{n-2},[M] \rangle \; dt \;\;\; \text{for} \;\;\; A \in \mathbb{R} \end{align}\tag{13}\] where \(e_0 \in H^2(M,\mathbb{R})\) is \((-1)\) times the real Euler class, \([M]\) is the fundamental class of \(M\) and \(c_0 \in H^2(M,\mathbb{R})\) is such that \(p^*c_0 = C_0 \in H^2(\Sigma, \mathbb{R})\). It follows from 13 that \[\begin{align} \label{montonicity95of95zoll95poly} \frac{dP(A)}{d A}\Big|_{A=0} = \langle c_0^{n-2}, [M] \rangle >0 \;\;\; \end{align}\tag{14}\] where the last inequality follows from the fact that \(c_0\) can be represented by a symplectic form.

1.5 Applications↩︎

1.5.1 Systolic inequalities for hypersurfaces in symplectic manifolds↩︎

In this section, we illustrate an application Theorem 1 by using the relative complexity of the dynamics on hypersurfaces in a symplectic manifold to compare the growth of volumes of tubular neighborhoods of such hypersurfaces. We preface the statement of the main results of this section with a brief outline of the geometric setup.

Given a connected symplectic manfiold \((W,\omega)\) and a closed, connected and embedded hypersurface \(\tilde{\Sigma} \hookrightarrow W\), by slightly abusing terminology, we will refer to the characteristic foliation of the odd-symplectic form \(\omega\vert_{\Sigma}\) as the characteristic foliation of \(\tilde{\Sigma}\). Since any embedded hypersurface of a symplectic manifold \((W,\omega)\) naturally inherits an orientation form the symplectic manifold, we define a Zoll hypersurface to be one whose oriented characteristic foliation corresponds with the foliation induced by the fibers of an oriented \(S^1\)-bundle under an orientation preserving diffeomorphism. Given another closed, connected and oriented hypersurface \(\Sigma\) that is \(C^2\)-close to a Zoll one, we pick a closed leaf of the characteristic foliation of \(\Sigma\) and define its action to be the following: \[\begin{align} \mathcal{A}_{\Sigma}(\gamma):= \int_{\Gamma_{\gamma}} \omega \end{align}\] where \(\Gamma_{\gamma} \hookrightarrow W\) is an embedded cylinder interpolating between \(\gamma\) and a leaf \(\gamma_0\) of the characteristic foliation of the Zoll hypersurface that is \(C^1\)-close to \(\gamma\). It is shown in Section 7.1 that the action is well defined and bounded when \(\Sigma\) is sufficiently close to a Zoll hypersurface. We will also denote by \(\text{sys}(\Sigma)\) the infimum of the action of \(\Sigma\) over closed leaves of the characteristic foliation of \(\Sigma\) that are \(C^1\)-close to those of a fixed Zoll hypersurface. We will denote by \(\tilde{D}(\Sigma)\) the region in \(W\) between \(\Sigma_0\) and \(\Sigma\).

If \(\Sigma_0\) is Zoll, then there exists an \(\epsilon>0\) such that a tubular neighborhood of \(\Sigma_0\) of radius \(\epsilon>0\) is foliated by Zoll hypersurfaces. We will denote the leaves of this foliation by \(\hat{\Sigma}_t\) for \(t \in (-\epsilon,\epsilon)\) and we will assume that the parameter \(t\) was chosen so that \(\text{sys}(\hat{\Sigma}_t) = t\) (see Section 7.1 for more details). We observe here that it is a priori unclear if Zoll hypersurfaces exist in an arbitrary symplectic manifold, especially if one is interested in hypersurfaces that lie outside the paradigm of Weinstein’s contact type condition, as we are in this work. We present the construction of such examples in Theorem 7.

The following is the first main result of this section; its proof can be found in Section 7.1:

1.5.2 Systolic inequalities for magnetic systems↩︎

Given a Riemannian manifold \((W,g)\) of dimension at least two, the metric Hamiltonian (or kinetic energy) of \(g\) is the following: \[\begin{align} H_g: T^*W \rightarrow \mathbb{R}\\\nonumber (q,p)\mapsto \frac{\vert\vert p\vert\vert^2_g}{2} \end{align}\] Upon picking a closed \(2\)-form \(\sigma \in \Omega^2(\Sigma)\), the following is a symplectic form on \(T^*W\): \[\begin{align} \label{mag95form95def95intro} \Omega_{g,\sigma}:= d\lambda - \pi^*\sigma \end{align}\tag{15}\] where \(\lambda \in \Omega^1(T^*W)\) is the Liouville \(1\)-form and \(\pi:T^*W \rightarrow W\) is the usual foot-point projection. The Hamiltonian vector field determined by \(H_g\) and \(\Omega_{g,\sigma}\), that is, the unique solution to the following differential equation, is called the magnetic (or twisted) geodesic flow of the pair \((g,\sigma)\): \[\begin{align} \iota_{X_{g,\sigma}}\Omega_{g,\sigma} = dH_{g} \end{align}\] The Hamiltonian dynamical system determined on the manifold \(T^*W\) by the flow \(\Phi_{g,\sigma}\) of the vector field \(X_{g,\sigma}\) is called the magnetic system on \((W,g)\) determined by the pair \((g,\sigma)\); we will refer to such pairs as magnetic pairs. The projections \(\pi(\Phi_{g,\sigma})\) to \(W\) are called the magnetic geodesics of the (magnetic) pair \((g,\sigma)\) when they are parametrized by arc length. Observe that if \(\sigma\) was chosen to be the \(2\)-form that is constantly zero everywhere on \(W\) then the magnetic geodesic flow of the pair \((g,\sigma)\) coincides with the geodesic flow of the Riemannian manifold \((W,g)\) and the magnetic geodesics on \(W\) are just the geodesics of the metric \(g\). The study of magnetic systems was initiated by V. Arnol’d [21] and S.P Novikov [22] and has since grown into a vibrant area of study in symplectic dynamics; see [23] for a nice introduction to the subject.

From the point of view of theoretical physics, magnetic dynamical systems can be thought of as modeling the motion of a charged particle in a magnetic field represented by \(\sigma\). (cf. [23]).

Given a \(k \in \mathbb{R}\), the flow of the vector field \(X_{g,\sigma}\) on a regular energy level \(H_{\text{kin}}^{-1}(k)\) can be conjugated to that of the Hamiltonian vector field determined by the metric Hamiltonian of \(g\) and the following symplectic form on \(S_g^*W\) [24]: \[\begin{align} \Omega^s_{g,\sigma}:= d\lambda -s\pi^*\sigma \;\;\; \text{for s = \frac{1}{\sqrt{2k}}} \end{align}\] The constant \(s\) is called the strength of the magnetic system determined by the pair \((g,\sigma)\).

A magnetic pair \((g,\sigma) \in \mathcal{M}(W) \times \Omega^2(W)\) will be called Zoll if for some \(s \in \mathbb{R}\), the magnetic geodesic flow of the pair \((g,\sigma)\) of strength \(s\) induces a free \(S^1\)-action on \(S^*_gW\). In such a setting, given another pair \((\tilde{g},\tilde{\sigma}) \in \mathcal{M}(W) \times \Omega^2(W)\), we will denote by \(\chi(\tilde{g}, \tilde{\sigma})\) the set of closed magnetic geodesics of the pair \((\tilde{g},\tilde{\sigma})\) of strength \(s\) that are \(C^2\)-close to those of the pair \((g,\sigma)\) of the same strength. In this section, since we are interested in applications of Theorem 1, we will always fix a Zoll magnetic pair and consider the dynamics of another magnetic pair relative to this fixed Zoll magnetic pair. In such a setting, we will always that both magnetic systems are being considered at a fixed strength where the fixed (Zoll) pair is Zoll.

It is also worth observing here that there are examples of magnetic systems that are Zoll at some energies but not at others. One such class of examples was constructed by Bimmerman in [25] (see also [26]), we recall the relevant version of Bimmerman’s statement here as Theorem 12.

The first result of this section is the following; its proof can be found in Section 8.1:

As observed by Benedetti and Kang in [4], the integral homology class, i.e., the class in \(H_1(W,\mathbb{Z})\) represented by the magnetic geodesics of a Zoll magnetic pair, is torsion. If this class vanishes then by for every Zoll magnetic geodesic \(\gamma\) there exists an immersed surface \(\mathbb{D}_{\gamma} \hookrightarrow W\) whose boundary coincides with \(\gamma\) [7]. We will call such a surface a capping surface for \(\gamma\).

We now fix a Zoll magnetic pair \((g,\sigma)\) such that the integral homology class represented by its magnetic geodesics is the zero class. Given another magnetic pair \((\tilde{g},\tilde{\sigma}) \in \mathcal{M}(W)\times \Omega^2(W)\) such that the set \(\chi(\tilde{g},\tilde{\sigma})\) is non-empty, we define the magnetic length of the magnetic geodesics in \(\chi(\tilde{g},\tilde{\sigma})\) in the following way:

\[\begin{align} \label{mag95length95functional95def} &l_{\text{mag}}^{\tilde{g},\tilde{\sigma}}: \chi(\tilde{g},\tilde{\sigma}) \rightarrow \mathbb{R}\\\nonumber \gamma &\mapsto \text{length}_{\tilde{g}}(\gamma) - \int_{\mathbb{D}_{\gamma}} \tilde{\sigma} \end{align}\tag{17}\] where \(\mathbb{D}_{\gamma}\) is a capping surface for the magnetic geodesic \(\gamma\) obtained by concatenating to a capping surface \(\mathbb{D}_{\gamma_0}\) of a magnetic geodesic \(\gamma_0 \in \chi(g,\sigma)\), the projection to \(W\) of a cylinder \(\Gamma_{\gamma} \hookrightarrow S^*_gW\) parameterizing a short homotopy between the lifts of \(\gamma\) and \(\gamma_0\) to \(S^*_gW\); a canonical lift exists for each such magnetic geodesic, see Section 8 for more details. In this setting, the inequality in 16 can now be expressed in the following succinct form: \[\begin{align} l_{\text{min}}^{\tilde{g},\tilde{\sigma}} \leq c(g,\sigma) \end{align}\] where the constant \(c(g,\sigma)\) is the value of the magnetic length functional \(l^{g,\sigma}_{\text{mag}}\) evaluated on a geodesic of the \((g,\sigma)\)-magnetic system and \(l_{\text{min}}^{\tilde{g},\tilde{\sigma}}\) is defined in the following way: \[\begin{align} \label{l95min95def95equation} l_{\text{min}}^{\tilde{g},\tilde{\sigma}}:= \inf_{\gamma\in \chi(\tilde{g},\tilde{\sigma})} l_{\text{mag}}^{\tilde{g},\tilde{\sigma}}(\gamma) \end{align}\tag{18}\] When the manifold \(W\) in Theorem 5 is a Kähler manifold where the associated Kähler metric \(g_0\) has constant holomorphic sectional curvature denoted by \(\kappa\) (see Section 9 for precise definitions), it was shown by Bimmermann (see Theorem 12 below) that the magnetic pair \((g_0,\sigma_0)\) where \(\sigma_0\) is the Kähler form, is Zoll at strengths satisfying: \(s^2+\kappa >0\). A consequence of this result of Bimmermann’s is also the fact that the integral homology class represented by Zoll magnetic geodesics is always zero. In this setting, we also have enough topological information to compute the Zoll polynomial explicitly, as a consequence, we obtain the next main result of this section; in it we will always assume that all the magnetic systems are being considered at a strength satisfying the condition \(s^2+\kappa>0\). This result can be seen as a generalization of the main result in [27] and its proof can be found in Section 9.1:

When the Kähler manifold \(M\) above is of dimension two, Theorem 6 is a generalization of the main result in [28], and when \(\text{Vol}_{\tilde{g}}(M) = \text{Vol}_{g_0}(M)\), it reduces to this result. This is because the following is true: \[\begin{align} \label{bk95mag953} a^2(1)\pi = \frac{2\pi}{\sqrt{s^2+\kappa}-s} \end{align}\tag{20}\] Combining 19 with the above equation and accounting for the sign change due to the assumption [28], we see that 19 reduces to the first inequality in [28] as required.

Acknowledgments↩︎

I am very grateful to my advisors, to Gabriele Benedetti for suggesting the problem to me and for his valuable help and guidance while working on the project and writing it down, and to Umberto Hryniewicz for numerous helpful discussions, for his advice on writing, and for the environment at MathGA. I am also very grateful to B.Albach, L.Asselle, V.Assenza, D.Bechara, J.Bimmermann, L.Dahinden U.Fuchs, R.Loiola, F.Morabito, and M.Voguel for several useful discussions. I am supported by the DFG SFB/TRR 191 “Symplectic Structures in Geometry, Algebra and Dynamics”, Projektnummer 281071066-TRR 191.

2 Proof of Theorem 1↩︎

We begin by fixing an \(S^1\)-bundle \(p: \Sigma \rightarrow M\) associated with \(\Omega_0\) and a connection \(1\)-form \(\alpha_0\). We now denote by \(\mathcal{U} \subset \Xi^2_{C_0}(\Sigma)\) the neighborhood of \(\Omega_0\) provided by Theorem 3. Upon denoting by \(\tilde{\mathcal{U}} \subset \Xi^2_{C_0}(\Sigma)\) the neighborhood of \(\Omega_0\) provided by Theorem 2, we can assume without any loss of generality that \(\mathcal{U}\) was chosen so that the following holds: \[\begin{align} \mathcal{U} \subset \tilde{\mathcal{U}} \end{align}\] We now pick an arbitrary odd-symplectic form \(\Omega \in \mathcal{U}\) and suppose that the following holds: \(\text{Vol}_{\Omega_0}(\Omega) =0\). In this case, the required local systolic inequality is the following: \[\begin{align} \label{sys95ineq95proof951} \inf \mathcal{A}_{\Omega} \leq 0 \leq \sup\mathcal{A}_{\Omega} \end{align}\tag{21}\] It is shown in [4] that if \(\Omega\) is sufficiently close to \(\Omega_0\) then both \(\inf \mathcal{A}_{\Omega}\) and \(\sup\mathcal{A}_{\Omega}\) lie in a neighborhood of zero and that the size of the neighborhood of zero can be controlled by the distance between \(\Omega\) and \(\Omega_0\). As a consequence, we can assume without any loss of generality that the neighborhood \(\mathcal{U}\) was chosen so that \(\inf \mathcal{A}_{\Omega}\) and \(\sup\mathcal{A}_{\Omega}\) lie in a neighborhood of zero where the Zoll polynomial is monotonic. 21 can then be seen to follow from [4] and Corollary 1.

Suppose now that \(\text{Vol}_{\Omega_0}(\Omega) \neq 0\) then, by Theorem 3, the following holds: \[\begin{align} \label{sys95ineq95volume95expansion} Vol_{\Omega_0}(\Omega) = \int^1_0 \int_{\Sigma} S\alpha_0 \wedge (\Omega_0 +rSd\alpha_0)^{n-1} \; dr + \int^1_0 \frac{1}{r} \int_{\Sigma} D(x,rS(x)) \; \alpha_0 \wedge \Omega_0^{n-1} \; dr \end{align}\tag{22}\] where \(S\) is the \(S^1\)-invariant function provided by Theorem 2. We claim that the above expression for the volume functional is monotonic in the variable \(S\). To prove this claim, we first consider the case where \(S\) is constant. In this case, by Theorem 3, the second summand in 22 vanishes. Let \(\bar S \in C^{\infty}(M)\) be the quotient of \(S\) by the relevant \(S^1\)-action on \(\Sigma\) and we now project the first summand in 22 to \(M\) by integrating along the fibers (see [29] or [4]) to obtain the following: \[\begin{align} \label{monotonic95in95S951} \int^1_0 \int_{\Sigma} S\alpha_0 \wedge (\Omega_0 +rSd\alpha_0)^n \; dr = \int^1_0 \int_{M} \bar{S}(\omega_0 +r\bar{S} \nu)^n dr \end{align}\tag{23}\] where \(\omega_0 \in \Omega^2(M)\) is a symplectic form satisfying: \(p^*\omega_0 = \Omega_0\) and \(\nu \in \Omega^2(M)\) is the \(2\)-form satisfying the relation \(p^*\nu = d\alpha_0\).

It can now readily be seen that the integral in 23 is the Zoll polynomial and monotonicity of the expression 22 in the variable \(S\) follows by shrinking the neighborhood \(\mathcal{U}\) of \(\Omega_0\) if necessary to ensure that \(S\) is contained in the neighborhood of zero where the Zoll polynomial is monotonic.

On the other hand, if \(S\) is not constant then there exists a smooth function \(Q: \Sigma \times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}\) defined by the following relation: \[\begin{align} \label{sys95ineq95proof95Q} Q(x,S(x),r) \; \alpha_0 \wedge \Omega_0^{n-1} \; dr = S\alpha_0 \wedge (\Omega_0 +rSd\alpha_0)^{n-1} \; dr \end{align}\tag{24}\] Upon differentiating both sides of 24 with respect to \(S\), we obtain the following: \[\begin{align} \frac{\partial Q}{ \partial S} \big|_{S=0} \; \alpha_0 \wedge \Omega^{n-1}_0 = \alpha_0 \wedge \Omega_0^{n-1} \end{align}\] We can therefore conclude that the following holds: \[\begin{align} \label{sys95ineq95partial95Q} \frac{\partial Q}{ \partial S} \Big|_{S=0} =1 \end{align}\tag{25}\] By Theorem 3, given an \(\epsilon >0\), there exists a \(\delta >0\) such that if \(\vert S\vert <\delta\) then, the following holds: \[\begin{align} \Bigg\vert \Bigg\vert\frac{1}{r} \frac{\partial D(x,rS)}{\partial S}\Bigg\vert\Bigg\vert_{C^0(\Sigma \times [-\delta,\delta])} < \epsilon \end{align}\] The above inequality combined with the observation in 25 shows that \(Vol_{\Omega_0}(\Omega)\) is monotonic in \(S\) if \(\vert S\vert <\delta\), a fact we can always guarantee by shrinking the neighborhood \(\mathcal{U} \subset\Xi^2_{C_0}(\Sigma)\) is necessary.

Since the expression in 12 is monotonic in the variable \(S\), we obtain the following chain of inequalities: \[\begin{align} \label{sys95ineq951} \int^1_0 \int_{\Sigma} &S_{\text{min}}\alpha_0 \wedge (\Omega_0 +rS_{\text{min}}d\alpha_0)^{n-1} \; dr \\\nonumber \leq \;\; &Vol_{\Omega_0}(\Omega) \\\nonumber \leq \int^1_0 \int_{\Sigma} &S_{\text{max}}\alpha_0 \wedge (\Omega_0 +rS_{\text{max}}d\alpha_0)^{n-1} \; dr \end{align}\tag{26}\] where \(S_{\text{min}}:= \inf_{x\in \Sigma} S(x)\) and \(S_{\text{max}}:= \sup_{x \in \Sigma} S(x)\) and any of these inequalities is an equality if and only if \(S\) is constant.

Since the \(2\)-form \(\nu \in \Omega^2(M)\) was chosen so that \(p^*\nu = d\alpha_0\), upon projecting to the base \(M\) of the \(S^1\)-bundle associated with \(\Omega_0\) by integrating along the \(S^1\)-fibers, for every constant \(S_0\) we obtain: \[\begin{align} \label{sys95ineq95zoll95poly95vol95relation95leq} \int^1_0 \int_{\Sigma} S_{0}\alpha_0 \wedge &(\Omega_0 +rS_{0}d\alpha_0)^{n-1} \; dr = \int^1_0 \int_{M} \bar{S}_{0}(\omega_0 +r\bar{S}_{0} \nu)^{n-1} dr = \\\nonumber &\int^1_0 \langle (c_0+r\bar{S}_{0}e_0)^{n-1} , [M]\rangle \;dr = P(\bar{S}_{0}) \end{align}\tag{27}\] where \(c_0 \in H^2(M, \mathbb{R})\) is the cohomology class represented by \(\omega_0 \in \Omega^2(M)\) and \(e_0 \in H^2(\Sigma, \mathbb{R})\) is \((-1)\) times the real Euler class of the \(S^1\)-bundle associated with \(\Omega_0\).

Replacing the constant \(S_0\) in 27 with \(S_{\text{min}}\) and \(S_{\text{max}}\) and using 26 and noting that the action and the volume are invariant under diffeomorphisms isotopic to the identity, we obtain: \[\begin{align} P(\inf \mathcal{A}_{\Omega}) \leq P(\bar{S}_{\text{min}}) \leq Vol_{\Omega_0}(\Omega) \leq P(\bar{S}_{\text{max}}) \leq P(\sup \mathcal{A}_{\Omega}) \end{align}\] with equality holding if and only if \(S\) is constant.

Finally, recall from Corollary 1 that \(S\) is constant if and only if \(\Omega\) is Zoll and in this case \(\inf \mathcal{A}_{\Omega} = \min S = \max S = \sup \mathcal{A}_{\Omega}\).

3 Examples of Zoll odd-symplectic forms↩︎

This section is dedicated to a result in which we present the construction of a large class of examples of Zoll odd-symplectic forms that are not necessarily of contact type. We preface the statement of this result by giving a brief outline of the geometric setup: let \((W,\omega)\) be a connected symplectic manifold of an arbitrary dimension and \(Q \hookrightarrow W\) be a closed, connected, symplectic submanifold. Then, \(Q\) can be realized as the unique Bott non-degenerate minimum of a smooth function \(H_0:W \rightarrow \mathbb{R}\). In this setting, the following is the main result of this section:

It is worth observing here that in the three-dimensional setting, as observed in [2], the trajectories of any volume-preserving flow on a closed \(3\)-manifold can be realized as the characteristic foliation of an odd-symplectic form. Therefore, if such a flow induces a free \(S^1\)-action on the underlying manifold, the corresponding odd-symplectic form will be Zoll. Such odd-symplectic forms will not be of contact type if and only if the associated \(S^1\)-bundle is trivial and they have been completely classified up to orientation-preserving diffeomorphisms of the associated \(S^1\)-bundles, see [4].

In Theorem 7 above, if the submanifold \(Q\) has exactly half the dimension of \(W\), the example falls into the class of Hamiltonian dynamical systems called magnetic systems In [31], Bimmermann constructs examples of Zoll odd-symplectic forms that are not of contact type within the paradigm of such Hamiltonian systems. In fact, the construction of Zoll odd-symplectic forms in Theorem 7 can be thought of as a generalization of the ideas leading to the proof of Bimmermann’s result to a more general geometric context.

4 Proof of the normal form theorem↩︎

Recall the setup of Theorem 2: \((\Sigma,\Omega_0)\) is a connected \(2n-1\)-dimensional Zoll odd-symplectic manifold; we denote by \(C_0 \in H^2(\Sigma,\mathbb{R})\) the class represented by \(\Omega_0\). We also fix an \(S^1\)-bundle \(p: \Sigma \rightarrow M\) associated with \(\Omega_0\) and a connection \(1\)-form \(\alpha_0 \in \Omega^1(\Sigma)\) and we denote by \(R_0\) the vector field fixed by \(\alpha_0\) and generating the relevant \(S^1\)-action on \(\Sigma\). In particular \(R_0\) is the Hamiltonian vector field of the pair \((\alpha_0,\Omega_0)\). We also fix a Riemannian metric \(g\) on \(\Sigma\) that is invariant with respect to the flow of \(R_0\).

Since we are working with smooth (i.e \(C^{\infty}\)) objects, the set \(\Xi^2_{C_0}(\Sigma)\) of representatives of the cohomology class \(C_0\) can be given the \(C^k\)-topology for any \(k \geq 0\). So, we fix the \(C^1\)-topology on \(\Xi^2_{C_0}(\Sigma)\) and note that by Lemma 1 there exists a neighborhood \(\mathcal{U} \subset \Xi^2_{C_0}(\Sigma)\) of \(\Omega_0\) such that \(\alpha_0\) is positively transverse to \(\ker(\Omega)\) for any odd-symplectic form \(\Omega \in \mathcal{U}\). So, upon fixing such a neighborhood \(\mathcal{U}\) of \(\Omega_0\) we denote by \(R_{\Omega}\) the Hamiltonian vector field of the pair \((\alpha_0,\Omega)\) for an odd-symplectic form \(\Omega \in \mathcal{U}\). Lemma 1 in the Appendix tells us that for each \(k \geq 0\) there exists a monotonically increasing continuous function \(\sigma_k: \mathbb{R} \rightarrow \mathbb{R}_{\geq 0}\) such that: \[\begin{align} ||R_{\Omega} - R_0||_{C^k} \leq \sigma_{k}(||\Omega - \Omega_0||_{C^{k}}) \end{align}\] for \(\Omega \in \mathcal{U}\); we refer to such functions as moduli of continuity. Using this fact and [5] we can assume without loss of generality that the neighbourhood \(\mathcal{U}\) is chosen such that for any \(\Omega \in \mathcal{U}\) the following holds: \[\begin{align} \label{bottkol95decomp95for95reeb} hu^*R_{\Omega} = R_{0} - \mathcal{P}_{u}[V] \end{align}\tag{31}\] such that:

1. \(u:\Sigma \rightarrow \Sigma\) is a diffeomorphism isotopic to the identity.

2. \(V\) is a vector field on \(\Sigma\) satisfying \(\mathcal{L}_{R_{0}} V =0\) and \(g(R_{0}, V) =0\).

3. \(h: \Sigma \rightarrow \mathbb{R}\) is a function satisfying \(\mathcal{L}_{R_{0}} h =0\).

4. \(\mathcal{P}_{u}: T\Sigma \rightarrow T\Sigma\) is a bundle isomorphism dependent on the diffeomorphism \(u\) and lifting the identity.

5. For each integer \(k \geq 0\) there exist moduli of continuity \(\tilde{\sigma}_k\) satisfying: \[\begin{align} \label{normal95form95bounds951} \max \Big\{dist_{C^{k+1}}(u,Id_{\Sigma}), &\vert\vert h-1\vert\vert_{C^{k+1}}, \vert\vert V\vert\vert_{C^{k+1}}, \vert\vert\mathcal{P}_u -Id \vert\vert_{C^{k}} \Big\} \\\nonumber \leq \;\; &\tilde{\sigma}_{k+1}(\vert\vert R_{\Omega} - R_{0} \vert\vert_{C^{k+1}}) \end{align}\tag{32}\]

Recall that we denote by \(\mathfrak{h}\) the homotopy class of the fibers of \(\Sigma \xrightarrow{p} M\) and by \(\Lambda_{\mathfrak{h}}(\Sigma)\) the set of short loops in \(\Sigma\) (c.f Section 1.4.1). Since we have fixed a neighborhood \(\mathcal{U} \subset \Xi^2_{C_0}(\Sigma)\) of \(\Omega_0\) such that \(\alpha_0\) is positively transverse to any odd-symplectic form \(\Omega \in \mathcal{U}\) and 31 holds for the Hamiltonian vector field \(R_{\Omega}\), we can now fix an arbitrary odd-symplectic form \(\Omega \in \mathcal{U}\) and define \(\tilde{\Omega}:= u^*\Omega\) so that \(hR_{\tilde{\alpha_0}, \tilde{\Omega}} = R_{0} - \mathcal{P}_{u}(V)\) where \(\tilde{\alpha_0}:= u^*\alpha_0\). By [5] the following bounds hold: \[\begin{align} \label{bounds95of95omega95tilde95minus95omega} &\vert \vert \tilde{\Omega} - \Omega_0\vert \vert_{C^{k}} \leq \vert \vert\tilde{\Omega} - \Omega\vert \vert_{C^{k}} + \vert \vert\Omega - \Omega_0\vert \vert_{C^k } \\ \leq \nonumber \vert \vert\Omega\vert \vert_{C^{k+1}} \Big(1+ &\vert \vert du\vert \vert^{k+2}_{C^{k}} \Big) dist_{C^{k+1}}(u,\text{Id}_{\Sigma}) +\vert \vert\Omega - \Omega_0\vert \vert_{C^k} \leq \sigma_{k+1}(\vert \vert\Omega - \Omega_0\vert \vert_{C^{k+1}}) \end{align}\tag{33}\] The last inequality above is justified by 32 and by the fact that by Lemma 1 for each such \(k\), there exists a modulus of continuity \(\sigma_{k}\) such that: \[\begin{align} \tilde{\sigma}_{k}(\vert \vert R_{\Omega} - R_{0} \vert \vert_{C^{k}}) \leq \sigma_{k}( \vert \vert \Omega - \Omega_0\vert \vert_{C^{k}}) \end{align}\] The function \(S_{\tilde{\Omega}}: \Sigma \rightarrow \mathbb{R}\) is defined in the following way: \[\begin{align} \label{def95of95S} S_{\tilde{\Omega}}(x):= \mathcal{A}_{\tilde{\Omega}}(\bar{\Gamma}_{x}) \;\;\;\text{where \bar{\Gamma}_{x} is the homotopy \bar{\Gamma}_x(\cdot,t) = p_x \forall t \in [0,1] } x \in \Sigma \end{align}\tag{34}\] where \(\mathcal{A}_{\tilde{\Omega}}\) is defined in 11 .

Given a representation \(\tilde{\Omega} = \Omega_0 + d\alpha\) it is easy to see that \(S_{\tilde{\Omega}}\) has the following convenient expression: \[\begin{align} \label{def95of95S95ind} S_{\tilde{\Omega}}(x):= \int_{p_x} \alpha \;\;\; \text{for } x\in \Sigma \end{align}\tag{35}\] By [4] the representation of the function \(S_{\tilde{\Omega}}\) in 35 is independent of the choice of parametrizing \(1\)-form \(\alpha\) since our assumption that \(C_0^{n-1} =0\) implies that the fibers of the bundle \(\Sigma \xrightarrow{p} M\) are null-homologous [4]. Since we have fixed \(\tilde{\Omega}\), we suppress the subscript in \(S_{\tilde{\Omega}}\) for ease of notation. The function \(S\) is invariant under the flow of \(R_0\) and note that for any \(1\)-form \(\alpha\) such that \(\tilde{\Omega} = \Omega_0 + d\alpha\), the difference \(\alpha - S\alpha_0\) satisfies: \[\begin{align} \int_{\mathfrak{p}_x} \alpha - S\alpha_0 =0 \;\; \text{forall x \in \Sigma} \end{align}\]

Therefore, by [5] the following is true: \[\begin{align} \label{alpha95eta95alpha950} \alpha = S\alpha_0 + \eta +df \end{align}\tag{36}\] where \(f\) is some smooth function and \(\eta \in \Omega^1(\Sigma)\) satisfies \(\iota_{R_{0}} \eta =0\). Therefore \(\tilde{\Omega}\) has the following expression: \[\begin{align} \label{Omega95tilde95full95expansion} \tilde{\Omega} = \Omega_0 + dS \wedge \alpha_0 + Sd\alpha_0 + d\eta \end{align}\tag{37}\] We will denote by \(\Phi^t_0\) the flow of the vector field \(R_0\) and we now make the following definition: \[\begin{align} \label{def95of95K} K_{u}: R_{0}^{\perp} &\DOTSB\relbar\joinrel\relbar\joinrel\relbar\joinrel\rightarrow [R^{\perp}_{0}]^* \nonumber\\ X &\DOTSB\mapstochar\DOTSB\relbar\joinrel\relbar\joinrel\relbar\joinrel\rightarrow \int_{S^1} u^*\Omega(\mathcal{P}_{u}(d\Phi^t_0(X)), \; d\Phi^t_0\;\cdot \;) \; dt \end{align}\tag{38}\] Since \(\Omega\) is \(C^1\)-close to \(\Omega_0\) Lemma 1 tells us that \(R_{\Omega}\) is \(C^1\)-close to \(R_0\) therefore, by the argument in [19] the map \(K_{\text{Id}_{\Sigma}}\) is invertible. Since the neighborhood \(\mathcal{U}\) was chosen so that \(u\) is \(C^1\)-close to the identity on \(\Sigma\), using Neumann’s series argument, we know that non-degeneracy is an open condition. So, we see can conclude that the map \(K_u\) is also invertible.

We know from [4] the function \(\mathcal{A}_{\tilde{\Omega}}\) is a primitive of the following \(1\)-form: \[\begin{align} \label{action95form95def95eqn} \mathfrak{a}_{\tilde{\Omega}}(\xi(t))|_{\gamma}:= \int_{S^1} \tilde{\Omega}(\xi(t), \Dot{\gamma}(t)) \text{ } \forall \gamma \in \Lambda_{\mathfrak{h}}(\Sigma) \text{ and } \xi \in T_{\gamma}\Lambda_{\mathfrak{h}}(\Sigma) \end{align}\tag{39}\] So, it is clear that \(dS\) has the following form: \[\begin{align} \label{dS61K40V41} dS|_x[w] = d\mathcal{A}_{\tilde{\Omega}}\big|_{\bar{\Gamma}_x}[d\Phi^t_0 (w)] \stackrel{\eqref{action95form95def95eqn}}{=} \int_{S^1} \tilde{\Omega}(d\Phi^t_0(w) ,\frac{ \partial p_x}{\partial t}) \; dt \\\nonumber \stackrel{\eqref{bottkol95decomp95for95reeb}}{=} \int_{S^1} \tilde{\Omega}( \mathcal{P}_u((V \circ \Phi^t_0)) - h(\Phi^t_0) R_{\tilde{\Omega}}(u \circ \Phi^t_0), \; d\Phi^t_0(w)) \; dt \\\nonumber = \int_{S^1} \tilde{\Omega}( \mathcal{P}_u((V \circ \Phi^t_0)), \; d\Phi^t_0 (w) \;) \; dt = K_u(V)[w] \end{align}\tag{40}\] where \(\frac{ \partial p_x}{\partial t} = \frac{ \partial \Phi^t_0(x)}{\partial t}\) and the final equality follows from the fact that \(\mathcal{L}_{R_0}(V) =0\).

So using 37 , 38 and 40 , the contraction \(\iota_{R_{0}} d\eta\) can be written as: \[\begin{align} \label{decompositon95of95odd95symp95forms95nft95proof952} \iota_{R_{0}} d\eta = \iota_{R_0} \tilde{\Omega} + \iota_{R_0} Sd\alpha_0 -\iota_{R_0} \alpha_0 \wedge dS \stackrel{\eqref{bottkol95decomp95for95reeb}}{=} \iota_{\mathcal{P} \circ K_{u}^{-1}(dS)} \tilde{\Omega} - dS \end{align}\tag{41}\] where in the second equality above, we have used the fact that since \(\alpha_0\) is a connection \(1\)-form for the \(S^1\)-bundle \(p: \Sigma \rightarrow M\), there exists a \(2\)-form \(\nu \in \Omega^2(M)\) such that \(p^*\nu = d\alpha_0\). As a consequence, the contraction \(\iota_{R_0} d\alpha_0\) vanishes.

We now define the endomorphism \(\mathcal{F}: T^*\Sigma \rightarrow T^*\Sigma\) in the statement of Theorem 2 in the following way \[\begin{align} \label{def95of95F} \mathcal{F}(\lambda) = \iota_{\mathcal{P}\circ K^{-1}_u(\lambda|_{[R_0]^{\perp}})} \tilde{\Omega} - \lambda|_{[R_0]^{\perp}} \;\; \text{for \lambda \in T^*\Sigma} \end{align}\tag{42}\] where \([R_0]^{\perp} \subset T\Sigma\) is the distribution that is perpendicular to the linear span of \(R_0\) with respect to the fixed invariant metric \(g\).

Clearly, \(\mathcal{F}\) is a linear endomorphism and since \(\mathcal{P}\) and \(K_u\) lift the identity on \(\Sigma\), so does \(\mathcal{F}\).

Putting 36 , 41 and 42 together we see that: \[\begin{align} \label{tilda95omega95final95form} \tilde{\Omega} = \Omega_0 + d(S\alpha_0 + \eta) \end{align}\tag{43}\] such that \(\eta \in \Omega^1(\Sigma)\) satisfies: \[\begin{align} \iota_{R_0} \eta =0 \;\;\; \& \;\;\; \iota_{R_0} d\eta = \mathcal{F}(dS) \end{align}\] So, in order to finish the proof of Theorem 2, we need to only establish 7 .

Towards that end, we first remind the reader that by Lemma 1 we can assume without any loss of generality that the neighborhood \(\mathcal{U}\) is chosen such that for each \(k \geq 1\) there exists a modulus of continuity \(\sigma_k\) such that: \[\begin{align} \label{normal95form95bounds952} \tilde{\sigma}_{k}(\vert\vert R_{\Omega} - R_{0}\vert\vert_{C^{k}}) \leq \sigma_{k}(\vert\vert\Omega - \Omega_0\vert\vert_{C^{k}}) \end{align}\tag{44}\] In each step of the sequel below, we will successively obtain new moduli of continuity by shrinking the previous ones but for ease of notation, we will continue to denote these by \(\sigma_k\).

By Proposition 16 and 33 there exists an \(\Hat{\alpha} \in \Omega^1(\Sigma)\) and moduli of continuity \(\sigma_k\) for \(k \geq 1\) such that \[\begin{align} \label{nft95proof95alpha95hat95bound} &\tilde{\Omega} = \Omega_0 + d\hat{\alpha} \\\nonumber \vert\vert \hat{\alpha}\vert\vert_{C^k}& \leq \sigma_{k}(\vert\vert\Omega - \Omega_0\vert\vert_{C^{k}}) \end{align}\tag{45}\] By [4], our assumption that \(C_0^{n-1}=0\) implies that the representation of \(S\) in 35 is independent of the \(1\)-form \(\alpha\), therefore: \[\begin{align} S_{\tilde{\Omega}}(x) = \int_{p_x} \hat{\alpha} \end{align}\] So, for the sake of continuity, we will continue to write \(\alpha\) for \(\hat{\alpha}\).

We therefore obtain the following: \[\begin{align} \label{bound95on95S} \vert\vert S_{\tilde{\Omega}}\vert\vert_{C^{k}} \leq \sigma_{k+1}(\vert\vert\Omega - \Omega_0\vert_{C^{k+1}}) \end{align}\tag{46}\] We know from 40 that \(dS = K_u(V)\) for \(V\) as in 31 . In addition, by [5] the following holds: \[\begin{align} \label{nft95bound953} \vert\vert \tilde{\Omega}\vert\vert_{C^k} \leq \vert\vert\Omega\vert\vert_{C^k} \cdot \vert\vert du\vert\vert^2_{C^k}(1+\vert\vert du\vert\vert_{C^{k-1}}) \end{align}\tag{47}\] Using 32 and combining 38 with 47 we obtain the following: \[\begin{align} \label{bound95on95dS} \vert\vert dS_{\tilde{\Omega}}\vert\vert_{C^k} \leq \vert\vert K_u\vert\vert_{C^k} \vert\vert V\vert\vert_{C^k} \leq \sigma_{k+1}(\vert\vert \Omega - \Omega_0\vert\vert_{C^{k+1}}) \end{align}\tag{48}\] Using the bound 45 on the \(1\)-form we now call \(\alpha\) and [5], the following holds: \[\begin{align} \label{nft95eta95bound951} \vert\vert \eta\vert\vert_{C^k} \leq \vert\vert \alpha -S\alpha_0\vert\vert_{C^k} + \vert\vert d\iota_{R_0}( \alpha -S\alpha_0)\vert\vert_{C^k} \end{align}\tag{49}\] In light of this fact, [5] allows us to deduce the following bounds: \[\begin{align} \label{nft95eta95bound952} \vert\vert\alpha -S\alpha_0\vert\vert_{C^k} \leq &\vert\vert\alpha - u^*\alpha\vert\vert_{C^k} + \vert\vert u^*\alpha - S\alpha_0\vert\vert_{C^k} \\ \nonumber \leq \Big[\vert\vert\alpha\vert\vert_{C^{k+1}} \text{ } dist_{C^{k+1}}(u,Id_{\Sigma})&(1+\vert\vert du\vert\vert^{k+1}_{C^k}) \Big] + \vert\vert u^*\alpha\vert\vert_{C^k} + \vert\vert S\alpha_0\vert\vert_{C^k} \\\nonumber \leq &\sigma_{k+1}(\vert\vert\Omega -\Omega_0\vert\vert_{C^{k+1}}) \end{align}\tag{50}\] Similarly, we arrive at the following bounds for \(\vert\vert d(\iota_{R_0} \alpha -S\alpha_0)\vert\vert_{C^k}\): \[\begin{align} \label{nft95eta95bound953} \vert\vert d(\iota_{R_0} \alpha -S\alpha_0)\vert\vert_{C^k} \leq &\vert\vert d(\iota_{R_0}\alpha)\vert\vert_{C^k} + \vert\vert dS \wedge \alpha_0 \vert\vert_{C^k} + \vert\vert S d\alpha_0\vert\vert_{C^k} \\\nonumber \leq &\sigma_{k+1}(\vert\vert \Omega -\Omega_0\vert\vert_{C^{k+1}}) \end{align}\tag{51}\] Plugging 48 , 50 and 51 into 49 we obtain: \[\begin{align} \vert\vert \eta\vert\vert_{C^k} \leq \sigma_{k+1}(\vert\vert \Omega -\Omega_0\vert\vert_{C^{k+1}}) \end{align}\] For the required bound on \(\vert\vert d\eta\vert\vert_{C^k}\) observe that: \[\begin{align} \label{d95eta95bounds} \vert\vert d\eta\vert\vert_{C^{k}} \leq \vert\vert \tilde{\Omega} - \Omega_0\vert\vert_{C^{k}} + \vert\vert dS\wedge \alpha_0\vert\vert_{C^{k}} + \vert\vert Sd\alpha_0\vert\vert_{C^{k}} \end{align}\tag{52}\] Applying the bounds in 33 and 48 to 52 we obtain \[\begin{align} \label{nft95deta95bound95final} \vert\vert d\eta\vert\vert_{C^k} \leq \sigma_{k+1}(\vert\vert \Omega - \Omega_0\vert\vert_{C^{k+1}}) \end{align}\tag{53}\] We now only need to show that \(\vert \vert \mathcal{F}\vert\vert_{C^k} \leq \sigma_{k+1}(\vert\vert\Omega -\Omega_0\vert\vert_{C^{k+1}})\). Towards this end, we make the following definitions: \[\begin{align} \mathcal{F}_1:= \iota_{K^{-1}_u|_{[R_0]^{\perp}}} \Omega -Id|_{[R_0]^{\perp}} \\\nonumber \mathcal{F}_0:= \iota_{K^{-1}_0|_{[R_0]^{\perp}}} \Omega_0 -Id|_{[R_0]^{\perp}} \end{align}\] where \(K_0: R_0^{\perp} \rightarrow [R^{\perp}_{0}]^*\) is defined by: \[\begin{align} X \mapsto \int_{S^1} \Omega_0(d\Phi^t_0 (X),\; d\Phi^t_0 \;\cdot) \end{align}\] Now observe that: \[\begin{align} \label{mathcal95F95bound951} \vert\vert\mathcal{F}\vert\vert_{C^k} \leq \vert\vert\mathcal{F} -\mathcal{F}_1\vert\vert_{C^k} + \vert\vert\mathcal{F}_1 -\mathcal{F}_0\vert\vert_{C^k} + \vert\vert\mathcal{F}_0\vert\vert_{C^k} \end{align}\tag{54}\] Note that by 32 and Lemma 1, \(\vert\vert\mathcal{P}_u -Id\vert\vert_{C^k} \leq \sigma_k(\vert\vert\Omega- \Omega_0\vert\vert_{C^k})\); combining this observation with 33 we see that: \[\begin{align} \label{mathcal95F95bound952} \vert\vert\mathcal{F} -\mathcal{F}_1\vert\vert_{C^k} \leq \omega_k(\vert\vert\Omega- \Omega_0\vert\vert_{C^{k}}) \\\nonumber \vert\vert\mathcal{F}_1 -\mathcal{F}_0\vert\vert_{C^k} \leq \omega_k(\vert\vert\Omega- \Omega_0\vert\vert_{C^k}) \end{align}\tag{55}\] Plugging 55 in 54 and observing that \(\vert\vert \mathcal{F}_0\vert\vert_{C^k} = 0\) we obtain: \[\begin{align} \vert\vert\mathcal{F}\vert\vert_{C^k} \leq \sigma_{k+1}(\vert\vert\Omega- \Omega_0\vert\vert_{C^{k+1}}) \end{align}\]

5 Proof of Corollary 1↩︎

We begin by fixing an \(S^1\)-bundle \(p:\Sigma \rightarrow M\) associated with the Zoll odd-symplectic form \(\Omega_0\) and pick a connection \(1\)-form \(\alpha_0\). Let \(\mathcal{U} \subset \Xi^2_{C_0}(\Sigma)\) be the neighborhood of \(\Omega_0\) provided by Theorem 2. Then, for any odd-symplectic form \(\Omega \in \mathcal{U}\), there exists a diffeomorphoism \(u:\Sigma \rightarrow \Sigma\) such that \(u^*\Omega\) has the form 6 . Denoting by \(\tilde{\Omega}:= u^*\Omega\) and using the same notation as in the statement of Theorem 2, we see that 6 implies the following chain of equalities: \[\begin{align} \label{var95princip951} \iota_{R_0} \tilde{\Omega} = \iota_{R_0} d\eta - dS = \mathcal{F}[dS] - dS \end{align}\tag{56}\] where \(R_0\) is the generating vector field for the \(S^1\)-action on \(\Sigma\) that interests us and \(S\) is the invariant function whose existence is guaranteed by Theorem 2. Denoting by \(\bar{S}\) the quotient of \(S\) by the \(S^1\)-action on \(\Sigma\), we observe that by 56 , for any \(y \in \Sigma\) contained in an \(S^1\)-fiber over a critical point \(x\) of \(\Bar{S}\) the following holds: \[\begin{align} \iota_{R_0} \tilde{\Omega}|_{y} =0 \end{align}\] So, any vector field that spans the characteristic distribution of \(\tilde{\Omega}\) at \(y\) is a scalar multiple of \(R_0\). As a consequence, the following is true: pick an arbitrary vector field \(R\) spanning the characteristic distribution of \(\tilde{\Omega}\) then, for any \(y \in \text{Crit}(S)\) the trajectory of \(R\) passing through \(y\) is the image \(u(p_y)\) of the \(S^1\)-fiber \(p_y\) through \(y\). From the definition of the quotient map \(\bar S\), we can then conclude that any vector field spanning the characteristic distribution of \(\tilde{\Omega}\) must have at least as many closed trajectories as the number of critical points of the function \(\bar S\). Moreover, by the Lusternik–Schnirelmann theorem [30], the following chain of inequalities hold: \[\begin{align} \text{Crit \bar{S}} \geq \text{cl}(M) \end{align}\] where \(\text{cl}(M)\) is the cup-length of \(M\) computed for the singular cohomology functor with real coefficients.

To finish the proof, we only need to show that \(\Omega\) is Zoll if and only if \(S\) is constant. Toward that end, observe that if \(S\) and therefore \(\bar{S}\) is constant, then the characteristic foliation of \(\Omega\) can be conjugated to that of \(\Omega_0\) by the diffeomorphism \(u\) and \(\Omega\) is therefore Zoll.

If, on the other hand, \(\Omega\) is Zoll then by [4] we know that \(\inf\mathcal{A}_{\Omega} = \sup\mathcal{A}_{\Omega}\). As a consequence, since function \(S\) is defined in 34 below to be the following: \[\begin{align} S(x) = \mathcal{A}_{\Omega}(\bar{\Gamma}_x) \end{align}\] where \(\Bar{\Gamma}: S^1 \times [0,1] \rightarrow \Sigma\) is just the constant homotopy \(\bar{\Gamma}(\cdot, t) = p_x\) \(\forall t \in [0,1]\), the inequalities in 8 now follow.

6 Proof of Theorem 3↩︎

Recall the setup of Theorem 3: \((\Sigma,\Omega_0)\) is a connected Zoll odd-symplectic manifold of dimension \(2n-1\) and the cohomology class \(C_0 \in H^2(\Sigma,\mathbb{R})\) represented by \(\Omega_0\) satisfies the condition: \(C_0^{n-1} =0\). The volume functional (c.f 12 ) was defined as: \[\begin{align} Vol_{\Omega_0}(\Omega) = \int^1_0 \int_{\Sigma} \alpha \wedge (\Omega_0+rd\alpha)^{n-1} \;dr \end{align}\] where \(\alpha \in \Omega^1(\Sigma)\) is such that \(\Omega = \Omega_0+d\alpha\).

As always, denote the \(S^1\)-bundle associated with \(\Omega_0\) by \(\Sigma \xrightarrow{p} M\) and let \(\alpha_0\) be a connection \(1\)-form of this \(S^1\)-bundle that is, the \(1\)-form satisfying: \(\alpha_0 (R_0) =1\) and \(\mathcal{L}_{R_0}\alpha_0 =0\) where \(R_0\) is the vector field generating the relevant \(S^1\)-action on \(\Sigma\). Then, by Theorem 2 there exists a \(C^2\)-neighbourhood \(\mathcal{U} \subset \Xi^2_{C_0}(\Sigma)\) of \(\Omega_0\) such that for any odd-symplectic form \(\Omega \in \mathcal{U}\) there exists a diffeomrophism \(u\) of \(\Sigma\) isotopic to the identity and a smooth function \(\tilde{S}: \Sigma \rightarrow \mathbb{R}\) such that: \[\begin{align} \tilde{\Omega}:= u^*\Omega = \Omega_0 + d(\tilde{S}\alpha_0 + \tilde{\eta}) \end{align}\] where the \(1\)-form \(\tilde{\eta}\) is as in the statement of Theorem 2. Since \(u\) is isotopic to the identity, it follows from [4] that \(Vol_{\Omega_0}(\Omega) = Vol_{\Omega_0}(\tilde{\Omega})\). Therefore, \[\begin{align} \label{raw95vol95integral95vol95formula95compuattion} Vol_{\Omega_0}(\Omega) = \int^1_0 \int_{\Sigma} (\tilde{S}&\alpha_0 + \tilde{\eta} ) \wedge (\Omega_0 + r d(\tilde{S}\alpha_0 + \tilde{\eta}))^{n-1} dr \\\nonumber = &\int^1_0 \frac{1}{r} \int_{\Sigma} (S\alpha_0 +\eta) \wedge (\Omega_0 + d(S\alpha_0 +\eta))^{n-1} \;\; dr \end{align}\tag{57}\] where \(S:= r \tilde{S}\) and \(\eta:= r \tilde{\eta}\).

We now consider the following two summands in 57 :

1. \(S\alpha_0 \wedge (\Omega_0 + d(S\alpha_0 + \eta))^{n-1}\)

2. \(\eta \wedge (\Omega_0 + d(S\alpha_0 + \eta))^{n-1}\)

A formula for \(\int_{\Sigma} S\alpha_0 \wedge (\Omega_0 + d(S\alpha_0 + \eta))^{n-1}\)

Step 1: We now claim that the following expansion follows from the binomial theorem: \[\begin{align} \label{formula95for95omega95to95the95n} (\Omega_0 + d(S\alpha_0 + \eta ))^{n-1} = (\Omega_0 + d\eta +Sd\alpha_0)^{n-1} + (n-1)(\Omega_0 + d\eta +Sd\alpha_0)^{n-2} \wedge (dS \wedge \alpha_0) \end{align}\tag{58}\] To see that this is true, we first write out the binomial expansion of \((\Omega_0 + d(S\alpha_0 + \eta ))^{n-1}\): \[\begin{align} \label{expansion95of95omega95to95the95n} (\Omega_0 + d(S\alpha_0 + \eta ))^{n-1} = \sum^{n-1}_{k=0} \binom{n-1}{k} (\Omega_0 + d\eta +Sd\alpha_0)^{n-1-k} \wedge (dS \wedge \alpha_0)^k \end{align}\tag{59}\] it is now easy to see that for \(k >1\) the product \((dS \wedge \alpha_0)^k =0\) by skew-symmetry of the wedge product. So, only the first two summands in 59 remain, thereby justifying the validity of 58 . Since \(S\alpha_0 \wedge (n-1)(\Omega_0 + d\eta +Sd\alpha_0)^{n-2} \wedge (dS \wedge \alpha_0) =0\) by skew-symmetry of the wedge product. We therefore obtain the following: \[\begin{align} \label{alpha95wedge951} S\alpha_0 &\wedge (\Omega_0 + d(S\alpha_0 + \eta))^{n-1} = S\alpha_0 \wedge (\Omega_0 +Sd\alpha_0)^{n-1} \\\nonumber + &S\Bigg[ \sum_{k=1}^{n-1} \binom{n-1}{k} \alpha_0 \wedge (\Omega_0 +Sd\alpha_0)^{n-1-k} \wedge d\eta^k \Bigg] \end{align}\tag{60}\]

We now focus just on the second summand in 60 and write: \[\begin{align} \label{def95beta95k} \beta_k := \alpha_0 \wedge (\Omega_0 +Sd\alpha_0)^{n-1-k} \wedge d\eta^k \;\;\; for \;\; 1\leq k \leq n-1 \end{align}\tag{61}\] Using the binomial theorem to expand \((\Omega_0 +Sd\alpha_0)^{n-1-k}\) in \(\beta_k\) we get: \[\begin{align} \label{beta95k95def} \beta_k = \sum^{n-1-k}_{j=0} \binom{n-1-k}{j} S^j \alpha_0 \wedge d\alpha^j_0 \wedge (\Omega_0)^{n-1-k-j} \wedge d\eta^k \end{align}\tag{62}\] By definition, the exterior derivative satisfies: \[\begin{align} &S^j \cdot d\big(\alpha_0 \wedge d\alpha^j_0 \wedge (\Omega_0)^{n-1-k-j}\wedge \eta \wedge d\eta^{k-1}\big) = \\\nonumber S^j \big[ d\alpha^{j+1}_0 &\wedge (\Omega_0)^{n-1-k-j} \wedge \eta \wedge d\eta^{k-1}\big] - S^j \alpha_0 \wedge d\alpha^j_0 \wedge (\Omega_0)^{n-1-k-j} \wedge d\eta^k \end{align}\] The term in 62 is: \[\begin{align} \label{alpha95wedge952} &\sum^{n-1-k}_{j=0} \binom{n-1-k}{j} S^j \big[ d\alpha^{j+1}_0 \wedge (\Omega_0)^{n-1-k-j} \wedge \eta \wedge d\eta^{k-1}\big] \\\nonumber - \sum^{n-1-k}_{j=0} & \binom{n-1-k}{j} S^j \cdot d\big(\alpha_0 \wedge d\alpha^j_0 \wedge (\Omega_0)^{n-1-k-j}\wedge \eta \wedge d\eta^{k-1}\big) \end{align}\tag{63}\] Plugging 63 into 60 we obtain: \[\begin{align} \label{alpha95wedge953} &S\alpha_0 \wedge (\Omega_0 + d(S\alpha_0 + \eta))^{n-1} = S\alpha_0 \wedge (\Omega_0 +Sd\alpha_0)^{n-1} \\\nonumber + S \Bigg[ \sum_{k=1}^{n-1} \binom{n-1}{k} &(\Omega_0 + Sd\alpha_0)^{n-1-k} \wedge \eta \wedge d\eta^{k-1} \wedge d\alpha_0 - \sum^{n-1}_{k=1} \sum^{n-1-k}_{j=0} S^j \cdot \tilde{p}^k_j \alpha_0 \wedge (\Omega_0)^{n-1} \Bigg] \end{align}\tag{64}\] where for each \(k,j\) satisfying \(1\leq k \leq n\) and \(0 \leq j \leq n-k\), the functions \(\tilde{p}^k_j: \Sigma \rightarrow \mathbb{R}\) are defined by: \[\begin{align} \label{def95of95p95tilde} \tilde{p}^k_j \alpha_0 \wedge (\Omega_0)^{n-1} = \binom{n-1}{k} \binom{n-1-k}{j} d\big(\alpha_0 \wedge d\alpha^j_0 \wedge (\Omega_0)^{n-1-k-j}\wedge \eta \wedge d\eta^{k-1}\big) \end{align}\tag{65}\] Step 2: We now focus on the first part of the second summand in 64 and write it as \(\sigma_{k,j}\): \[\begin{align} \sigma_{k,j} = \sum^{n-1-k}_{j=0} \binom{n-1-k}{j} S^{j+1} \big[ d\alpha^{j+1}_0 \wedge (\Omega_0)^{n-1-k-j} \wedge \eta \wedge d\eta^{k-1}\big] \end{align}\] Since \(\alpha_0\) is the connection \(1\)-form of the \(S^1\)-bundle \(\Sigma \xrightarrow{p} M\) associated with \(\Omega_0\) and \(R_0\) is the vector field generating the \(S^1\)-action, it is clear that the operator \([\alpha_0 \wedge \iota_{R_0}] : \Omega^{2n-1}(\Sigma) \rightarrow \Omega^{2n-1}(\Sigma)\) acts as the identity. Therefore, on applying the operator \(\alpha_0 \wedge \iota_{R_0}\) to \(\sigma_{k,j}\) we obtain: \[\begin{align} \sigma_{k,j} = \sum^{n-1-k}_{j=0} \binom{n-1-k}{j} S^{j+1} [\alpha_0 \wedge \iota_{R_0}] ( d\alpha^{j+1}_0 \wedge \Omega^{n-1-k-j}_0 \wedge \eta \wedge & d\eta^{k-1}) \\\nonumber = \sum^{n-1-k}_{j=0} \binom{n-1-k}{j} S^{j+1}[\alpha_0 \wedge d\alpha^{j+1}_0 \wedge \Omega^{n-1-k-j}_0 \wedge \eta \wedge (k-1) \mathcal{F}(dS&) \wedge d\eta^{k-2}] \end{align}\] where the last equality follows from the fact that \(\iota_{R_0} d\alpha_0 = \iota_{R_0} \eta = \iota_{R_0} \Omega_0 =0\) and \(\iota_{R_0} d\eta = \mathcal{F}(dS)\) (c.f Theorem 2) So, we can write \(\sigma_{k,j}\) as: \[\begin{align} \label{sigma95kj951} \sigma_{k,j} = \sum^{n-1-k}_{j=0} \binom{n-1-k}{j} (k-1) S^{j+1} \mathcal{F}(dS) \wedge [\alpha_0 \wedge d\alpha^{j+1}_0 \wedge \Omega^{n-1-k-j}_0 \wedge \eta \wedge d\eta^{k-2}] \end{align}\tag{66}\] Before going further, we recall the following observation extracted from [5]: the wedge product induces an invertible map \(L: \Omega^{2n-2}(\Sigma) \times \Omega^1(\Sigma) \rightarrow \Omega^{2n-1}(\Sigma)\) in the following way: \[\begin{align} (\omega_1,\omega_2) \mapsto \omega_1 \wedge \omega_2 \text{ }\;\;\forall\;\; (\omega_1,\omega_2) \in \Omega^{2n-2}(\Sigma) \times \Omega^1(\Sigma) \end{align}\] So, the endomorphism \(\mathcal{F}:\Omega^1(\Sigma) \rightarrow \Omega^1(\Sigma)\) in Theorem 2 has a dual \(\hat{\mathcal{F}}: \Omega^{2n-1}(\Sigma) \rightarrow \Omega^{2n-1}(\Sigma)\) defined in the following way: \[\begin{align} \label{dual95of95endo95F} \mathcal{F}(\omega_1)\wedge \omega_2 = \omega_1 \wedge \hat{\mathcal{F}}(\omega_2) \text{ }\;\;\forall (\omega_1,\omega_2) \in \Omega^{2n-2}(\Sigma) \times \Omega^1(\Sigma) \end{align}\tag{67}\] Combining this observation with 66 we obtain: \[\begin{align} \sigma_{k,j} = \sum^{n-1-k}_{j=0} \binom{n-1-k}{j}(k-1)\frac{1}{j+2} dS^{j+2} \wedge \hat{\mathcal{F}}[\alpha_0 \wedge d\alpha^j_0 \wedge \Omega^{n-1-k-j}_0 \wedge \eta \wedge d\eta^{n-1-k-2}] \end{align}\] where \(\hat{\mathcal{F}}\) is the dual to \(\mathcal{F}\) defined in 67 .

After applying Stokes’ theorem, for each fixed \(k\) we can now write the integral \(\int_{\Sigma} \sigma_{k,j}\) as: \[\begin{align} \label{sigma95k95final95form} \int_{\Sigma} \sigma_{k,j} = \sum^{n-1-k}_{j=0} \binom{n-1-k}{j}(k-1) \frac{1}{j+2} \int_{\Sigma} S^{j+2} d\hat{\mathcal{F}}(\alpha_0 \wedge d\alpha^{j+1}_0 \wedge \Omega^{n-1-k-j}_0 \wedge \eta \wedge d\eta^{k-2}) \end{align}\tag{68}\] Step 3:

Finally, because \(\sigma_{k,j}\) is a top-rank form on \(\Sigma\), for each fixed \(k,j\) there exist smooth functions \(p^k_j: \Sigma \rightarrow \mathbb{R}\) given by: \[\begin{align} \label{sigma95k95polynomial} p^k_j \cdot \alpha_0 \wedge (\Omega_0)^{n-1} = \binom{n-1}{k} \binom{n-1-k}{j}\frac{1}{j+2}(k-1) d\hat{\mathcal{F}}(\alpha_0 \wedge d\alpha^{j+1}_0 \wedge \Omega^{n-1-k-j}_0 \wedge \eta \wedge d\eta^{k-2}) \end{align}\tag{69}\]
Plugging 69 into 64 we get: \[\begin{align} \label{part95295final95form} &\int_{\Sigma}S\alpha_0 \wedge (\Omega_0 + Sd\alpha_0 + d\eta)^{n-1} \\\nonumber = \int_{\Sigma} S\alpha_0 \wedge (\Omega_0 +Sd\alpha_0)^{n-1} + &\int_{\Sigma} \Big[\sum^{n-1}_{k=1} \sum^{n-1-k}_{j=0} S^{j+2} \cdot p^k_j + \sum^{n-1}_{k=1} \sum^{n-1-k}_{j=0} S^{j+1} \cdot \tilde{p}^k_j\Big] \alpha_0 \wedge (\Omega_0)^{n-1} \\\nonumber = \int_{\Sigma} S\alpha_0 &\wedge (\Omega_0 +Sd\alpha_0)^{n-1} + \int_{\Sigma} \tilde{P}(x,S) \; \alpha_0 \wedge (\Omega_0)^{n-1} \end{align}\tag{70}\] where \(\tilde{P}(x,s):= \sum_{k=0}^{n-1} \sum_{j=0}^{n-1-k} \tilde{P}_{k,j}(x,s)\) and the functions \(\tilde{P}_{k,j}(x,s)\) are given by \[\begin{align} \label{def95of95Tilde95P} \tilde{P}_{k,j}(x,s):= \sum^{n-1}_{k=1} \sum^{n-1-k}_{j=0} s^{j+2} \cdot p^k_j + \sum^{n-1}_{k=1} \sum^{n-1-k}_{j=0} s^{j+1} \cdot \tilde{p}^k_j \end{align}\tag{71}\] Part 3: A formula for \(\int_{\Sigma} \eta \wedge (\Omega_0 + d(S\alpha_0 + \eta ))^{n-1}\)

Step 1:
First note that by 58 , we can write \(\eta \wedge (\Omega_0 + d(S\alpha_0 + \eta))^{n-1}\) as: \[\begin{align} \label{eta95wedge951} \eta \wedge (\Omega_0 +Sd\alpha_0 + d\eta)^{n-1} + n-1 [\eta \wedge (\Omega_0 +Sd\alpha_0 + d\eta)^{n-2} \wedge (dS \wedge \alpha_0)] \end{align}\tag{72}\] Looking just at the second summand above, see that we can write it as: \[\begin{align} \label{part395formula951} &(n-1)dS \wedge \alpha_0 \wedge \eta \wedge (\Omega_0 +Sd\alpha_0 + d\eta)^{n-2} \nonumber\\ = \sum^{n-2}_{k=0} \sum^{n-k-2}_{j=0} &\binom{n-2}{k} \binom{n-k-2}{j} \frac{n-1}{j+1} dS^{j+1} \wedge \alpha_0 \wedge \eta \wedge (\Omega_0)^{n-1-k-j} \wedge (d\alpha_0)^j \wedge (d\eta)^{k} \end{align}\tag{73}\] After applying Stokes’ theorem, the integral of 73 above can be written as: \[\begin{align} \label{part95395step951} - \sum^{n-2}_{k=0} \sum^{n-1-k}_{j=0} \binom{n-2}{k} \binom{n-1-k}{j} \frac{n-1}{j+1} \int_{\Sigma} S^{j+1} d(\alpha_0 \wedge \eta \wedge (\Omega_0)^{n-1-k-j} \wedge (d\alpha_0)^j \wedge (d\eta)^{k}) \end{align}\tag{74}\] For each fixed \(k\) and \(j\) we can write the top-rank form in 74 as: \[\begin{align} \label{part95395final951} \tilde{Q}^{k,j}_2 \cdot \alpha_0 \wedge (\Omega_0)^{n-1} = \binom{n-1}{k} \binom{n-1-k}{j} \frac{-(n-1)}{j+1} d(\alpha_0 \wedge \eta \wedge (\Omega_0)^{n-1-k-j} \wedge (d\alpha_0)^j \wedge (d\eta)^{k}) \end{align}\tag{75}\] where \(\tilde{Q}^{k,j}_2: \Sigma \rightarrow \mathbb{R}\) are smooth functions for each \(k\) and \(j\).
Step 2:
Applying the operator \([\alpha_0 \wedge \iota_{R_0}]\) to the first summand \(\eta \wedge (\Omega_0 +Sd\alpha_0 + d\eta)^{n-1}\) in 72 we get: \[\begin{align} \sum_{k=1}^{n-1} \sum_{j=0}^{n-1-k} \binom{n-1}{k} \binom{n-1-k}{j} k S^j \mathcal{F}[dS] \wedge \alpha_0 \wedge \eta \wedge (\Omega_0)^{n-1-k-j} \wedge (d\alpha_0)^{j} \wedge d\eta^{k-1} \end{align}\] On applying Stokes’ theorem, the integral of the above differential form can be written as the integral of the following differential form: \[\begin{align} \label{part95395step95295eq952} \int_{\Sigma} \sum_{k=1}^{n-1} \sum_{j=0}^{n-1-k} \binom{n-1}{k} \binom{n-1-k}{j} \frac{-k}{j+1} S^{j+1} \cdot d\hat{\mathcal{F}}[\alpha_0 \wedge \eta \wedge (\Omega_0)^{n-1-k-j} \wedge (d\alpha_0)^{j} \wedge d\eta^{k-1}] \end{align}\tag{76}\] For each fixed \(k,j\) we define smooth functions \(\tilde{Q}^{k,j}_1: \Sigma \rightarrow \mathbb{R}\) satisfying: \[\begin{align} \label{part95395final952} \tilde{Q}^{k,j}_1 \cdot \alpha_0 \wedge (\Omega_0)^{n-1} = \binom{n-1}{k} \binom{n-1-k}{j} \frac{-k}{j+1} \cdot d\hat{\mathcal{F}}[\alpha_0 \wedge \eta \wedge (\Omega_0)^{n-1-k-j} \wedge (d\alpha_0)^{j} \wedge d\eta^{k-1}] \\\nonumber \end{align}\tag{77}\] \(\hat{\mathcal{F}}\) is again the dual to \(\mathcal{F}\) defined in 67 .

Plugging 75 and 77 into 72 we see that the integral \(\int_{\Sigma} \eta \wedge (\Omega_0 + d(S\alpha_0+\eta))^{n-1}\) can be written as: \[\begin{align} \label{part95395final95final} \int_{\Sigma}\eta \wedge (\Omega_0 + d(S\alpha_0+\eta))^{n-1} = \int_{\Sigma} Q_1(x,s) \; \alpha_0 \wedge \Omega_0^{n-1} + \int_{\Sigma} Q_2(x,s) \; \alpha_0 \wedge \Omega_0^{n-1} \end{align}\tag{78}\] where \[\begin{align} \label{def95of95Q95195and95Q952} Q_1: &\Sigma \times \mathbb{R} \rightarrow \mathbb{R} \\\nonumber &(x,s) \mapsto \sum_{k=1}^{n-1} \sum_{j=0}^{n-1-k} s^{j+1} \cdot \tilde{Q}^{k,j}_1(x) \\\nonumber Q_2: &\Sigma \times \mathbb{R} \rightarrow \mathbb{R} \\\nonumber &(x,s) \mapsto \sum_{k=0}^{n-1} \sum^{n-1-k}_{j=0} s^{j+1} \cdot \tilde{Q}^{k,j}_2(x) \end{align}\tag{79}\] Part 4: Putting the previous steps together

By 57 \(Vol_{\Omega_0}(\Omega)\) can be written as: \[\begin{align} Vol_{\Omega_0}(\Omega) = \int^1_0 \frac{1}{r} \int_{\Sigma} (S\alpha_0 + \eta) \wedge (\Omega_0 + r d(S\alpha_0 + \eta ))^{n-1} dr \end{align}\] By 71 , 79 we see that: \[\begin{align} Vol_{\Omega_0}(\Omega) = \int^1_0 \frac{1}{r}\int_{\Sigma} S\alpha_0 &\wedge (\Omega_0 +Sd\alpha_0)^{n-1} dr + \int^1_0 \frac{1}{r} \int_{\Sigma} D(x,S) \; \alpha_0 \wedge \Omega^{n-1}_0 dr \end{align}\] where: \[\begin{align} \label{def95of95D} D:= \tilde{P}+Q_1+Q_2 \end{align}\tag{80}\] We now note the following:

1. For each \(k\) and \(j\) the following is true just from the definition of the functions \(\tilde{Q}^{k,j}_1\) (c.f 79 ) : \[\begin{align} &Q_1 = \sum_{k=1}^{n-1} \sum_{j=0}^{n-1-k} S^{j+1} \cdot \tilde{Q}^{k,j}_1 \cdot \alpha_0 \wedge (\Omega_0)^{n-1} \\\nonumber = \sum_{k=0}^{n-1} \sum^{n-1-k}_{j=0} \binom{n-1}{k} &\binom{n-1-k}{j} (j+1) S^{j+1} d\hat{\mathcal{F}}[\alpha_0 \wedge \eta \wedge (\Omega_0)^{n-1-k-j} \wedge (d\alpha_0)^{j} \wedge d\eta^{k-1}] \end{align}\] It follows that the functions \(\tilde{Q}_1\) integrates to zero over \(\Sigma\) when \(S\) is constant.
   

2. Similarly for the functions \(\tilde{Q}^{k,j}_2\) we see that for each \(k\) and \(j\) the following equality follows from their definition (c.f 79 ) : \[\begin{align} &Q_2 = \sum_{k=0}^{n-1} \sum^{n-1-k}_{j=0} S^{j+1} \cdot \tilde{Q}^{k,j}_2 \cdot \alpha_0 \wedge (\Omega_0)^{n-1} \\\nonumber = \sum_{k=0}^{n-1} \sum^{n-1-k}_{j=0} \binom{n-1}{k} &\binom{n-1-k}{j} S^{j+1} \cdot d( \alpha_0 \wedge \eta \wedge (\Omega_0)^{n-1-k-j} \wedge (d\alpha_0)^j \wedge (d\eta)^{k}) \end{align}\] The function \(\tilde{Q}_2\) integrates to \(0\) over \(\Sigma\) when \(S\) is constant because the left-hand side of the above equation is exact.

3. for each \(k,j\) the functions \(\tilde{P}_{k,j}(x,S)\) integrates to zero over \(\Sigma\) when \(S\) is constant because they are defined by: \[\begin{align} \tilde{P}_{k,j}(x,S):= \sum^{n-1}_{k=1} \sum^{n-1-k}_{j=0} S^{j+2} \cdot p^k_j + \sum^{n-1}_{k=1} \sum^{n-1-k}_{j=0} S^{j+1} \cdot \tilde{p}^k_j \end{align}\] Where for each fixed \(k\) and \(j\) the functions \(p^k_j(x)\) satisfy: \[\begin{align} p^k_j \cdot \alpha_0 \wedge (\Omega_0)^{n-1} = d\hat{\mathcal{F}}(\alpha_0 \wedge d\alpha^{j+1}_0 \wedge \Omega^{n-1-k-j}_0 \wedge \eta \wedge d\eta^{k-2}) \end{align}\] And the functions \(\tilde{p}^k_j\) satisfy: \[\begin{align} \tilde{p}^k_j \cdot \alpha_0 \wedge (\Omega_0)^{n-1} = d\big(\alpha_0 \wedge d\alpha^j_0 \wedge (\Omega_0)^{n-1-k}\wedge \eta \wedge d\eta^{k-1}\big) \end{align}\]

4. It also follows from the definition of \(\tilde{P}\), \(Q_1\) and \(Q_2\) that \(D(x,0) = 0\) \(\forall\) \(x \in \Sigma\).

To finish the proof of Theorem 3 we need only to establish 10 . This is the context of the following result.

7 Hamiltonian dynamics on hypersurfaces↩︎

7.1 Proof of Theorem 4↩︎

Given a symplectic manifold \((W,\omega)\), the \(2n\)-form \(\omega^{2n}\) defines an orientation on \(W\), which in turn induces a natural orientation on an embedded hypersurface \(\Sigma \hookrightarrow W\). The characteristic foliation of the odd-symplectic form \(\omega\vert_{\Sigma}\) will be called the characteristic foliation of \(\Sigma\). In this setting, we suppose that the symplectic manifold \((W,\omega)\) admits a Zoll hypersurface \(\Sigma_0 \hookrightarrow W\), to this Zoll hypersurface, one can associate a smooth function \(H \in C^{\infty}(A)\) where \(A \subset W\) is a neighborhood of \(\Sigma_0\) such that the following holds: \[\begin{align} H^{-1}(0)= \Sigma_0 \end{align}\] Given another embedded hypersurface \(\tilde{\Sigma} \hookrightarrow W\), there similarly exists a smooth function \(\tilde{H} \in C^{\infty}(\tilde{A})\) where \(\tilde{A} \subset W\) is a neighborhood of \(\tilde{\Sigma}\). Recall that in the introduction, we defined \(\tilde{\Sigma}\) as being \(C^2\)-close to \(\Sigma\) if the two functions \(H\) and \(\tilde{H}\) are \(C^2\)-close; this definition is motivated by the observation 96 below.

We now fix a connected symplectic manifold \((W,\omega)\) of dimension \(2n\) for \(n\geq 2\) admitting a Zoll hypersurface \(\Sigma_0 \hookrightarrow W\) and pick another closed, connected and embedded hypersurface \(\Sigma \hookrightarrow W\) that is \(C^2\)-close to \(\Sigma_0\). In this setting, there exists a smooth function \(H_0 \in C^{\infty}(A)\) defined in a neighborhood \(A \subset W\) of the union \(\Sigma \cup \Sigma_0\) such that for some \(a \in \mathbb{R}\) the following hold: \[\begin{align} H^{-1}_0(0) = \Sigma_0 \qquad H^{-1}(a) = \Sigma \end{align}\] Since \(\Sigma\) is assumed to be \(C^2\)-close to \(\Sigma_0\), we can assume without any loss of generality that \(H_0\) was chosen so that \(a \in (-\epsilon_0,\epsilon_0)\) where \(\epsilon_0>0\) was chosen so that the interval \((-\epsilon_0,\epsilon_0)\) contains no critical values of \(H_0\). In this setting, we make the following definition: \[\begin{align} \label{D40Sigma4195region95def} \tilde{D}(\Sigma):= \{x \in W\;\big|\; H(x)\in[0,a] \} \end{align}\tag{86}\] For \(t \in (-\epsilon_0,\epsilon_0)\) we will denote by \(\varphi^{t}\) the flow of the following vector field \[\begin{align} X:= \frac{\nabla H}{\vert\vert \nabla H\vert\vert^2} \end{align}\] In particular, \(H(\varphi^t)=t\) for \(t \in (-\epsilon_0,\epsilon_0)\). This allows us to define the following diffeomorphism: \[\begin{align} \label{hyp95def95of95Phi} \Phi: \Sigma_0 \times [0,a] \longrightarrow \tilde{D}(\Sigma)\\\nonumber (x,t)\longmapsto \varphi^t(x) \end{align}\tag{87}\] Upon denoting by \(\phi^t\) the pullback \(\Phi^*(\varphi^t)\), we see that \(\phi^t\) satisfies the following properties: \[\begin{align} \phi^0\equiv \text{Id} \qquad \phi^t: \Sigma_0 \times\{0\} \rightarrow \Sigma_0\times \{t\} \end{align}\] It then follows that the following pullback is cohomologous with \(\Omega_0:= \omega\big\vert_{\Sigma_0}\) \[\begin{align} \Omega_t:= \big(\phi^t\big)^*(\Phi^*\omega) \qquad \text{for each t \in [0,a]} \end{align}\] where by a slight abuse of notation, we denote by \(\omega\) the restriction \(\tilde{D}(\Sigma)\big\vert_{\omega}\). Indeed, a simple computation using Cartan’s magic formula and [7] shows that the following holds for each \(t \in [0,a]\): \[\begin{align} \label{eta95t95hyp95def95eqn} \Omega_t -\Omega_0 = \Phi^*\bigg( d\Big(\int^t_0 (\varphi^s)^* (\iota_{X} \omega) \; ds\Big)\bigg) = d\Big(\int^t_0 \eta_s \;ds\Big) \end{align}\tag{88}\] The above equation motivates us to define the following family of \(1\)-forms on \(\Sigma_0\): \[\begin{align} \alpha_t:= \int^t_0 \eta \qquad \text{for t \in [0,a]} \end{align}\] So that 88 can be expressed in the following succinct way: \[\begin{align} \Omega_t -\Omega_0 = d\alpha_t \end{align}\] The following proposition shows that the Volume of the region \(\tilde{D}(\Sigma)\) has a convenient expression in terms of the Volume functional defined in 12 :

We now define the action and systole of an arbitrary regular level set of \(H\).

The Hamiltonian vector field of \(H_0\) is the unique vector field satisfying the following differential equation: \[\begin{align} \iota_{X_{H_0}}\omega = dH_0 \end{align}\] We denote by \(\tilde{X}_H\) the pullback: \(\Phi^*(X_H)\). When restricted to \(\Sigma_0 \times \{t\}\) for some \(t \in [0,1]\), the flow of \(\tilde{X}_H\) generates the characteristic distribution of \(\tilde{\Omega}_t:= \Phi^*\omega\big\vert_{\Sigma_0 \times \{t\}}\).

We now pick a closed leaf \(\gamma\) of the characteristic foliation of \(\tilde{\Omega}_a\) and since \(\Sigma\) is assumed to be \(C^2\)-close to \(\Sigma_0\), we can find a leaf \(\gamma_0\) of the characteristic foliation of \(\Omega_0= \Phi^*\omega\big\vert_{\Sigma_0\times \{0\}}\) that is \(C^1\)-close to \(\gamma\). In such a setting, we will denote by \(\Gamma_{\gamma}\) a cylinder parameterizing a homotopy between \(\gamma\) and \(\gamma_0\) and we define the action of \(\gamma\) to be the following: \[\begin{align} \mathcal{A}_{\Sigma}(\gamma):= \int_{\Gamma_{\gamma}}\Phi^*\omega \end{align}\] Since \(\omega\) is closed, the above integral is invariant under homotopies with a fixed boundary. We therefore choose a short cylinder in the following way: let \(\tilde{\Gamma}_{\gamma}\) be a projection of \(\Gamma_{\gamma}\) to \(\Sigma_0\) and consider the cylinder defined to be the concatenation \(\tilde{\Gamma}_{\gamma} \# (\phi^t)^*\gamma\). Using 89 , we observe that the above integral has the following expression: \[\begin{align} \label{hyp95action95decomp951} \int_{\Gamma_{\gamma}}\Phi^*\omega = \int_{\Gamma_{\gamma}} dt \wedge \eta_t + \int_{\Gamma_{\gamma}} \tilde{\Omega}_t =\int^a_0\int_{(\phi^t)^*\gamma} \eta_t \; dt + \int_{\tilde{\Gamma}_{\gamma} \# (\phi^t)^*\gamma} \Omega_t \end{align}\tag{93}\] By definition of the \(1\)-form \(\alpha_t\), the first factor in 93 can be expressed in the following form: \[\begin{align} \label{hyp95action95decomp952} \int^a_0\int_{(\phi^t)^*\gamma} \eta_t \; dt = \int_{(\phi^a)^*\gamma} \alpha_a \end{align}\tag{94}\] Putting 93 and 94 together, we obtain the following expression for \(\mathcal{A}_{\Sigma}(\gamma)\): \[\begin{align} \mathcal{A}_{\Sigma}(\gamma) = \int_{\Gamma_{\gamma}}\Phi^*\omega = \int_{(\phi^a)^*\gamma} \alpha_a + \int_{\tilde{\Gamma}_{\gamma} \# (\phi^t)^*\gamma} \Omega_0 = \mathcal{A}_{\Omega_a}((\phi^a)^*\gamma) \end{align}\] where \(\mathcal{A}_{\Omega_a}\) is the action defined in Section 1.4.1. Denoting by \(\mathfrak{h} \in \pi_1(\Sigma_0)\) the homotopy class represented by the leaves of the characteristic foliation of \(\Sigma_0\), we recall that the set of short loops in \(\Sigma_0\) were denoted by \(\Lambda_{\mathfrak{h}}(\Sigma_0)\) (c.f Section 1.4.1). We can now conveniently express the systole of \(\Sigma\) in the following way: \[\begin{align} \label{sys95in95odd-symp} \text{sys}(\Sigma)= \inf_{\bar \gamma \in \Lambda_{\mathfrak{h}}(\Sigma_0)} \mathcal{A}_{\Omega_a}(\bar \gamma) \end{align}\tag{95}\] Since the pair \((\Sigma_0,\Omega_0)\) is a Zoll odd-symplectic manifold, there exists an oriented \(S^1\)-bundle \(p: \Sigma_0 \rightarrow M\) whose leaves are the leaves of characteristic foliation of \(\Omega_0\). Given a connection \(1\)-form \(\alpha_0 \in \Omega^1(\Sigma_0)\) for such an \(S^1\)-bundle, it is easy see that the following realtions are automatically satisfied \[\begin{align} \alpha_0 \in \mathcal{C}(\Sigma_0,\Omega_0) \;\;\; \text{and} \;\;\; \ker \Omega_0 \subset \ker d\alpha_0 \end{align}\] Therefore, the odd-symplectic form \(\Omega_0\) and \(1\)-form \(\alpha_0\) determine a stable Hamiltonian structure on \(\Sigma_0\) in the sense of [23]. As a consequence, there exists an \(\tilde{\epsilon}_0 >0\) such that the neighborhood \(\Sigma_0 \times (-\tilde{\epsilon}_0,\tilde{\epsilon}_0)\) is foliated by Zoll hypersurfaces. By re-normalizing if necessary we can parametrize the leaves of this foliation in the following way: we denote the leaves by \(\hat{\Sigma}_t\) for \(t \in (-\tilde{\epsilon}_0,\tilde{\epsilon}_0)\) such that the following holds for every \(t \in (-\tilde{\epsilon}_0,\tilde{\epsilon}_0)\): \[\begin{align} \text{sys}(\hat{\Sigma}_t) = t = \mathcal{A}_{\hat{\Sigma}_t}(\tilde{\gamma}_t) \;\;\; \text{\small for any leaf \tilde{\gamma}_t of the characteristic foliation of \hat{\Sigma}_t} \end{align}\] We can also assume with any loss of generality that \(\tilde{\epsilon}_0\) was chosen so that \(\text{sys}(\Sigma) \in [0,\tilde{\epsilon}_0)\). In such a situation, since the hypersurface \(\hat{\Sigma}_{\text{sys}(\Sigma)}\) is Zoll, by Proposition 9 the following holds: \[\begin{align} \text{Vol}\big(\tilde{D}(\hat{\Sigma}_{\text{sys}(\Sigma)})\big) = P\big(\text{sys}(\Sigma)\big) \end{align}\] where the volume of the manifold \(\tilde{D}(\hat{\Sigma}_{\text{sys}(\Sigma)})\) is computed with respect to \(\omega\). The \(1\)-form \(\eta_a\) defined in 88 has the following Taylor expansion: \[\begin{align} \label{eta95taylor} \eta_a = \eta_0 + a\dot{\eta}_0 + \frac{a^2}{2}\ddot \eta_0 + \frac{a^3}{6} \partial_a\ddot \eta_0 + \mathcal{O}(a^4) \end{align}\tag{96}\] Therefore, by 88 , the odd-symplectic form \(\Omega_a\) will be \(C^2\)-close to \(\Omega_0\) if \(a>0\) chosen so that \(a^3\) is close to zero. So, we can now use Theorem 1 to obtain the following inequality: \[\begin{align} Vol\big(\tilde{D}(\hat{\Sigma}_{\text{sys}(\Sigma)})\big) = P\big(\text{sys}(\Sigma)\big) \leq Vol_{\Omega_0}\big(\Omega_a\big) = Vol\big(\tilde{D}(\Sigma)\big) \end{align}\] with equality holding if and only if the action of every leaf of the characteristic foliation of \(\Omega_a\) equals \(\text{sys}(\Sigma)\), which happens if and only if \(\Sigma\) is a Zoll hypersurface as required.

8 Magnetic dynamics on Riemannian manifolds↩︎

We begin this section with a justification of the fact that odd-symplectic forms model the dynamics of a magnetic system on a regular energy level.

Observe that given a magnetic pair \((g,\sigma)\) on a Riemannian manifold \((W,g)\) of dimension \(n\geq 3\), the trajectories of the associated magnetic geodesic flow \(\Phi^t_{g,\sigma}\) on a regular level set \(H^{-1}_{g}(k)\) can be viewed as the leaves of the characteristic foliation of the following odd-symplectic form defined on the manifold \(S_g^*W \cong H^{-1}_{g}(\frac{1}{2})\): \[\begin{align} \label{omega95s95def} \Omega^s_{g,\sigma}:= d\lambda-\pi^*\sigma_s \;\;\; \text{where \sigma_s:= s\sigma and s = \frac{1}{\sqrt{2k}}} \end{align}\tag{97}\] The above odd-symplectic form is called the odd-symplectic form determined by the magnetic pair \((g,\sigma)\). It is also easy to see from 97 that the odd-symplectic form defined above will be Zoll if and only if the magnetic system determined by the pair \((g,\sigma)\) is Zoll at strength \(s\). In the sequel, if the level set of the metric Hamiltonian is obvious, in the interest of a more concise presentation of the formulas, we will drop the constant \(s\) from the notation.

In this section, we intend to compare the dynamics of the two different magnetic systems on a fixed odd-symplectic manifold. To do so, we begin by picking two magnetic pairs \((g,\sigma) , (\tilde{g},\tilde{\sigma}) \in \mathcal{M}(W) \times \Omega^2(W)\) on the Riemannian manifold \((W,g)\), and define the following function: \[\begin{align} \label{def95of95the95function95f} f_{\tilde{g}}:\;&T^*W \rightarrow \mathbb{R}\\\nonumber &(p,q)\mapsto \frac{\vert\vert p\vert\vert_{\tilde{g}}}{\vert\vert p\vert\vert_{g}} \end{align}\tag{98}\] On the unit sphere sub-bundle \(S_g^*W\) of the Riemannian manifold \((W,g)\), the trajectories of the magentic geodesic flow \(\Phi_{\tilde{g},\tilde{\sigma}}\) are the leaves of the characteristic foliation of the following odd-symplectic form: \[\begin{align} \label{odd-symp95determiend95by95triple} \tilde{\Omega}^{s}_{\tilde{g},\tilde{\sigma}}:= d(f_{\tilde{g}}\lambda)-s\pi^*\tilde{\sigma} \Big|_{S_g^*W} \end{align}\tag{99}\] On the manifold \(S_g^*W\), the odd-symplectic form defined by the magnetic pair \((g,\sigma)\) has the form 97 and that defined by the magnetic pair \((\tilde{g},\tilde{\sigma})\) has the form 99 . Since the first summand in 99 is exact, it is easy to see that for each fixed \(s\in \mathbb{R}\), the two odd-symplectic forms determiend on the manifold \(S_g^*W\) by the two magnetic pairs \((g,\sigma)\) and \((\tilde{g},\tilde{\sigma})\) are cohomologous if and only if \(\sigma\) and \(\tilde{\sigma}\) are.

We now fix a closed and connected Riemannian manifold \((W,g)\) of dimension \(n\geq 3\), a Zoll magnetic pair \((g,\sigma)\) and strength \(s \in \mathbb{R}\) at which the pair is Zoll. We will also assume that the class \(C_0 \in H^2(W,\mathbb{R})\) represented by \(\sigma\) satisfies the following condition: \[\begin{align} \label{cohomology95assumption95mag95systems} 0=\pi^*\big[C^{n-1}_0\big] \in H^{2n-2}(S^*_gW,\mathbb{R}) \end{align}\tag{100}\] We also fix an \(S^1\)-bundle associated with the Zoll odd-symplectic form determined by the fixed Zoll magnetic pair \((g,\sigma)\) of strength \(s\) and denote by \(\mathfrak{h} \in \pi_1(S_g^*W)\) the homotopy class represented by the fibers of this \(S^1\)-bundle.

Moreover, observe that since \(W\) has dimension \(n\geq 3\), the fibers of the following bundle are contractible: \[\begin{align} \label{S94m95fibration} S^{n-1}\hookrightarrow S_g^*W \xrightarrow{\pi} W \end{align}\tag{101}\] where by a slight abuse of notation, we denote by \(\pi\) the restriction of foot-point projection to \(S_g^*W\). As a consequence, using the long exact sequence for homotopy groups of the fibration 101 , the following holds: \[\begin{align} \pi_1(S_g^*W) \cong \pi_1(W) \end{align}\] Motivated by the above observation, we will hence forth identify \(\pi_1(S_g^*W)\) with \(\pi_1(W)\).

In this setting, using the odd-symplectic action defined in Section 1.4.1, the magnetic action of an arbitrary magnetic pair \((\tilde{g},\tilde{\sigma}) \in \mathcal{M}(W) \times \Xi^2_{C_0}(W)\) is defined to be the following: \[\begin{align} \label{mag95action95def95eqn} \mathcal{A}^{\text{mag}}_{\tilde{g},\tilde{\sigma}}:\; \Lambda_{\mathfrak{h}}(S^*&W) \rightarrow \mathbb{R}\\\nonumber &\gamma\mapsto \mathcal{A}_{\tilde{\Omega}_{\tilde{g},\tilde{\sigma}}}(\gamma) \end{align}\tag{102}\] where \(\tilde{\Omega}_{\tilde{g},\tilde{\sigma}}:= d(f_{\tilde{g}}\lambda)-\pi^*\tilde{\sigma}\big|_{S_g^*W}\). We will also denote by \(\tilde{\chi}^{s}(\tilde{g}, \tilde{\sigma})\) the set of closed leaves of the characteristic foliation of \(\tilde{\Omega}^{s}_{\tilde{g},\tilde{\sigma}}\) that can be joined to the leaves of the characteristic foliation of the Zoll odd-symplectic form \(\big(d\lambda-s\pi^*\sigma\big)\big\vert_{S_g^*W}\) by a short homotopy (c.f Section 1.4.1). Recall that in Section 1.5.2, we introduced the notation \(\chi^{s}(\tilde{g},\tilde{\sigma})\) for the set of closed magnetic geodesics of the pair \((\tilde{g},\tilde{\sigma})\) of strength \(s\) that are \(C^2\)-close to those of \((g,\sigma)\). Since these geodesics are parametrized by arc length, there is a canonical lift map \[\begin{align} L_{\tilde{g},g}:\chi^{s}(\tilde{g},\tilde{\sigma})\longrightarrow \tilde{\chi}^{s}(\tilde{g},\tilde{\sigma}) \end{align}\] obtained by first lifting \[\begin{align} \gamma \mapsto \big(t\mapsto \tilde{g}_{\gamma(t)}(\dot{\gamma}(t),\cdot)\big) \end{align}\] to \(S^*_{\tilde{g}}W\) and then composing with the radial identification \(S^*_{\tilde{g}}W \rightarrow S^*_gW\) obtained via the function \(f_{\tilde{g}}\). With this convention, the following holds: \[\begin{align} \tilde{\chi}^{s}(\tilde{g},\tilde{\sigma}) = L_{\tilde{g},g}\big(\chi^{s}(\tilde{g},\tilde{\sigma})\big) \end{align}\] Thus, for \(\gamma\in \chi^{s}(\tilde{g},\tilde{\sigma})\), we denote its canonical lift by \(\tilde{\gamma}:= L_{\tilde{g},g}(\gamma) \in \tilde{\chi}^{s}(\tilde{g},\tilde{\sigma})\). The justification of the fact that the set \(\tilde{\chi}^{s}(\tilde{g},\tilde{\sigma})\) is non-empty when the magnetic pair \((\tilde{g},\tilde{\sigma})\) is sufficiently \(C^3\)-close to the pair fixed Zoll pair \((g,\sigma)\) is provided in Section 8.1. Since the strength \(s\) is fixed throughout and no ambiguity can arise, we will suppress it from the notation in the sequel. For the benefit of the reader, we recall from the introduction the following convention that was introduced: when picking a Zoll magnetic pair, we will always implicitly assume that its dynamics is being considered at a fixed strength where it is Zoll. In addition, in the sequel, we would like to compare the dynamics of such a pair with another appropriately chosen magnetic pair. In such a situation, we will always assume that both pairs are being considered at a fixed strength, where the Zoll magnetic pair is Zoll.

In this context, the following proposition furnishes a convenient expression for the magnetic action.

Before proceeding further, we recall some facts about the geometry of cotangent bundles of Riemannian manifolds. Given a Riemannian manifold \((W,g)\), a point \(q \in W\) and some \(p \in T_q^*W\), the tangent space \(T_{p}TW\) can we written as the direct sum of two vector spaces \(\mathcal{H}(p)\) and \(\mathcal{V}(p)\) called the horizontal and vertical components respectively. This isomorphism is constructed by combining the following two endomorphisms: \[\begin{align} pr_{\mathcal{H}}: &T_{p}T^*W \rightarrow T_q W \cong \mathcal{\mathcal{H}}(p) \\\nonumber &\xi \mapsto \xi_{\mathcal{H}} := d_{p}\pi(\xi) \end{align}\] \[\begin{align} pr_{\mathcal{V}}: &T_{p}T^*W \rightarrow T^*_q W \cong \mathcal{V}(p) \\\nonumber &\xi \mapsto \xi_\mathcal{V} := \nabla_{t}(Z(0)) \end{align}\] where \(Z:(-\epsilon, \epsilon) \rightarrow T^*W\) is a curve such that \(Z(0) = p\) and \(\Dot{Z}(0) = \xi\) and \(\nabla_t\) represents the covariant derivative along the curve \((\pi \circ Z)\). We now define \(\mathcal{H}(p)\) to be the set of all \(\xi \in T_pT^*W\) such that \(\xi_{\mathcal{V}} =0\) and \(\mathcal{V}(p)\) to be the set of all \(\xi \in T_{p}T^*W\) such that \(\xi_{\mathcal{H}} =0\). We also have the following identifications given by the maps \(pr_{\mathcal{H}}\) and \(pr_{\mathcal{V}}\) respectively: \(\mathcal{H}(p) \cong T_qW\) and \(\mathcal{V}(p) \cong T^*_qW\). The Sasaki metric on \(T^*W\) is defined with respect to this direct sum decomposition in the following way: \[\begin{align} \Hat{g} : = g \oplus g \end{align}\]

8.1 Proof of Theorem 5↩︎

We begin by recalling the geometric setup of Theorem 5: \((W,g)\) is a closed and connected Riemannian manifold of dimension \(n\geq 3\) and \(C_0 \in H^2(W,\mathbb{R})\) is a cohomology class satisfying condition 100 and admitting a representative \(\sigma \in \Xi^2_{C_0}(W)\) with the property that the magnetic system determined by the pair \((g,\sigma)\) is Zoll; we now consider the following Zoll odd-symplectic form on the manifold \(S_g^*W\), defined by the pair \((g,\sigma)\): \[\begin{align} \Omega_{g,\sigma} := d\lambda - \pi^*\sigma\big\vert_{S^*_gW} \end{align}\] The odd-symplectic form defined on the manifold \(S^*_gW\) by another magnetic pair \((\tilde{g},\tilde{\sigma}) \in \mathcal{M}(W) \times \Xi^2_{C_0}(W)\) has the following expression: \[\begin{align} \tilde{\Omega}_{\tilde{g},\tilde{\sigma}}= d(f_{\tilde{g}}\lambda)-\pi^*\tilde{\sigma} \big\vert_{S_g^*W} \end{align}\] where \(f_{\tilde{g}}\) is the function defined in 98 .

We now set \(\delta>0\) to be the following: \[\begin{align} \label{def95delta95mag95ineq95gen} \delta:= \max\{\vert \vert g - \tilde{g}\vert\vert_{C^3}, \vert\vert \sigma -\tilde{\sigma}\vert\vert_{C^3}\} \end{align}\tag{112}\] And claim that the following holds, for some real constant \(C>0\): \[\begin{align} \label{mag95ineq95gen951} \big\vert\big\vert \Omega^{s}_{g,\sigma} - \tilde{\Omega}^{s}_{\tilde{g},\tilde{\sigma}}\big\vert\big\vert_{C^2} < C\delta \end{align}\tag{113}\] To prove the above claim, we begin by considering the following difference: \[\begin{align} \Omega_{g,\sigma} - \tilde{\Omega}_{\tilde{g},\tilde{\sigma}} = -d\big( (f_{\tilde{g}} -1) \lambda\big) - \pi^*\big(\sigma -\tilde{\sigma}\big) \end{align}\] Taking the \(C^2\)-norms, we obtain the following: \[\begin{align} \big\vert\big\vert \Omega_{g,\sigma} - \tilde{\Omega}_{\tilde{g},\tilde{\sigma}}\big\vert\big\vert_{C^2} \leq C_1 \big\vert\big\vert f_{\tilde{g}}-1\big\vert\big\vert_{C^3} + C_2 \big\vert\big\vert \sigma -\tilde{\sigma}\big\vert\big\vert_{C^2} \end{align}\] The loss of one derivative above comes from applying the exterior derivative \(d\) to the the \(1\)-form \((f_{\tilde{g}} -1) \lambda\). It remains to estimate the difference \(f_{\tilde{g}} -1\) in terms of \(\delta\). Towards this end, we observe that on \(S_g^*W\), \(\vert\vert p\vert\vert^2_g =1\), therefore, the following holds locally: \[\begin{align} f_{\tilde{g}}(q,p)= \sqrt{\tilde{g}^{ij}(q)p_ip_j } \end{align}\] And the above map is smooth on compact subsets of \(W\). Hence, by the mean value theorem in the \(C^3\)-norm, the following holds for some real constant \(C_3>0\): \[\begin{align} \big\vert\big\vert f_{\tilde{g}} -1\big\vert\big\vert_{C^3} \leq C_3\; \big\vert\big\vert g -\tilde{g}\big\vert\big\vert_{C^3} \end{align}\] Therefore, we can conclude that there exists a positive real constant \(C'>0\) such that the following holds: \[\begin{align} \big\vert\big\vert \Omega_{g,\sigma} - \tilde{\Omega}_{\tilde{g},\tilde{\sigma}}\big\vert\big\vert_{C^2} \leq C'\big( \big\vert\big\vert \tilde{g}-g \big\vert\big\vert_{C^3} + \big\vert\big\vert \sigma -\tilde{\sigma}\big\vert\big\vert_{C^3}\big) \end{align}\] By the definition of \(\delta\) in 112 , we now obtain the following: \[\begin{align} \big\vert\big\vert \Omega_{g,\sigma} - \tilde{\Omega}_{\tilde{g},\tilde{\sigma}}\big\vert\big\vert_{C^2} \leq 2C'\delta \end{align}\] setting \(C:= 2C'\), we obtain bound claimed in 113 .

Therefore, we can conclude that there exists a \(C^3\)-neighborhood \(\tilde{\mathcal{U}} \subset \mathcal{M}(W) \times \Xi^2_{C_0}(W)\) of the pair \((g,\sigma)\) with the property that the odd-symplectic form defined by any magnetic pair \((\tilde{g},\tilde{\sigma}) \in \tilde{\mathcal{U}}\) is contained in the neighborhood \(\mathcal{U} \subset \Xi^2_{\tilde{C}_0}(S_g^*W)\) of \(\Omega_{g,\sigma}\) satisfying the conditions of Theorem 1 where \(\tilde{C}_0 \in H^2(S_g^*W,\mathbb{R})\) is the cohomology class represented by the odd-symplectic form \(\Omega_{g,\sigma} \in \Omega^2(S_g^*W)\) (c.f the proof of Proposition 10).

We now pick a pair \((\tilde{g},\tilde{\sigma}) \in \tilde{\mathcal{U}}\) such that the following holds: \[\begin{align} \label{mag95sys95gen95vol95proof951} \text{Vol}_g(W) = \text{Vol}_{\tilde{g}}(W) \end{align}\tag{114}\] And denote by \(\tilde{\Omega}_{\tilde{g},\tilde{\sigma}}\) the odd-symplectic form defined on \(S^*_gW\) by the magnetic pair \((\tilde{g},\tilde{\sigma})\).

Using the assumption 114 , in Proposition 11, we see that the following holds: \[\begin{align} \label{mag95sys95gen95vol95proof952} \text{Vol}_{\Omega_{g,\sigma}}(\tilde{\Omega}_{\tilde{g},\tilde{\sigma}})=0 \end{align}\tag{115}\] And since \((\tilde{g},\tilde{\sigma}) \in \tilde{\mathcal{U}}\), by the discussion above, \(\tilde{\Omega}_{\tilde{g},\tilde{\sigma}} \in \mathcal{U}\). Using 115 , we can conclude from Theorem 1 that the following holds: \[\begin{align} \inf\mathcal{A}_{\tilde{\Omega}_{\tilde{g},\tilde{\sigma}}} \leq 0 \end{align}\] By Proposition 10, the action of a loop \(\gamma \in \chi(\tilde{g},\tilde{\sigma})\) has the following expression: \[\begin{align} \text{length}_{\tilde{g}}(\gamma) - \int_{\pi_*\Gamma_{ \gamma}} \tilde{\sigma} - \text{length}_{g}(\gamma) + \int_{\gamma_0}\eta \end{align}\] where \(\eta \in \Omega^1(W)\) is any \(1\)-form such that: \(\sigma= \tilde{\sigma} + d\eta\).

Taking the infumum over all loops in \(\chi(\tilde{g},\tilde{\sigma})\), we obtain the following, as required: \[\begin{align} \label{mag95sys95general95proof951} \inf_{\gamma \in \chi(\tilde{g},\tilde{\sigma})}\Big(\text{length}_{\tilde{g}}(\gamma)-\int_{\pi_*\Gamma_{\gamma}} \tilde{\sigma}\Big) \leq \text{length}_{g}(\gamma_0) - \int_{\gamma_0}\eta \end{align}\tag{116}\] with equality holding if and only if the pair \((\tilde{g},\tilde{\sigma})\) is Zoll.

9 Magnetic dynamics on Kähler manifolds↩︎

To make this work self-contained, we begin with some relevant preliminaries about Kähler manifolds. A Kähler manifold is defined to be a triple \((M, \sigma_0, J)\) where \(M\) is a smooth manifold, \(\sigma_0 \in \Omega^2(M)\) a symplectic form and \(J\) an integrable almost complex structure that is compatible with \(\sigma_0\). Recall that an almost complex structure \(J\) is said to be integrable if it induces complex charts on \(M\) and it is compatible with \(\sigma_0\) if the pair \((\sigma_0, J)\) determines a Riemannian metric on \(M\) in the following way: \[\begin{align} \label{kahler95metric} g_0(u,v):= \sigma_0(u,Jv) \;\;\; \forall\; u,v \in TM. \end{align}\tag{117}\] The Riemannian metric determined by the symplectic form and complex structure corresponding to a Kähler manifold will be called the associated Kähler metric. The holomorphic sectional curvature of the Kähler metric \(g_0\) associated with a given Kähler manifold \((M,\sigma_0,J)\) is defined to be: \[\begin{align} \label{def95of95hol95sectional95curvature} &\text{Hol}: TM \backslash \{0\} \rightarrow \mathbb{R}\\\nonumber (q,v) &\longmapsto \frac{g_0(v, R(v,Jv)Jv)}{\vert\vert v\vert\vert_{g_0}^4} \end{align}\tag{118}\] where \(R\) is the Riemannian curvature tensor of \(g_0\). We observe here that if \(u,v \in TM\) span the same complex line, then \(\text{Hol}(u)= \text{Hol}(v)\). Therefore, the holomorphic sectional curvature can be considered a function from the complex projectivization \(\mathbb{P}(TM)\) to the real line. In this section, when working with a Kähler manifold, we will always make use of the convention of denoting its dimension by \(2n\).

A recent result of J. Bimmermann [31] asserts the existence of Zoll magnetic systems on Kähler manifolds of constant holomorphic sectional curvature (c.f Theorem 12). We fix some notational conventions before we state the relevant version of Bimmermann’s result here. Given a Kähler manifold \((M, \sigma_0, J)\) the following vector field generates the fiber-wise rotation on \(T^*M\): \[\begin{align} \label{angular95vf95cpn95def} V_{(q,p)}:= (Jp)^{\mathcal{V}} \end{align}\tag{119}\] where the \(\mathcal{V}\) in the superscript above denotes the lift of the relevant vector to the vertical distribution in \(TT^*M\). We will denote by \(\tau\) the dual to the above vector field with respect to the Sasaki metric. \[\begin{align} \mathbb{D}_rW:=\{(q,p) \in TW\;\big\vert \;\;\vert\vert p\vert\vert^2_{g} = r\} \;\; \text{for r \in \mathbb{R}_{+}} \end{align}\]

In addition to this, to stress the strength of the magnetic system being considered, given some \(s \in \mathbb{R}\), we will denote by \(X^{s}_{g_0,\sigma}\) the vector field generating the magnetic geodesic flow of the magnetic pair \((g_0,\sigma_0)\) on the Riemannain manifold \((M,g_0)\) of strength \(s\). The relevant version of the result of Bimmermann referenced above is the following:

In the theorem above, if the Kähler manifold \(M\) has constant negative holomorphic sectional curvature, then there exists a finite \(s_c \in \mathbb{R}\) such that \(s_c^2= -\kappa\) [31]. It is well known that at this strength, the magnetic geodesic flow can be conjugated to a Horocycle flow, see for instance [31], [23] and [26].

By the discussion in Section 8, the odd-symplectic forms defined on the manifold \(S^*_{g_0}M\) by Bimmerman’s Zoll magnetic pairs \((g_0,\sigma_0)\) in Theorem 12 above are Zoll. Therefore, by Theorem 1, these odd-symplectic forms satisfy a local systolic inequality. The purpose of this section is calculate this systolic inequality. We begin by fixing some notational conventions, for the remainder of this section, we will fix a closed Kähler manifold \((M, \sigma_0,J)\) of dimension \(2n\) for \(n\geq 1\), with constant holomorphic sectional curvature \(\kappa\) and denote the associated Kähler metric by \(g_0\). As observed earlier, the pair \((M,g_0)\) determines a Riemannian manifold; we will use this fact in the sequel without any further qualification. In this setting, when talking about a Zoll magnetic pair \((g_0,\sigma_0)\) we will always assume that the resulting magnetic system is being considered at a fixed strength \(s\) such that \(s^2+\kappa >0\).

By Bimmermann’s theorem, on a Kähler manifold \((M,\sigma_0,J)\) of constant holomorphic sectional curvature \(\kappa\), the magnetic geodesic flow of the magnetic pair \((g_0,\sigma_0)\) can be conjugated to the fiber-wise rotation \(e^{2\pi Jt}: T^*M \rightarrow T^*M\) at appropriate strengths. We now claim that this fiber-wise rotation is an isometry for the Sasaki metric on \(T^*M\). Indeed, observe that upon fixing a \(q_0 \in M\) and \(p_0 \in T_{q_0}^*M\), the horizontal lift of an arbitrary \(u \in T^*M\) is defined by: \[\begin{align} u^{\mathcal{H}}:= \frac{d}{dt}\Bigg|_{t=0} (q(t),v(t)) \end{align}\] where \((q(t),v(t))\) is a path in \(T^*M\) satisfying the following conditions: \[\begin{align} q(0) = q_0& \;\;\;\;\;\; \dot{q}(0)=u \\\nonumber v(0) =p_0& \;\;\;\; \nabla_{t}v(0) =0 \end{align}\] and \(\nabla_{t}\) denotes the covarinat derivative of the Kähler metric \(g_0\) along the curve \(\pi \circ v(t)\). From the definition of the curve \((q(t),v(t))\), it is clear that \(u^{\mathcal{H}} \in \mathcal{H}(p_0)\). Moreover, using the explicit expression for the flow \(\Phi_{g_0,\sigma_0}^s\), we compute: \[\begin{align} d \Phi_{g_0,\sigma_0}^s\big|_{(q(0),v(0)} u^{\mathcal{H}} = \frac{d}{dt}\Bigg|_{(q(0),v(0))} \Phi_{g_0,\sigma_0}^s(q(t),v(t)) = \frac{d}{dt}\Bigg|_{(q(0),v(0))} (q(t), e^{2\pi J s}v(t)) \end{align}\] Since the complex structure \(J\) of a Kähler manifold is parallel with respect to the Kähler metric, that is, \(\nabla J \equiv 0\), the endomorphism \(e^{2\pi J s}\) is also parallel. As a consequence, the following holds: \[\begin{align} \nabla_{t}\Big(e^{2\pi J s}v(t)\Big) = \Big(\nabla_t e^{2\pi J s}\Big)v(t) + e^{2\pi J s}\nabla_t v(t) = 0. \end{align}\] Therefore, the curve \[\begin{align} t \mapsto (q(t), e^{2\pi J s}v(t)) \end{align}\] represents a horizontal lift at the point \(\Phi_{g_0,\sigma_0}^s\vert_{q(0),v(0)}\). Hence, \[\begin{align} d \Phi_{g_0,\sigma_0}^s\big|_{(q(0),v(0))} u^{\mathcal{H}} \in \mathcal{H}\big({\Phi_{g_0,\sigma_0}^s(q(0),v(0))}\big) \end{align}\] In particular, the flow \(\Phi_{g_0,\sigma_0}^s\) preserves the horizontal distribution. Therefore, with respect to the orthogonal splitting \(TT^*M \cong \mathcal{H} \bigoplus \mathcal{V}\) the map \(d\Phi_{g_0,\sigma_0}^s\) can be written in the following way: \[\begin{align} \label{rotation95is95killing} d\Phi_{g_0,\sigma_0}^s = \begin{bmatrix} Id & 0\\ 0 & e^{2\pi J s} \end{bmatrix} \end{align}\tag{122}\] It is worth noting here that in [31] it is observed that upon denoting by \(\eta:= J^*\lambda\), the \(2\)-form \(d\tau\) has the following expression on \(T^*M\): \[\begin{align} \label{dtau95decomop} d\tau = +2\sigma_{0}^{\mathcal{V}} - \frac{\kappa}{2}\Bigg[\lambda \wedge \eta + 2H_{g_0}\pi^*(\sigma_{0})\Bigg] \end{align}\tag{123}\] where \(\sigma_{\text{0}}^{\mathcal{V}}\) is the vertical lift of \(\sigma_{\text{0}}\) and \(H_{g_0}: T^*M \rightarrow \mathbb{R}\) is the metric Hamiltonian. We note here that the formula in 123 differs from the one appearing in [31] by a sign due to a difference in conventions with respect to the definition of curvature used.

Since the Zoll magnetic pair \((g_0,\sigma_0)\) on the Riemannian manifold \((M,g_0)\) defines a Zoll odd-symplectic form on the manifold \(S_{g_0}^*M\), after a picking an \(S^1\)-bundle associated with this Zoll odd-symplectic form, we can define a Zoll polynomial as in Section 1.4.3. We will refer to the resulting Zoll polynomial as the Zoll polynomial of the magnetic pair \((g_0,\sigma_0)\). The following proposition furnishes a connivent way to compute this Zoll polynomial in Bimmermann’s examples:

Recall that the magnetic length of a magnetic geodesic \(\gamma_0\) of the magnetic system determined by the Zoll pair \((g_0,\sigma_0)\) was defined in 17 to be the following: \[\begin{align} \label{mag95length95kahler95def95Zoll} l^{g_0,\sigma_0}_{\text{mag}}(\gamma_{0}) = \text{length}_{g_0}(\gamma_{0}) - \int_{\mathbb{D}_{\gamma_{0}} } \sigma_{0} \end{align}\tag{136}\] We now describe in more detail how the capping surface \(\mathbb{D}_{\gamma_0}\) is obtained. We begin by observing that the magnetic geodesic flow of the Zoll magnetic pair \((g_0,\sigma_0)\) is contractible in the disk bundle \(\mathbb{D}_1M\) by Theorem 12. And since the magnetic geodesics of the Zoll pair \((g_0,\sigma_0)\) are precisely the curves obtained by projecting to \(M\) these contractible trajectories in \(\mathbb{D}_1M\), the magnetic geodesics of the Zoll pair \((g_0,\sigma_0)\) are also null-homotopic in \(M\). We now fix a universal covering \(\tilde{p}: \tilde{M} \rightarrow M\) and pick a magnetic geodesic \(\gamma_0 \in \chi(g_0,\sigma_0)\). Let \(\tilde{x} \in \tilde{M}\) be a point such that \(\tilde{p}(\tilde{x}) =\gamma_0(0)\). The magnetic geodesic \(\gamma_0\) then has a unique lift \(\tilde{\gamma}_0 \subset \tilde{M}\) such that \(\tilde{\gamma}_0(0)=\tilde{x}\) and since the magnetic geodesic \(\gamma_0\) is contractible, the curve \(\tilde{\gamma}_0\) is closed. In addition, the homology class \([\tilde{\gamma}_0] \in H_1(\tilde{M},\mathbb{Z})\) represented by the curve \(\tilde{\gamma}_0\) is the trivial or zero class. This is because the universal cover \(\tilde{M}\) is simply connected and by the Hurewicz theorem, the following holds: \[\begin{align} H_1(\tilde{M},\mathbb{Z}) \cong \pi_1(\tilde{M})_{\text{ab}} \cong \{e\} \end{align}\] Therefore, by [7], the curve \(\tilde{\gamma}_0\) admits a capping disk \(\mathbb{D}_{\tilde{\gamma}_0}\). The capping surface \(\mathbb{D}_{\gamma_0}\) in 136 is then obtained by just projecting the disk \(\mathbb{D}_{\tilde{\gamma}_0}\) to \(M\). If \(\kappa \leq 0\) then, \(\pi_2(\tilde{M}) = \{e\}\) as a result, the following integral is independent of the choice of capping disk \(\mathbb{D}_{\tilde{\gamma}_0}\) \[\begin{align} \int_{\mathbb{D}_{\tilde{\gamma}_0}} \tilde{p}^*\sigma_0 \end{align}\] However if \(\kappa >0\) then \(\pi_2(\tilde{M}) \cong \mathbb{Z}\). So, in this case, we need to specify which disk we choose as a capping disk for the the lifted curve \(\tilde{\gamma}_0\), the disk we wish to use is obtained in the following way: upon lifting a given magnetic geodesic \(\gamma_0\) to the universal cover \(\tilde{M} \cong \mathbb{CP}^n\) of \(M\) in the way specified above, as observed in [32], there exists an immersion \(\mathbb{CP}^1 \hookrightarrow \mathbb{CP}^n\) with the property that this projectivised complex line completely contains the lift of the chosen magnetic geodesic \(\gamma_0\). In addition to this, there is a well-defined normal direction to the lifted magnetic geodesic \(\tilde{\gamma}_0\) determined by the orientation on \(\mathbb{CP}^1\) induced by the lift of \(\sigma_0\). We now define the capping disk \(\mathbb{D}_{\tilde{\gamma}_0}\) of the lifted curve \(\tilde{\gamma}_0\) to be the one for which this normal vector points inward.

The following proposition furnishes a convenient expression for the magnetic length considered in 136 .

We will denote by \(B_0 \in H^2(M,\mathbb{R})\) the cohomology class represented by the Kähler form \(\sigma_0 \in \Omega^2(M)\) and we now pick an arbitrary pair \((g,\sigma) \in \mathcal{M}(M)\times \Xi^2_{B_0}(M)\). Recall that after fixing the Zoll magnetic pair \((g_0,\sigma_0)\) on the Riemannian manifold \((M,g_0)\), the set \(\chi(g,\sigma)\) was introduced earlier to denote the set of closed magnetic geodesics of the magnetic pair \((g,\sigma)\) that are \(C^2\)-close to those of the Zoll pair \((g_0,\sigma_0)\).

As explained in Section 8, the two magnetic pairs \((g_0,\sigma_0)\) and \((g,\sigma)\) define odd-symplectic forms on the manifold \((S^*_{g_0}M)\). These odd-sympelctic forms are the following: \[\begin{align} \Omega_{g_0,\sigma_0}= \big(d\lambda-\pi^*\sigma_0\big)\vert_{S^*_{g_0}M} \end{align}\] And, \[\begin{align} \Omega_{g,\sigma}:= \big(d(f_g\lambda)-\pi^*\sigma\big)\vert_{S^*_{g_0}M} \end{align}\] where \(f_g\) is the function defined in 98 .

Since the first summand in the definition of \(\Omega_{g,\sigma}\) and \(\Omega_{g_0,\sigma_0}\) is an exact \(2\)-form and \(\sigma\) was chosen from the set \(\Xi^2_{B_0}(M)\), we see that the odd-symplectic forms \(\Omega_{g,\sigma}\) and \(\Omega_{g_0,\sigma_0}\) and cohomologous. We can therefore define their magnetic action as in 102 . The following proposition furnishes a connivent expression for the magnetic action \(\mathcal{A}^{\text{mag}}_{{g,\sigma}}\) in terms of the magnetic length functional defined in 17 .

9.1 Proof of Theorem 6↩︎

Let \((M,\sigma_0,J)\) be a closed connected Kähler manifold such that the associated Kähler metric \(g_0\) has constant holomorphic sectional curvature \(\kappa\). By Theorem 12, the following is a Zoll odd-symplectic form when \(s^2+\kappa>0\): \[\begin{align} \label{kahler95sys95ineq95proof95os95195def} \Omega_{g_0,\sigma_0}:= \big(d\lambda -s\pi^*\sigma_0\big)\big\vert_{S^*_{g_0}M} \end{align}\tag{157}\] In the sequel, we will fix one \(s\in\mathbb{R}\) satisfying the condition \(s^2+\kappa>0\) and drop it from the notation.

Let \(\tilde{B}_0 \in H^2(S_{g_0}^*M,\mathbb{R})\) be the cohomology class represented by the odd-symplectic form 157 . Then, by Theorem 1, there exists a \(C^2\)-neighborhood \(\mathcal{U}\subset \Xi^2_{\tilde{B}_0}(S_{g_0}^*M)\) with the property that for any odd-symplectic form \(\Omega \in \mathcal{U}\), a local systolic inequality of the form 1 holds.

We now denote by \(B_0 \in H^2(M,\mathbb{R})\) the cohomology class represented by the Kähler form \(\sigma_0\), and pick a pair \((\tilde{g},\tilde{\sigma}) \in \mathcal{M}(M) \times \Xi^2_{B_0}(M)\). As discussed in Section 8, the pair \((\tilde{g},\tilde{\sigma})\) defines the following odd-symplectic form on the manifold \(S_{g_0}^*M\): \[\begin{align} \tilde{\Omega}_{\tilde{g},\tilde{\sigma}}:= \big(d(f_{\tilde{g}}\lambda) -\pi^*\tilde{\sigma}\big)\big\vert_{S^*_{g_0}M} \end{align}\] In Section 8.1, it is shown that there exists a \(C^3\)-neighborhood \(\tilde{\mathcal{U}} \subset \mathcal{M}(M) \times \Xi^2_{B_0}(M)\) of the pair \((g_0,\sigma_0)\) such that the odd-symplectic form defined by any pair \((\tilde{g},\tilde{\sigma}) \in \tilde{\mathcal{U}}\) is contained in the neighborhood \(\mathcal{U} \subset \Xi^2_{\tilde{B}_0}(S^*_{g_0}M)\). We now fix one such pair \((\tilde{g},\tilde{\sigma}) \in \tilde{\mathcal{U}}\) and denote by \(\tilde{\Omega}_{\tilde{g},\tilde{\sigma}}\) the odd-symplectic form it defines on the manifold \((S_{g_0}^*M)\). By Theorem 1, the following holds: \[\begin{align} \label{mag95final95ineq951} P\big(\inf_{\gamma \in \chi(\tilde{g},\tilde{\sigma})}\mathcal{A}^{\text{mag}}_{\tilde{g},\tilde{\sigma}}(\gamma)\big) \leq \text{Vol}_{\Omega_{g_0,\sigma_0}}(\tilde{\Omega}_{\tilde{g},\tilde{\sigma}}) \end{align}\tag{158}\] with equality holding if and only if the magnetic system determined by the pair \((\tilde{g},\tilde{\sigma})\) is Zoll at strength \(s\); where \(\mathcal{A}^{\text{mag}}_{\tilde{g},\tilde{\sigma}}\) is the magnetic action defined in 102 . By Proposition 15, the following holds: \[\begin{align} \label{kahler95proof95action95final} \inf_{\gamma \in \chi(\tilde{g},\tilde{\sigma})}\mathcal{A}^{\text{mag}}_{\tilde{g},\tilde{\sigma}}(\gamma) = \inf_{\gamma \in \chi(\tilde{g},\tilde{\sigma})} l_{\text{mag}}^{\tilde{g},\tilde{\sigma}}(\gamma) - a^2(1)\pi =l_{\text{min}}^{\tilde{g},\tilde{\sigma}} - a^2(1)\pi \end{align}\tag{159}\] where \(l_{\text{mag}}^{\tilde{g},\tilde{\sigma}}\) is the magnetic length functional defined in 17 .

By Proposition 13, the Zoll polynomial of the magnetic pair \((g_0,\sigma_0)\) has the following expression: \[\begin{align} \label{zoll95poly95final95expression} P(\tilde{A}) = \begin{cases} n!\pi^{n-1}[a(1)]^{2n-2} \;\text{Vol}_{g_0}(M)\; \big(A(\tilde{A})C(\tilde{A})\big)^{n-1}\bigg(C(\tilde{A}) -\frac{B(\tilde{A})}{n} \bigg) \;\;\;&\text{if \kappa \neq 0} \\ \\ (-s)^n A(\tilde{A})^{n-1}\; n!\pi^{n-1}[a(1)]^{2n-2} \;\text{Vol}_{g_0}(M)\;\;\; &\text{if \kappa =0} \end{cases} \end{align}\tag{160}\] where \(s \in \mathbb{R}\) is any constant satisfying the condition of Theorem 12, \(a\) is the function defined in that theorem and : \[\begin{align} \label{def95of95K95tilde95final95proof} A(\tilde{A})=1+\frac{\tilde{A}}{\pi a^2(1)},\qquad B(\tilde{A})=\frac{\kappa\bigl(\tilde{A}+\pi a^2(1)\bigr)}{4\pi a^2(1)},\qquad C(\tilde{A})=\frac{\kappa\bigl(\tilde{A}+\pi a^2(1)\bigr)}{4\pi}-s \end{align}\tag{161}\] We now abbreviate \(\inf_{\gamma \in \chi(\tilde{g},\tilde{\sigma})}\mathcal{A}^{\text{mag}}_{\tilde{g},\tilde{\sigma}}(\gamma)\) as \(\inf \mathcal{A}^{\text{mag}}_{\tilde{g},\tilde{\sigma}}\) and plug 159 into 161 to obtain the following: \[\begin{align} \label{ABC95final95proof} A(\inf \mathcal{A}^{\text{mag}}_{\tilde{g},\tilde{\sigma}}) = \frac{l_{\text{min}}^{\tilde{g},\tilde{\sigma}}}{\pi a^2(1)}, \qquad B(\inf \mathcal{A}^{\text{mag}}_{\tilde{g},\tilde{\sigma}})= \frac{\kappa\big(l_{\text{min}}^{\tilde{g},\tilde{\sigma}}\bigr)}{4\pi a^2(1)},\qquad C(\inf \mathcal{A}^{\text{mag}}_{\tilde{g},\tilde{\sigma}})= \frac{\kappa\big(l_{\text{min}}^{\tilde{g},\tilde{\sigma}}\bigr)}{4\pi} -s \end{align}\tag{162}\] Plugging 162 into 160 , we obtain the following expression for \(P(\inf \mathcal{A}^{\text{mag}}_{\tilde{g},\tilde{\sigma}})\) when \(\kappa \neq 0\): \[\begin{align} \label{zoll95poly95final951} \frac{n!}{4^n\pi^n a^2(1)} \Bigg( l_{\text{min}}^{\tilde{g},\tilde{\sigma}} \Big( \kappa l_{\text{min}}^{\tilde{g},\tilde{\sigma}} -4\pi^2s a^2(1) \Big) \Bigg)^{n-1} \Bigg( \kappa l_{\text{min}}^{\tilde{g},\tilde{\sigma}} \big(na^2(1)-1\big) -4\pi s a^2(1) \Bigg) \;\text{Vol}_{g_0}(M). \end{align}\tag{163}\] Similarly, when \(\kappa =0\), we obtain the following expression for \(P(\inf \mathcal{A}^{\text{mag}}_{\tilde{g},\tilde{\sigma}})\): \[\begin{align} (-s)^n \Bigg(&\frac{l_{\text{min}}^{\tilde{g},\tilde{\sigma}}}{\pi a^2(1)} \Bigg)^{n-1}n!\pi^{n-1}[a(1)]^{2n-2} \;\text{Vol}_{g_0}(M) \\\nonumber = &(-s)^n \Big(l_{\text{min}}^{\tilde{g},\tilde{\sigma}}\Big)^{n-1}n! \;\text{Vol}_{g_0}(M) \end{align}\] From Proposition 11, the right hand side of 158 has the following expression: \[\begin{align} \label{vol95kahler95final951} Vol_{\Omega_{g_0,\sigma_0}}(\tilde{\Omega}_{\tilde{g}, \tilde{\sigma}}) = 2n\text{Vol}(B^{2n}) \Big( \text{Vol}_{\tilde{g}}(M) - \text{Vol}_{g_0}(M) \Big) \end{align}\tag{164}\] where \(\text{Vol}(B^{2n})\) is the Euclidean volume of the \(2n\)-dimensional ball. It has the following expression: \[\begin{align} \text{Vol}(B^{2n}) = \frac{\pi^{n}}{(2n-1)!} \end{align}\] Plugging the above expression into 164 and plugging the resulting expression into 158 along with 163 , we obtain the following systolic inequality when \(\kappa \neq 0\): \[\begin{align} \Bigg( l_{\text{min}}^{\tilde{g},\tilde{\sigma}} \Big( \kappa l_{\text{min}}^{\tilde{g},\tilde{\sigma}} -&4\pi^2s a^2(1) \Big) \Bigg)^{n-1} \Bigg( \kappa l_{\text{min}}^{\tilde{g},\tilde{\sigma}} \big(na^2(1)-1\big) -4\pi s a^2(1) \Bigg) \\\nonumber \leq& \frac{2^{2n+1}\pi^{2n}a^2(1)}{n!(2n-1)!} \Bigg(\frac{\text{Vol}_{\tilde{g}}(M)}{\text{Vol}_{g_0}(M)}-1\Bigg) \end{align}\] with equality holding if and only if the magnetic system determined by the pair \((\tilde{g},\tilde{\sigma})\) is Zoll at strength \(s\).

Similarly, when \(\kappa=0\), we obtain the following systolic inequality: \[\begin{align} \Big(l_{\text{min}}^{\tilde{g},\tilde{\sigma}}\Big)^{n-1} \leq \frac{\pi^n}{(-s)^n\; n!\;(2n-1)! }\Bigg(\frac{\text{Vol}_{\tilde{g}}(M)}{\text{Vol}_{g_0}(M)}-1\Bigg) \end{align}\] We now observe that if \(\text{Vol}_{g_0}(M) = \text{Vol}_{\tilde{g}}(M)\) then, by 164 , the following holds: \[\begin{align} Vol_{\Omega_{g_0,\sigma_0}}(\tilde{\Omega}_{\tilde{g}, \tilde{\sigma}}) =0 \end{align}\] Theorem 1 then implies that the local systolic inequality 158 reduces to the following: \[\begin{align} \inf \mathcal{A}^{\text{mag}}_{\tilde{g},\tilde{\sigma}} \leq 0 \end{align}\] Plugging 159 into the above equation, we then obtain the following: \[\begin{align} l_{\text{min}}^{\tilde{g},\tilde{\sigma}} \leq a^2(1)\pi \end{align}\] with equality holding if and only if the magnetic system determined by the pair \((\tilde{g},\tilde{\sigma})\) is Zoll at strength \(s\).

10 Appendix: Some facts about differential forms↩︎

10.1 The size of a \(k\)-form in terms of its differential↩︎

10.2 The size of Reeb vector fields of odd-symplectic forms↩︎

References↩︎