3 A Duality Theorem

Proof. In case of \(2\pi\)-periodic geodesic flows the first statement is already contained in [11]. To prove the general statement let \(\Sigma \subset TS^2\) be a \(K=1\) Finsler structure with \(2\pi p\)-periodic geodesic flow \(\phi : \Sigma \times \mathbb{R}\to \Sigma\), i.e. the flow factorizes through a smooth, almost free \(S^1\)-action \(\phi : \Sigma \times S^1 \to \Sigma\). The Cartan coframe will be denoted by \((\chi,\eta,\nu)\) and the dual vector fields by \((X,H,V)\). Since \(\Sigma\cong SO(3)\) is an \(L(2,1)\) lens space, the quotient map \(\lambda\) for the \(S^1\)-action is a smooth orbifold submersion onto a spindle orbifold \(\mathcal{O}=S^2(a_1,a_2)\), with \(a_1\geqslant a_2\), \(2|(a_1+a_2)\), \(c:=\gcd(a_1,a_2) \in \{1,2\}\) and \(c^3|a_1a_2\) by Lemma 2.1. With Theorem 3.1 we see that \(a_1=q\) and \(a_2=2p-q\). Since \(X\hspace{3pt}\rule[0.5pt]{2mm}{0.5pt}\rule[0.5pt]{0.5pt}{4.5pt}\hspace{3pt}\eta=X\hspace{3pt}\rule[0.5pt]{2mm}{0.5pt}\rule[0.5pt]{0.5pt}{4.5pt}\hspace{3pt}\nu=0\), the \(1\)-forms \(\eta\) and \(\nu\) are semibasic for the projection \(\lambda\) and using the structure equations for the \(K=1\) Finsler structure, we compute the Lie derivative \[\mathrm{L}_{X}\left(\eta\otimes\eta+\nu\otimes\nu\right)=\nu\otimes \eta+\eta\otimes \nu-\eta\otimes \nu-\nu\otimes \eta=0.\] Likewise, we compute \(\mathrm{L}_X\left(\nu\wedge\eta\right)=0\). Hence the symmetric \(2\)-tensor \(\eta\otimes\eta+\nu\otimes\nu\) and the \(2\)-form \(\nu\wedge\eta\) are invariant under \(\phi\) and therefore there exists a unique Riemannian metric \(g\) on \(\mathcal{O}\) for which \(\lambda^*g=\eta\otimes\eta+\nu\otimes\nu\) where \(\lambda : \Sigma \to \mathcal{O}\) is the natural projection. We may orient \(\mathcal{O}\) in such a way that the pullback of the area form \(d\sigma_g\) of \(g\) satisfies \(\lambda^*d\sigma_g=\nu\wedge\eta\). The structure equations also imply that \(\chi,\eta,\nu\) are invariant under \((\phi_{2\pi})^{\prime}\) (cf. [9]). Therefore the map \[\begin{align} \Phi : \Sigma &\to T\mathcal{O}\\ v & \mapsto -\lambda^{\prime}_v\left(V(v)\right)\\ \end{align}\] and the forms \(\chi,\eta,\nu\) are invariant under the action of the cyclic subgroup \(\Gamma<S^1\) of order \(p\) on \(\Sigma\). Hence \(\Phi\) factors through a map \(\bar \Phi: \Sigma/\Gamma \rightarrow T\mathcal{O}\), and \(\chi,\eta,\nu\) descend to \(\Sigma/\Gamma\) where they define a generalized Finsler structure. The composition of \(\bar\Phi\) with the canonical projection onto the projective sphere bundle \(\mathbb{S}T\mathcal{O}:=\left(T\mathcal{O}\setminus\{0\}\right)/\mathbb{R}^{+}\) will be denoted by \(\tilde{\Phi}\). Note that \(\tilde{\Phi}\) is an immersion, thus a local diffeomorphism and by compactness of \(\Sigma/\Gamma\) and connectedness of \(\mathbb{S}T\mathcal{O}\) a covering map. Since by [24] both \(\Sigma/\Gamma\) and \(\mathbb{S}T\mathcal{O}\) have fundamental group of order \(2p\), it follows that \(\tilde{\Phi}\) is a diffeomorphism. Therefore, \(\bar\Phi\) is an embedding which sends \(\Sigma/\Gamma\) to the total space of the unit tangent bundle \(\pi : S\mathcal{O}\to\mathcal{O}\) of \(g\). Abusing notation, we also write \(\chi,\eta,\nu \in \Omega^1(S\mathcal{O})\) to denote the pushforward with respect to \(\bar\Phi\) of the Cartan coframe on \(\Sigma/\Gamma\) . Also we let \(\alpha,\beta,\zeta\in \Omega^1(S\mathcal{O})\) denote the canonical coframe of \(S\mathcal{O}\) with respect to the orientation induced by \(d\sigma_g\). More precisely, the pullback of \(g\) to \(S\mathcal{O}\) is \(\alpha\otimes\alpha+\beta\otimes \beta\) and \(\zeta\) denotes the Levi-Civita connection form. By construction, the map \(\Phi\) sends lifts of \(\Sigma\) geodesics onto the fibres of the projection \(\pi : S\mathcal{O}\to \mathcal{O}\). Moreover, for \(v\in \Sigma\) the projection \((\pi \circ\Phi)^*V(v)\) to \(T_{(\pi\circ \Phi)(v)}\mathcal{O}\), i.e. the horizontal component of \(\Phi^*V(v)\), is parallel to \(\Phi(v)\) and so the vertical vector field \(V\) on \(\Sigma\) is mapped into the contact distribution defined by the kernel of \(\beta\). Therefore, we see that \(\beta\) and \(\eta\) are linearly dependent and that \(\nu(\Phi^* V),\alpha(\Phi^* V)<0\). However, since both \((\alpha,\beta)\) and \((\nu,\eta)\) are oriented orthonormal coframes for \(g\), it follows that \(\beta=-\eta\) and \(\alpha=-\nu\). The structure equations for the coframing \((\alpha,\beta,\zeta)\) imply \[0=\mathrm{d}\alpha+\beta\wedge\zeta=-\mathrm{d}\nu-\eta\wedge\zeta=(\chi-J\nu)\wedge\eta-\eta\wedge\zeta=-\eta\wedge\left(\chi-J\nu+\zeta\right)\] and \[0=\mathrm{d}\beta+\zeta\wedge\alpha=-\mathrm{d}\eta-\zeta\wedge\nu=\nu\wedge\left(\chi-I\eta\right)-\zeta\wedge\nu=-\left(\chi-I\eta+\zeta\right)\wedge\nu,\] where again we abuse notation by also writing \(I\) and \(J\) for the pushforward of the functions \(I\) and \(J\) with respect to \(\bar{\Phi}\). It follows that the Levi-Civita connection form \(\zeta\) of \(g\) satisfies \[\zeta=I\eta+J\nu-\chi=-(J\alpha+I\beta)-\chi.\] Recall that \(\pi : S\mathcal{O}\to \mathcal{O}\) denotes the basepoint projection. Comparing with Lemma 2.7 we want to argue that there exists a unique \(1\)-form \(\theta\) on \(\mathcal{O}\) so that \(\pi^*(\star_g\theta)=-(J\alpha+I\beta)\). Since \(J\alpha+I\beta\) is semibasic for the projection \(\pi\), it is sufficient to show that \(J\alpha+I\beta\) is invariant under the \(\mathrm{SO}(2)\) right action generated by the vector field \(Z\), where \((A,B,Z)\) denote the vector fields dual to \((\alpha,\beta,\zeta)\). Denoting by \((X,H,V)\) the vector fields dual to \((\chi,\eta,\nu)\) on \(S\mathcal{O}\), the identities \[\begin{pmatrix} \alpha\\ \beta\\ \zeta\\ \end{pmatrix}=\begin{pmatrix} -\nu \\ -\eta \\ I\eta+J\nu-\chi\end{pmatrix}\] imply \(Z=-X\). Now observe the Bianchi identity \[0=\mathrm{d}^2\eta=\left(J\chi-\mathrm{d}I\right)\wedge\eta\wedge\nu\] so that \(XI=J\). Likewise we obtain \[0=\mathrm{d}^2\nu=-\left(\mathrm{d}J+I\chi\right)\wedge\eta\wedge\nu\] so that \(XJ=-I\). From this we compute \[\mathrm{L}_X\left(I\eta+J\nu\right)=J\eta+I\nu-I\nu-J\eta=0,\] so that \(-(J\alpha+I\beta)=\pi^*(\star_g\theta)\) for some unique \(1\)-form \(\theta\) on \(\mathcal{O}\) as desired. We obtain a Weyl structure defined by the pair \((g,\theta)\). Since \[\begin{aligned} \mathrm{d}(\pi^*(\star_g\theta)-\zeta)&=\mathrm{d}\left(-(J\alpha+I\beta)+(J\alpha+I\beta+\chi)\right)=\mathrm{d}\chi=\nu\wedge\eta\\ &=\pi^*\left((K_g-\delta_g\theta)d\sigma_g\right)=\left(K_g-\delta_g\theta\right)\circ \pi\,\nu\wedge\eta,\end{aligned}\] we see that \(K_g-\delta_g\theta=1\). Therefore \((g,\theta)\) is the natural gauge for the positive Weyl structure \([(g,\theta)]\). Finally, by construction, the Weyl structure \([(g,\theta)]\) is Besse. Conversely, let \(\mathcal{O}=S^2(a_1,a_2)\) be a spindle orbifold as in \((2)\) with a positive Besse–Weyl structure \([(g,\theta)]\). Let \((g,\theta)\) be the natural gauge of \([(g,\theta)]\) and let \(\pi : S\mathcal{O}\to \mathcal{O}\) denote the unit tangent bundle with respect to \(g\). By [24] the unit-tangent bundle \(S\mathcal{O}\) is a lens space of type \(L(a_1+a_2,1)\). The canonical coframe on \(S\mathcal{O}\) as explained in Example 2.3 will be denoted by \((\alpha,\beta,\zeta)\). By Lemma 2.7 the \(1\)-forms \(\chi,\eta,\nu\) on \(S\mathcal{O}\) given by \[\chi:=\pi^*(\star\theta)-\zeta, \qquad \eta:=-\beta, \qquad \nu:=-\alpha\] define a generalized Finsler structure on \(S\mathcal{O}\) of constant Finsler–Gauss curvature \(K=1\), i.e. they satisfy the structure equations \[\mathrm{d}\chi=-\eta\wedge\nu, \qquad \mathrm{d}\eta=-\nu\wedge(\chi-I\eta), \qquad \mathrm{d}\nu=-(\chi-J\nu)\wedge\eta,\\\] for some smooth functions \(I,J :S\mathcal{O}\to \mathbb{R}\). Moreover they parallelise \(S\mathcal{O}\) and have the property that the leaves of the foliation \(\mathcal{F}_g:=\left\{\chi,\eta\right\}^{\perp}\) are tangential lifts of maximal oriented geodesics of the Weyl connection \({}^{(g,\theta)}\nabla\) on \(\mathcal{O}\). Since this connection is Besse by assumption, all of these leaves are circles. It follows from a theorem by Epstein [15] that the leaves are the orbits of a smooth, almost free \(S^1\)-action. Since \(a_1+a_2\) is odd, \(S\mathcal{O}\) admits a normal covering by a space \(M\cong L(2,1)\cong \mathbb{RP}^3\) with deck transformation group \(\bar \Gamma\) isomorphic to \(\mathbb{Z}_{(a_1+a_2)/2}\). The lifts of \(\chi,\eta,\nu\) to \(M\), which we denote by the same symbols, define a generalized Finsler structure on \(M\) of constant Finsler–Gauss curvature 1. Moreover, the \(S^1\)-action on \(S\mathcal{O}\) lifts to a smooth, almost free \(S^1\)-action on \(M\) whose orbits are again the leaves of the foliation \(\left\{\chi,\eta\right\}^{\perp}\). The leaves of the foliation \(\mathcal{F}_t:=\left\{\eta,\nu\right\}^{\perp}\) correspond to (the lifts of) the fibres of the projection \(S\mathcal{O}\rightarrow\mathcal{O}\) (to \(M\)) and are in particular also all circles. We can cover the space \(M\) further by \(S^3\) and lift the \(S^1\)-action and the foliations \(\mathcal{F}_g\) and \(\mathcal{F}_t\) to \(S^3\). By the classification of Seifert fibrations of lens spaces quotienting out the foliations \(\mathcal{F}_t\) and \(\mathcal{F}_g\) of \(S\mathcal{O}\), \(M\) and \(S^3\) yields a diagram of maps as follows (cf. 2.1 and e.g. [18]) \[\begin{xy} \xymatrix { \tilde{\mathcal{O}} \cong S^2(a_1/c,a_2/c) \ar[d] & \tilde M \cong S^3 \ar[l] \ar[r] \ar[d] & \tilde{\mathcal{O}}_g \cong S^2(k_1,k_2) \ar[d] \\ \bar{\mathcal{O}} \cong S^2(a_1/a,a_2/a) \ar[d] & M\cong L(2,1) \ar[l] \ar[r]^{\tau} \ar[d] & \bar{\mathcal{O}}_g \cong S^2(k'k_1,k'k_2) \ar[d] \\ \mathcal{O}\cong S^2(a_1,a_2) &S\mathcal{O}\cong L(a_1+a_2,1) \ar[l] \ar[r] & \mathcal{O}_g \cong S^2(kk_1,kk_2) } \end{xy}\] with \(a| \gcd(a_1,a_2)=c\in \{1,2\}\), \(\gcd(k_1,k_2)=1\), \(k'|2\) and \(k | |\Gamma|= (a_1+a_2)/2\). Here the horizontal maps are smooth orbifold submersions, and the vertical maps are coverings (of manifolds in the middle and of orbifolds on the left and the right). Moreover, the deck transformation groups in the middle descend to deck transformation groups of the orbifold coverings. We claim that \(a=1\). To prove this we can assume that \(c=2\). In this case the co-prime numbers \(a_1/c\) and \(a_2/c\) have different parity by our assumption that \(c^3 | a_1a_2\). Since \(a_1/a+a_2/a\) has to be even by Lemma 2.1, it follows that \(a=1\) as claimed.

The involution \[\begin{array}{cccl} i: & S\mathcal{O}& \rightarrow & S\mathcal{O}\\ & (x,v) & \mapsto & (x,-v). \end{array}\] maps fibres of \(\mathcal{F}_g\) and \(\mathcal{F}_t\) to fibres of \(\mathcal{F}_g\) and \(\mathcal{F}_t\), respectively, and descends to a smooth orbifold involution \(i\) of \(\mathcal{O}_g\). We claim that the same argument as in [24] shows that \(i\) does not fix the singular points on \(\mathcal{O}_g\). Here we only sketch the ideas and refer to [24] for the details: If \(a_1\) and \(a_2\) are odd then \(i\) acts freely on \(\mathcal{O}_g\), and in this case nothing more has to be said. On the other hand, if \(a_1\) and \(a_2\) are even, then any geodesic that runs into a singular point is fixed by the action of \(i\) on \(\mathcal{O}_g\). In this case one first has to show that the lift \(\tilde i :\tilde M \rightarrow\tilde M\) of \(i\) to the universal covering of \(S\mathcal{O}\) commutes with the deck transformation group \(\Gamma\) of the covering \(\tilde M \rightarrow S\mathcal{O}\). This can be shown based on the observation that a fibre of \(\mathcal{F}_t\) on \(S^3\) over the singular point of \(\mathcal{O}\), together with its orientation, is preserved by both \(\Gamma\) and \(\tilde i\) (see [24] for the details). Now, if \(a_1\) and \(a_2\) are both even and a singular point on \(\mathcal{O}_g\) is fixed by \(i\), then there also exists a fibre of \(\mathcal{F}_g\) on \(S^3\) which is invariant under both \(\Gamma\) and \(\tilde i\). However, in this case only \(\Gamma\) preserves the orientation of this fibre, whereas \(\tilde i\) reverses its orientation. This leads to a contradiction to the facts that \(|\Gamma|=a_1+a_2>2\) and that \(\Gamma\) commutes with \(\tilde i\) (see [24] for the details).

Since \(i\) preserves the orbifold structure of \(\Gamma_g\) and does not fix its singular points, it has to interchange the singular points. In particular, this implies that \(kk_1=kk_2\), and hence \(k_1=k_2=1\). Therefore the foliation \(\mathcal{F}\) on \(S^3\) is the Hopf-fibration and we must have \(k'=1\) by Lemma 2.1. In other words, \(\bar{\mathcal{O}_g}\) is a smooth \(2\)-sphere without singular points and \(\tau : M \to \bar{\mathcal{O}_g}=S^2\) is a smooth submersion. Consider the map \[\begin{align} \Phi : M &\to TS^2\\ u &\mapsto -\tau^{\prime}_u(X(u)).\\ \end{align}\] Then by [9] \(\Phi\) immerses each \(\tau\)-fibre \(\tau^{-1}(x)\) as a curve in \(T_xS^2\) that is strictly convex towards \(0_x\). The number of times \(\Phi(\tau^{-1}(x))\) winds around \(0_x\) does not depend on \(x\). Since both \(M\) and \(\mathbb{S}TS^2\) are diffeomorphic to \(L(2,1)\), the same argument as above proves that \(\Phi\) is one-to-one, and so this number is one. Therefore, by [9] \(\Phi(M)\) is a Finsler structure on \(S^2\). Moreover, \(S^2\) can be oriented in such a way that the \(\Phi\)-pullback of the canonical coframing induced on \(\Phi(M)\) agrees with \((\chi,\eta,\nu)\). In particular, this implies that the Finsler structure satisfies \(K=1\) and has periodic geodesic flow. Moreover, because of \(a=1\) we have \(\bar{\mathcal{O}} = \mathcal{O}\) and therefore the preimages of the leaves of \(\mathcal{F}_t\) under the covering \(M \rightarrow S\mathcal{O}\) are connected. Since the covering \(M \rightarrow S\mathcal{O}\) is \((a_1+a_2)/2\)-fold, so is its restriction to the fibres of \(\mathcal{F}_t\). Therefore, \(p:=(a_1+a_2)/2\) is the minimal number for which the geodesic flow of the Finsler structure on \(S^2\) is \(2\pi p\)-periodic. The structure of \(\bar{\mathcal{O}}\) implies that all closed geodesics of \(S^2\) have length \(2\pi p\) except at most two exceptions, which are \(q:=a_1\) and \(2q-p=a_2\) times shorter than the regular geodesics. In particular, the shortest geodesic has length \(2\pi p/q=2\pi \frac{a_1+a_2}{2a_1}\) as claimed.

Finally, going through the proof shows that an isometry between two Finsler metrics as in the statement of Theorem induces a diffeomorphism between the corresponding spindle orbifolds that pulls back the two natural gauges onto each other, and vice versa. Hence, since such a pullback of a natural gauge is a natural gauge, the last statement of the Theorem follows from uniqueness of the natural gauge of a given Besse–Weyl structure.