DEFORMATIONS OF THE VERONESE EMBEDDING AND FINSLER 2-SPHERES OF CONSTANT CURVATURE

We establish a one-to-one correspondence between, on the one hand, Finsler structures on the 2-sphere with constant curvature 1 and all geodesics closed, and on the other hand, Weyl connections on certain spindle orbifolds whose symmetric Ricci curvature is positive deﬁnite and whose geodesics are all closed. As an application of our duality result, we show that suitable holomorphic deformations of the Veronese embedding CP ( a 1 ,a 2 ) → CP ( a 1 , ( a 1 + a 2 ) / 2 ,a 2 ) of weighted projective spaces provide examples of Finsler 2-spheres of constant curvature whose geodesics are all closed.


Background
Riemannian metrics of constant curvature on closed surfaces are fully understood, but a complete picture in the case of Finsler metrics is still lacking. Akbar-Zadeh [2] proved a first key result by showing that on a closed surface, a Finsler metric of constant negative curvature must be Riemannian, and locally Minkowskian in the case where the curvature vanishes identically (see also [16]). In the case of constant positive curvature, a Finsler metric must still be Riemannian, provided it is reversible [10], but the situation turns out to be much more flexible in the nonreversible case.
Katok [22] gave the first examples (later analysed by Ziller [41]) of nonreversible Finsler metrics of constant positive curvature, though it was only realised later that his examples actually have constant curvature. Meanwhile, Bryant [7] gave another construction of nonreversible Finsler metrics of constant positive curvature on the 2-sphere S 2 , and in subsequent work [8] classified all Finsler metrics on S 2 having constant positive curvature and being projectively flat. Bryant also observed that every Zoll metric on S 2 with positive Gauss curvature gives rise to a Finsler metric on S 2 with constant positive curvature [9]. Hence, already by the work of Zoll [42] in the beginning of the 20th century, the moduli space of constant-curvature Finsler metrics on S 2 was known to be infinite-dimensional. Its global structure, however, is not well understood.

A duality result
Recently in [11], Bryant et al. showed, inter alia, that a Finsler metric on S 2 with constant curvature 1 either admits a Killing vector field or has all of its geodesics closed. Moreover, in the first case all geodesics become closed, and even of the same length, after a suitable (invertible) Zermelo transformation. Hence, in this sense the assumption that all geodesics are closed is not a restriction. However, in the second case the geodesics can in general have different lengths, unlike the geodesics of the Finsler metrics that arise from Bryant's construction using Zoll metrics.
In this paper we generalise Bryant's observation about Zoll metrics to a one-to-one correspondence which covers all Finsler metrics on S 2 with constant curvature 1 and all geodesics closed. The correspondence arises from the classical notion of duality for so-called path geometries.
An oriented path geometry on an oriented surface M prescribes an oriented path γ ⊂ M for every oriented direction in T M. This notion can be made precise by considering the bundle π : S(T M) := (T M \ {0 M }) /R + → M , which comes equipped with a tautological co-orientable contact distribution C. An oriented path geometry is a 1-dimensional distribution P → S(T M) so that P together with the vertical distribution L = kerπ spans C.
The orientation of M equips P and L with an orientation as well, and following [8], a 3manifold N equipped with a pair of oriented 1-dimensional distributions (P,L) spanning a contact distribution is called an oriented generalised path geometry. In this setup the surface M is replaced with the leaf space of the foliation L defined by L, and the leaf space of the foliation P defined by P can be thought of as the space of oriented paths of the oriented generalised path geometry (P,L). We may reverse the role of P and L and thus consider the dual (−L, − P ) of the oriented generalised path geometry (P,L), where here the minus sign indicates reversing the orientation.
The unit circle bundle Σ ⊂ T M of a Finsler metric F on an oriented surface M naturally carries the structure of an oriented generalised path geometry (P,L). In the case where all geodesics are closed, the dual of the path geometry arising from a Finsler metric on the 2-sphere with constant positive curvature arises from a certain generalisation of a Besse 2-orbifold [24] with positive curvature. Here a 2-orbifold is called Besse if all its geodesics are closed. Namely, using the recent result [11] by Bryant et al. about such Finsler metrics (see Theorem 3.1), we show that the space of oriented geodesics is a spindle-orbifold Oor equivalently, a weighted projective line -which comes equipped with a positive Besse-Weyl structure. By this we mean an affine torsion-free connection ∇ on O which preserves some conformal structure -a so-called Weyl connection -and has the property that the image of every maximal geodesic of ∇ is an immersed circle. Moreover, the symmetric part of the Ricci curvature of ∇ is positive definite. Conversely, having such a positive Besse-Weyl structure on a spindle orbifold, we show that the dual path geometry yields a Finsler metric on S 2 with constant positive curvature whose geodesics are all closed. More precisely, we prove the following duality result, which generalises [9,Theorem 3] and [10, Proposition 6, Corollary 2] by Bryant: Theorem A. There is a one-to-one correspondence between, on the one hand, Finsler structures on S 2 with constant Finsler-Gauss curvature 1 and all geodesics closed and, on the other hand, positive Besse-Weyl structures on spindle orbifolds S 2 (a 1 ,a 2 ) with c := gcd(a 1 ,a 2 ) ∈ {1,2}, a 1 ≥ a 2 , 2|(a 1 + a 2 ) and c 3 |a 1 a 2 . More precisely, (1) such a Finsler metric with shortest closed geodesic of length 2π ∈ (π,2π], = p/q ∈ 1 2 ,1 , gcd(p,q) = 1, gives rise to a positive Besse-Weyl structure on S 2 (a 1 ,a 2 ), with a 1 = q and a 2 = 2p − q; and (2) a positive Besse-Weyl structure on such an S 2 (a 1 ,a 2 ) gives rise to such a Finsler metric on S 2 with shortest closed geodesic of length 2π a1+a2 2a1 ∈ (π,2π]; and these assignments are inverse to each other. Moreover, two such Finsler metrics are isometric if and only if the corresponding Besse-Weyl structures coincide up to a diffeomorphism.

Construction of examples
In [31], it is shown that Weyl connections with prescribed (unparametrised) geodesics on an oriented surface M are in one-to-one correspondence with certain holomorphic curves into the 'twistor space' over M . In Section 4 we make use of this observation to construct deformations of positive Besse-Weyl structures on the weighted projective line CP(a 1 ,a 2 ) in a fixed projective class, by deforming the Veronese embedding of CP(a 1 ,a 2 ) into the weighted projective plane with weights (a 1 ,(a 1 + a 2 )/2,a 2 ). Applying our duality result, we obtain a corresponding real 2-dimensional family of nonisometric, rotationally symmetric Finsler structures on the 2-sphere with constant positive curvature and all geodesics closed, but not of the same length. The length of the shortest closed geodesic of the resulting Finsler metric is unchanged for our family of deformations, and so it is of different nature from the Zermelo deformation used by Katok in the construction of his examples [22]. Moreover, we expect that not all of these examples are of Riemannian origin in the following sense (compare Remark 4.13). The construction of rotationally symmetric Zoll metrics on S 2 can be generalised to give an infinite-dimensional family of rotationally symmetric Riemannian metrics on spindle orbifolds whose geodesics are all closed [4,24]. Since every Levi-Civita connection is a Weyl connection, we obtain an infinite-dimensional family of rotationally symmetric positive Riemannian Besse-Weyl structures.
Furthermore, in [27,28] LeBrun and Mason construct a Weyl connection ∇ on the 2-sphere S 2 for every totally real embedding of RP 2 into CP 2 which is sufficiently close to the standard real linear embedding. The Weyl connection has the property that all of its maximal geodesics are embedded circles and hence define a Besse-Weyl structure. In addition, they show that every such Weyl connection on S 2 is part of a complex 5-dimensional family of Weyl connections having the same unparametrised geodesics (see also [32]). In particular, LeBrun and Mason's Weyl connections that arise from an embedding of RP 2 that is sufficiently close to the standard embedding provide examples of positive Besse-Weyl structures. The corresponding dual Finsler metrics on S 2 will have geodesics that are all closed and of the same length.
A complete local picture of the space of Finsler 2-spheres of constant positive curvature and with all geodesics closed likely requires extending the work of LeBrun and Mason to the orbifold setting. Our results in Section 4 lay the foundation for such an extension. We hope to be able to build upon it in future work.

Background on orbifolds
For a detailed account of different perspectives on orbifolds, we refer the reader to, for example, [1,5,25,36]. Here we only quickly recall some basic notions which are relevant for our purposes. An n-dimensional Riemannian orbifold O n can be defined as a length space such that for each point x ∈ O there exists a neighbourhood U of x in O, an ndimensional Riemannian manifold M and a finite group Γ acting by isometries on M such that U and M/Γ are isometric [25]. In this case we call M a manifold chart for O. Every Riemannian orbifold admits a canonical smooth structure -that is, roughly speaking, there exist equivariant, smooth transition maps between manifold charts. Conversely, every smooth orbifold is 'metrisable' in the sense already mentioned. For a point x on an orbifold, the linearised isotropy group of a preimage of x in a manifold chart is uniquely determined up to conjugation. Its conjugacy class is denoted as Γ x and is called the local group of O at x. A point x ∈ O is called regular if its local group is trivial, and singular otherwise.
For example, the metric quotient O a , a = (a 1 ,a 2 ), of the unit sphere S 3 ⊂ C 2 by the isometric action of S 1 ⊂ C defined by for coprime numbers a 1 ≥ a 2 is a Riemannian orbifold which is topologically a 2-sphere but which metrically has two isolated singular points with cyclic local groups of order a 1 and a 2 . We denote the underlying smooth orbifold as S 2 (a 1 ,a 2 ) and refer to it as an (a 1 ,a 2 )-spindle orbifold. The quotient map π from S 3 to O a is an example of an orbifold (Riemannian) submersion, in the sense that for every point z in S 3 , there is a neighbourhood V of z such that M/Γ = U = π(V ) is a chart, and π| V factors as Vπ −→ M −→M/Γ = U , whereπ is a standard submersion. The anti-Hopf action of S 1 on S 3 defined by z(z 1 ,z 2 ) = zz 1 ,z −1 z 2 commutes with the previous S 1 -action and induces an isometric S 1 -action on O a . Let Γ k be a cyclic subgroup of the anti-Hopf S 1 -action. The quotient S 3 /Γ k is a lens space of type L(k,1). By modding out such Γ k -actions on O a , we obtain spindle orbifolds S 2 (a 1 ,a 2 ) with arbitrary a 1 and a 2 as quotients. These spaces fit in the commutative diagram for some k |k. Here the left vertical map is an example of a (Riemannian) orbifold covering [25] for a metric definition). Thurston has shown that the theory of orbifold coverings works analogously to the theory of ordinary coverings [39]. In particular, there exist universal coverings, and one can define the orbifold fundamental group π orb 1 (O) of a connected orbifold O as the deck transformation group of the universal covering. For instance, the orbifold fundamental group of S 2 (a 1 ,a 2 ) is a cyclic group of order gcd(a 1 ,a 2 ). Moreover, the number k in the diagram is determined in [18,Theorem 4.10] to be .
More generally, in his fundamental monograph [38], Seifert studies foliations of 3manifolds by circles that are locally orbits of effective circle actions without fixed points (for a modern account, see, e.g., [37]). The orbit space of such a Seifert fibration naturally carries the structure of a 2-orbifold with isolated singularities. If both the 3-manifold and the orbit space are orientable, then the Seifert fibration can globally be described as a decomposition into orbits of an effective circle action without fixed points (see, e.g., [24,Section 2.4] and the references therein). In particular, in [38,Chapter 11], Seifert shows that any Seifert fibration of the 3-sphere is given by the orbit decomposition of a weighted Hopf action. The classification of Seifert fibrations of lens spaces, their quotients and their behaviour under coverings are described in detail in [18]. Let us record the following special statement, which will be needed later: Lemma 2.1. Let F be a Seifert fibration of RP 3 ∼ = L(2,1) with orientable quotient orbifold. Then the quotient orbifold is an S 2 (a 1 ,a 2 )-spindle orbifold, a 1 ≥ a 2 , with 2|(a 1 + a 2 ), c := gcd(a 1 ,a 2 ) ∈ {1,2} and c 3 |a 1 a 2 .
Proof. Since RP 3 and the quotient surface are orientable, the Seifert fibration is induced by an effective circle action without fixed points. It follows from the homotopy sequence that the orbifold fundamental group of the quotient is either trivial or Z 2 [37, Lemma 3.2]. In particular, the quotient has to be a spindle orbifold (see, e.g., [37,Chapter 3]  Usually notions that make sense for manifolds can also be defined for orbifolds. The general philosophy is to either define them in manifold charts and demand that they be invariant under the action of the local groups (and transitions between charts, as in the manifold case), like in the case of a Riemannian metric, or demand certain lifting conditions. For instance, a map between orbifolds is called smooth if it locally lifts to smooth maps between manifold charts. Let us also explicitly mention that the tangent bundle of an orbifold can be defined by gluing together quotients of the tangent bundles of manifold charts by the actions of local groups [1,Proposition 1.21]. In particular, if the orbifold has only isolated singularities, then its unit tangent bundle (with respect to any Riemannian metric) is in fact a manifold. For instance, the unit tangent bundle of an S 2 (a 1 ,a 2 )-spindle orbifold is an L(a 1 + a 2 ,1) lens space [24,Lemma 3.1]. General vector bundles on orbifolds can be similarly defined on the level of charts. We will only work with vector bundles on spindle orbifolds S 2 (a 1 ,a 2 ) which can be described as associated bundles SU(2) × S 1 V for some linear representation of S 1 on a vector space V .
In the sequel we liberally use orbifold notions which follow this general philosophy without further explanation, and refer to the literature for more details.

Besse orbifolds
The Riemannian spindle orbifolds O a ∼ = S 2 (a 1 ,a 2 ) constructed in the preceding section have the additional property that all their geodesics are closed -that is, any geodesic factors through a closed geodesic. Here an (orbifold) geodesic on a Riemannian orbifold is a path that can locally be lifted to a geodesic in a manifold chart, and a closed geodesic is a loop that is a geodesic on each subinterval. We call a Riemannian metric on an orbifold as well as a Riemannian orbifold Besse if all its geodesics are closed. The moduli space of (rotationally symmetric) Besse metrics on spindle orbifolds is infinite-dimensional [4,24]. For more details on Besse orbifolds we refer to [3,24].

Finsler structures
A Finsler metric on a manifold is, roughly speaking, a Banach norm on each tangent space varying smoothly from point to point. Instead of specifying the family of Banach norms, one can also specify the norm's unit vectors in each tangent space. Here we consider only oriented Finsler surfaces and use definitions for Finsler structures from [8]: A Finsler structure on an oriented surface M is a smooth hypersurface Σ ⊂ T M for which the base-point projection π : Σ → M is a surjective submersion which has the property that for each p ∈ M , the fibre Σ p = π −1 (p) = Σ∩ T p M is a closed, strictly convex curve enclosing the origin 0 ∈ T p M. Cartan [13] has shown how to associate a coframing to a Finsler structure on an oriented surface M . For a modern reference for Cartan's construction, the reader may consult [6]. Let Σ ⊂ T M be a Finsler structure. Then there exists a unique coframing P = (χ,η,ν) of Σ with dual vector fields (X,H,V ) which satisfies the structure equations for some smooth functions I,J,K : Σ → R. Moreover, the π-pullback of any positive volume form on M is a positive multiple of χ ∧ η, and the tangential lift of any Σ-curve γ satisfieṡ A Σ-curve γ is a Σ-geodesic -that is, a critical point of the length functional -if and only if its tangential lift satisfiesγ * ν = 0. The integral curves of X therefore project to Σ-geodesics on M , and hence the flow of X is called the geodesic flow of Σ.
For a Riemannian Finsler structure, the functions I,J vanish identically, as a result of which K is constant on the fibres of π : Σ → M and therefore the π-pullback of a function on M which is the Gauss curvature K g of g. Since in the Riemannian case the function K is simply the Gauss curvature, it is usually called the Finsler-Gauss curvature. In general, K need not be constant on the fibres of π : Σ → M .
Let Σ ⊂ T M andΣ ⊂ TM be two Finsler structures on oriented surfaces with coframings P andP. An orientation-preserving diffeomorphism Φ : M →M with Φ (Σ) =Σ is called a Finsler isometry. It follows that for a Finsler isometry, (Φ | Σ ) * P = P, and conversely any diffeomorphism Ξ : Σ →Σ which pulls backP to P, is of the form Ξ = Φ for some Finsler isometry Φ : M →M . Following [8, Def. 1], we use the following definition: A coframing (χ,η,ν) on a 3-manifold Σ satisfying the structure equations (2.2) for some functions I,J and K on Σ will be called a generalised Finsler structure.
As in the case of a Finsler structure, we denote the dual vector fields of (χ,η,ν) by (X,H,V ). Note that a generalised Finsler structure naturally defines an oriented generalised path geometry by defining P to be spanned by X while calling positive multiples of X positive and by defining L to be spanned by V while calling positive multiples of V positive.
is a manifold, and like in the case of a smooth Finsler structure, it can be equipped with a canonical coframing as well. In order to distinguish the Riemannian orbifold case from the smooth Finsler case, we will use the notation (α,β,ζ) instead of (χ,η,ν) for the coframing.

C. Lange and T. Mettler
The construction is as follows: a manifold chart M/Γ of O gives rise to a manifold chart SM/Γ of SO. In such a chart, the first two coframing forms are explicitly given by Here π : SO → O denotes the base-point projection and i : T M → T M the rotation of tangent vectors by π/2 in positive direction. Note that these expressions are invariant under the group action of Γ, and hence in fact define forms on OM . The third coframe form ζ is the Levi-Civita connection form of g, and we have the structure equations where K g denotes the Gauss curvature of g. Moreover, note that π * dσ g = α ∧ β, where dσ g denotes the area form of O with respect to g. Denoting the vector fields dual to (α,β,ζ) by (A,B,Z), we observe that the flow of Z is 2π-periodic. Finally, if O is a manifold, then the coframing (α,β,ζ) agrees with Cartan's coframing (χ,η,ν) on the Riemannian Finsler structure Σ = SO.

Weyl structures and connections
A Weyl connection on an orbifold O is an affine torsion-free connection on O preserving some conformal structure [g] on O in the sense that its parallel transport maps are anglepreserving with respect to [g]. An affine torsion-free connection ∇ is a Weyl connection with respect to the conformal structure [g] on O if for some (and hence any) conformal metric g ∈ [g] there exists a 1-form θ ∈ Ω 1 (O) such that Conversely, it follows from Koszul's identity that for every pair (g,θ) consisting of a Riemannian metric g and 1-form θ on O, the connection is the unique affine torsion-free connection satisfying equation (2.3). Here g ∇ denotes the Levi-Civita connection of g, and θ is the vector field dual to θ with respect to g. Notice that for u ∈ C ∞ (O) we have the formula from which one easily computes the identity Consequently, we define a Weyl structure to be an equivalence class [(g,θ)] subject to the equivalence relation Clearly, the mapping which assigns to a Weyl structure [(g,θ)] its Weyl connection (g,θ) ∇ is a one-to-one correspondence between the set of Weyl structures and the set of Weyl connections on O.
The Ricci curvature of a Weyl connection (g,θ) ∇ on O is where δ g denotes the codifferential with respect to g.
Definition 2.4. We call a Weyl structure [(g,θ)] positive if the symmetric part of the Ricci curvature of its associated Weyl connection is positive definite.
In the case where O is oriented, we may equivalently say the Weyl structure [(g,θ)] is positive if the 2-form (K g − δ g θ) dσ g -which depends only on the orientation and given Weyl structure -is an orientation-compatible volume form on O. Note that by the Gauss-Bonnet theorem [36], simply connected spindle orbifolds are the only simply connected 2-orbifolds carrying positive Weyl structures.
We now obtain the following: Lemma 2.5. Every positive Weyl structure contains a unique pair (g,θ) satisfying K g − δ g θ = 1.

Proof.
We have the following standard identity for the change of the Gauss curvature under conformal change: where Δ g = − (dδ g + δ g d) is the negative of the Laplace-de Rham operator. Also, we have the identity δ e 2u g = e −2u δ g for the codifferential acting on 1-forms. If [(g,θ)] is a positive Weyl structure, we may take any representative (g,θ), define u = 1 2 ln (K g − δ g θ) and consider the representative ĝ,θ = e 2u g,θ + du . Then we have Suppose the two representative pairs (g,θ) and ĝ,θ both satisfy Kĝ −δĝθ = K g −δ g θ = 1.
Since they define the same Weyl connection, the expression for the Ricci curvature implies thatĝ = Kĝ − δĝθ ĝ = (K g − δ g θ) g = g and hence alsoθ = θ, as claimed.
defines a generalised Finsler structure of constant Finsler-Gauss curvature K = 1 on SO.
Proof. We compute .
where on the right-hand side we think of θ as a real-valued function on SO. Since ν = −α, we thus have for I = −θ, again interpreted as a function on SO. Likewise, we obtain where J = Z(θ). The claim follows.
Remark 2.8. We remark that correspondingly, we have a natural gauge (g,θ) for a negative Weyl structure -that is, (g,θ) satisfies K g − δ g θ = −1. On a closed oriented surface (necessarily of negative Euler characteristic), the associated flow generated by the vector field A − Z(θ)Z falls into the family of flows introduced in [33]. In particular, its dynamics are Anosov.
The geometric significance of the form χ in Lemma 2.7 is described in the following statement. For a proof in the manifold case -which carries over, mutatis mutandis, to the orbifold case -the reader may consult [ We conclude this section with a definition: 10. An affine torsion-free connection ∇ on O is called Besse if the image of every maximal geodesic of ∇ is an immersed circle. A Weyl structure whose Weyl connection is Besse will be called a Besse-Weyl structure.
Note that the Levi-Civita connection of any (orientable) Besse orbifold O (see Section 2.2) gives rise to a Besse-Weyl structure on SO.

A duality theorem
Let us cite the following result from [11]: (1) Either = 1 and all geodesics have the same length 2π or 2 ,1 , with p,q ∈ N and gcd(p,q) = 1, and in this case all unit-speed geodesics have a common period 2πp. Furthermore, there exist at most two closed geodesics with length less than 2πp. A second one exists only if 2p − q > 1, and its length is 2πp/(2p − q) ∈ (2π,2pπ).
In particular, if all geodesics of a Finsler metric on S 2 are closed, then its geodesic flow is periodic with period 2πp for some integer p.
We now have our main duality result: There is a one-to-one correspondence between, on the one hand, Finsler structures on S 2 with constant Finsler-Gauss curvature 1 and all geodesics closed and, on the other hand, positive Besse-Weyl structures on spindle orbifolds S 2 (a 1 ,a 2 ) with c := gcd(a 1 ,a 2 ) ∈ {1,2}, a 1 ≥ a 2 , 2|(a 1 + a 2 ) and c 3 |a 1 a 2 . More precisely, (1) such a Finsler metric with shortest closed geodesic of length 2π ∈ (π,2π], = p/q ∈ Proof. In the case of 2π-periodic geodesic flows, the first statement is already contained in [10]. To prove the general statement, let Σ ⊂ T S 2 be a K = 1 Finsler structure with 2πp-periodic geodesic flow φ : Σ × R → Σ -that is, the flow factorises through a smooth, almost free S 1 -action φ : Σ × S 1 → Σ. The Cartan coframe will be denoted by (χ,η,ν) and the dual vector fields by (X,H,V ). Since Σ ∼ = SO (3) is an L(2,1) lens space, the quotient map λ for the S 1 -action is a smooth orbifold submersion onto a spindle orbifold O = S 2 (a 1 ,a 2 ), with a 1 ≥ a 2 , 2|(a 1 +a 2 ), c := gcd(a 1 ,a 2 ) ∈ {1,2} and c 3 |a 1 a 2 by Lemma 2.1. With Theorem 3.1, we see that a 1 = q and a 2 = 2p − q. Since X η = X ν = 0, the 1forms η and ν are semibasic for the projection λ, and using the structure equations for the K = 1 Finsler structure, we compute the Lie derivative Likewise, we compute L X (ν ∧ η) = 0. Hence the symmetric 2-tensor η ⊗ η + ν ⊗ ν and the 2-form ν ∧ η are invariant under φ, and therefore there exists a unique Riemannian metric g on O for which λ * g = η ⊗ η + ν ⊗ ν, where λ : Σ → O is the natural projection. We may orient O in such a way that the pullback of the area form dσ g of g satisfies λ * dσ g = ν ∧ η.
The structure equations also imply that χ,η,ν are invariant under (φ 2π ) (compare [8, p. 186]). Therefore the map , it follows thatΦ is a diffeomorphism. Therefore,Φ is an embedding which sends Σ/Γ to the total space of the unit tangent bundle π : SO → O of g. Abusing notation, we also write χ,η,ν ∈ Ω 1 (SO) to denote the push-forward with respect toΦ of the Cartan coframe on Σ/Γ. Also, we let α,β,ζ ∈ Ω 1 (SO) denote the canonical coframe of SO with respect to the orientation induced by dσ g . More precisely, the pullback of g to SO is α ⊗ α + β ⊗ β, and ζ denotes the Levi-Civita connection form. By construction, the map Φ sends lifts of Σ geodesics onto the fibres of the projection π : , and so the vertical vector field V on Σ is mapped into the contact distribution defined by the kernel of β. Therefore, we see that β and η are linearly dependent and that ν( However, since both (α,β) and (ν,η) are oriented orthonormal coframes for g, it follows that β = −η and α = −ν. The structure equations for the coframing (α,β,ζ) imply where again we abuse notation by also writing I and J for the push-forward of the functions I and J with respect toΦ. It follows that the Levi-Civita connection form ζ of g satisfies Recall that π : SO → O denotes the base-point projection. Comparing with Lemma 2.7, we want to argue that there exists a unique 1-form θ on O so that π * ( g θ) = −(Jα + Iβ).
Since Jα + Iβ is semibasic for the projection π, it is sufficient to show that Jα + Iβ is invariant under the SO (2)  so that −(Jα + Iβ) = π * ( g θ) for some unique 1-form θ on O, as desired. We obtain a Weyl structure defined by the pair (g,θ). Since we see that K g − δ g θ = 1. Therefore (g,θ) is the natural gauge for the positive Weyl structure [(g,θ)]. Finally, by construction, the Weyl structure [(g,θ)] is Besse.
Conversely, let O = S 2 (a 1 ,a 2 ) be a spindle orbifold as in statement (2) 1 + a 2 ,1). The canonical coframe on SO as explained in Example 2.3 will be denoted by (α,β,ζ). By Lemma 2.7 the 1-forms χ,η,ν on SO given by define a generalised Finsler structure on SO of constant Finsler-Gauss curvature K = 1that is, they satisfy the structure equations for some smooth functions I,J : SO → R. Moreover, they parallelise SO and have the property that the leaves of the foliation F g := {χ,η} ⊥ are tangential lifts of maximal oriented geodesics of the Weyl connection (g,θ) ∇ on 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, SO admits a normal covering by a space M ∼ = L(2,1) ∼ = RP 3 with deck transformation groupΓ isomorphic to Z (a1+a2)/2 . The lifts of χ,η,ν to M , which we denote by the same symbols, define a generalised Finsler structure on M of constant Finsler-Gauss curvature 1. Moreover, the S 1 -action on SO lifts to a smooth, almost free S 1 -action on M whose orbits are again the leaves of the foliation {χ,η} ⊥ . The leaves of the foliation 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 F g and F t to S 3 . By the classification of Seifert fibrations of lens spaces, quotienting out the foliations F t and F g of SO, M and S 3 yields a diagram of maps as follows (compare Section 2.1 and, e.g., [18]): with a| gcd(a 1 ,a 2 ) = c ∈ {1,2}, gcd(k 1 ,k 2 ) = 1, k |2 and k||Γ| = (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 coprime numbers a 1 /c and a 2 /c have different parities, by our assumption that c 3 |a 1 a 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 maps fibres of F g and F t to fibres of F g and F t , respectively, and descends to a smooth orbifold involution i of O g . We claim that the same argument as in [24] shows that i does not fix the singular points on O g . Here we only sketch the ideas, referring to [24] for the details: if a 1 and a 2 are odd, then i acts freely on 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 O g . In this case one first has to show that the liftĩ :M →M of i to the universal covering of SO commutes with the deck transformation group Γ of the coveringM → SO. This can be shown based on the observation that a fibre of F t on S 3 over the singular point of O, together with its orientation, is preserved by both Γ andĩ (see [24,Lemma 3.4] for the details). Now, if a 1 and a 2 are both even and a singular point on O g is fixed by i, then there also exists a fibre of F g on S 3 which is invariant under both Γ andĩ. However, in this case only Γ preserves the orientation of this fibre, whereasĩ reverses its orientation. This leads to a contradiction of the facts that |Γ| = a 1 + a 2 > 2 and that Γ commutes withĩ (see [24,Lemma 3.5] for the details).
Since i preserves the orbifold structure of Γ 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 F on S 3 is the Hopf-fibration and we must have k = 1 by Lemma 2.1. In other words,Ō g is a smooth 2-sphere without singular points and τ : M →Ō g = S 2 is a smooth submersion. Consider the map Φ : M → T S 2 u → −τ u (X(u)).
Then by [8, Proposition 1], Φ immerses each τ -fibre τ −1 (x) as a curve in T x S 2 that is strictly convex toward 0 x . The number of times Φ τ −1 (x) winds around 0 x does not depend on x. Since both M and ST S 2 are diffeomorphic to L(2,1), the same argument as before proves that Φ is one to one, and so this number is 1. Therefore, by [8, Proposition 2], Φ(M ) is a Finsler structure on S 2 . Moreover, S 2 can be oriented in such a way that the Φ-pullback of the canonical coframing induced on Φ(M ) agrees with (χ,η,ν). In particular, this implies that the Finsler structure satisfies K = 1 and has periodic geodesic flow. Moreover, because a = 1 we haveŌ = O, and therefore the preimages of the leaves of F t under the covering M → SO are connected. Since the covering M → SO is (a 1 + a 2 )/2-fold, so is its restriction to the fibres of 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 2πp-periodic. The structure ofŌ implies that all closed geodesics of S 2 have length 2πp 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 2πp/q = 2π a1+a2 2a1 , as claimed. Finally, going through the proof shows that an isometry between two Finsler metrics as in the statement of the 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 the uniqueness of the natural gauge of a given Besse-Weyl structure.

Construction of examples
In this section we exhibit our duality result to construct a 2-dimensional family of deformations of a given rotationally symmetric Finsler metric on S 2 of constant curvature and all geodesics closed through metrics with the same properties. On the Besse-Weyl side these deformations correspond to deformations through Besse-Weyl structures in a fixed projective equivalence class.

The twistor space
Inspired by the twistorial construction of holomorphic projective structures by Hitchin [19] and LeBrun [26], it was shown in [14,35]  ). In [31], it is shown that a Weyl connection in the projective equivalence class [∇] corresponds to a section of J + (M ) → M whose image is a holomorphic curve. In the case of the 2-sphere S 2 equipped with the projective structure arising from the Levi-Civita connection of the standard metric -or equivalently, CP 1 equipped with the projective structure arising from the Levi-Civita connection of the Fubini-Study metric -the twistor space J + S 2 is biholomorphic to CP 2 \ RP 2 .
Here we think of RP 2 as sitting inside CP 2 via its standard real linear embedding. As a consequence, one can show that the Weyl connections on S 2 whose geodesics are the great circles are in one-to-one correspondence with the smooth quadrics in CP 2 \ RP 2 [31]. Using our duality result, this recovers on the Finsler side Bryant's classification of Finsler structures on S 2 of constant curvature K = 1 and with linear geodesics [8].
The construction of the twistor space can still be carried out for the case of a projective structure [∇] on an oriented orbifold O. Again, sections of J + (O) → O having holomorphic image correspond to Weyl connections in [∇]. Since the spindle orbifold S 2 (a 1 ,a 2 ) may also be thought of as the weighted projective line CP(a 1 ,a 2 ) with weights (a 1 ,a 2 ) (see Section 4.2), one would expect that J + (CP(a 1 ,a 2 )) can be embedded holomorphically into the weighted projective plane, where we equip CP(a 1 ,a 2 ) with the projective structure arising from the Levi-Civita connection of the Fubini-Study metric. This is indeed the case, as we will show in Section 4.4. However, a difficulty that arises is that there is more than one natural candidate for the Fubini-Study metric on CP(a 1 ,a 2 ). We will next identify the correct metric for our purposes.

The Fubini-Study metric on the weighted projective line
The (complex) weighted projective space is the quotient of C n \ {0} by C * , where C * acts with weights (a 1 , . . . ,a n ) ∈ N n -that is, by the rule z · (z 1 , . . . ,z n ) = (z a1 z 1 , . . . ,z an z n ) for all z ∈ C * and (z 1 , . . . ,z n ) ∈ C n \ {0}. It inherits a natural quotient complex structure from C n . We denote the projective space with weights (a 1 , . . . ,a n ) by CP(a 1 , . . . ,a n ). Clearly, taking all weights equal to 1 gives an ordinary projective space, and for n = 2 with weights (a 1 ,a 2 ) we obtain the spindle orbifold S 2 (a 1 ,a 2 ). To omit case differentations we will henceforth restrict to the case where the pair (a 1 ,a 2 ) is coprime with a 1 ≥ a 2 and both numbers odd.
For what follows, we would like to have an explicit Besse orbifold metric on CP(a 1 ,a 2 ) which induces the quotient complex structure of CP(a 1 ,a 2 ). The quotient Besse orbifold metric on S 2 (a 1 ,a 2 ) described in Section 2.1 satisfies this condition if and only if a 1 = a 2 . Abstractly, the existence of such a metric follows from the uniformisation theorem for orbifolds [40] (see also [17,Theorem 7.8.]). In fact, since the biholomorphism group of CP(a 1 ,a 2 ) for (a 1 ,a 2 ) = (1,1) contains a unique subgroup isomorphic to S 1 , it even follows that such a metric can be chosen to be rotationally symmetric. We are now going to describe an orbifold metric with these properties which, in addition, will have strictly positive Gauss curvature. For this purpose it is convenient to describe the weighted projective line CP(a 1 ,a 2 ) as a quotient of SU (2). In particular, we will identify SU(2) as an (a 1 + a 2 )-fold cover of the unit tangent bundle of CP(a 1 ,a 2 ). for ϑ ∈ R and where L g and R g denote left and right multiplication by the group element g ∈ SU(2). Explicitly, we have and hence the corresponding quotient can be identified with the weighted projective line CP (a 1 ,a 2 ).

Recall that the Maurer-Cartan form is defined as
we also obtain dz = −wϕ − izκ and dw = zϕ − iwκ, as well as ϕ = zdw − wdz and κ = i(zdz + wdw) . In order to compute a basis for the 1-forms that are semibasic for the projection π a1,a2 : SU(2) → CP(a 1 ,a 2 ), we evaluate the Maurer-Cartan form on the infinitesimal generator Z := d dt t=0 T e it of the S 1 -action. We obtain Consequently, we see that the complex-valued 1-form satisfies ω(Z) = 0 and hence, by definition, is semibasic for the projection π a1,a2 : SU(2) →  CP(a 1 ,a 2 ). Because of the left-invariance of we have T * e iϑ = (R e i(a 1 +a 2 )ϑ/2 ) * and hence where we have used the equivariance property R * g = g −1 g, which holds for all g ∈ SU(2).
In particular, the structure equations for (α,β,ζ) imply the commutator relations where, by abuse of notation, we think of K g as a function on the quotient SU(2)/Z a1+a2 . These commutator relations in turn imply that the coframing α,β,ζ of SU(2)/Z a1+a2 that is dual to (A,B,Z) defines a generalised Finsler structure of Riemannian type. Exactly as in the proof of Theorem A, it follows that there exists a unique orientation and orbifold metric g on CP(a 1 ,a 2 ), so that π * g = α ⊗ α + β ⊗ β and so that the area form of g satisfies π * dσ g = α ∧ β. Here π : SU(2)/Z a1+a2 → CP(a 1 ,a 2 ) denotes the quotient projection with respect to the S 1 action T e iϑ . Moreover, the map Φ : SU(2)/Z a1+a2 → T CP(a 1 ,a 2 ), u → π u (A(u)), is a diffeomorphism onto the unit tangent bundle SCP(a 1 ,a 2 ) of g which has the property that the pullback of the canonical coframing on SCP(a 1 ,a 2 ) yields α,β,ζ . Thus, we will henceforth identify the unit tangent bundle SCP(a 1 ,a 2 ) of (CP(a 1 ,a 2 ),g) with SU(2)/Z a1+a2 .
We will next show that g is a Besse orbifold metric. For an element y in the Lie algebra su(2) of SU(2), we let Y y denote the vector field on SU(2) generated by the flow R exp(ty) . Recall that the Maurer-Cartan form satisfies (Y y ) = y for all y ∈ su (2). It follows that the basis e 1 = 0 −1 1 0 and e 2 = 0 i i 0 and e 3 = −i 0 0 i of su(2) yields a framing (Y e1 ,Y e2 ,Y e3 ) of SU(2) which is dual to the coframing (Re(ϕ),Im(ϕ),κ). Therefore, using equations (4.3) and (4.6) and the definition of α,β, we obtain A = Y e1 /U . The flow of Y e1 is given by R exp(te1) and hence periodic with period 2π. Recall that the geodesic flow of the metric g on CP(a 1 ,a 2 ) is A. Thinking of U as a function on SCP(a 1 ,a 2 ), we have and hence g is a Besse orbifold metric.
In complex notation, we have (π a1,a2 ) * g = ω • ω, where • denotes the symmetric tensor product and ω = α + iβ. The complex structure on CP(a 1 ,a 2 ) defined by g and the orientation is thus characterised by the property that its (1,0)-forms pull back to SU (2) to become complex multiples of ω. In particular, this complex structure coincides with the natural quotient complex structure of CP(a 1 ,a 2 ), since ω is a linear combination of dz and dw (see equation (4.2)). Finally, observe that K g is strictly positive. We have thus shown the following: There exists a Besse orbifold metric g and orientation on CP(a 1 ,a 2 ) so that (π a1,a2 ) * g = α ⊗ α + β ⊗ β and so that (π a1,a2 ) * dσ g = α ∧ β. This metric and orientation induce the quotient complex structure of CP(a 1 ,a 2 ). Moreover, g has strictly positive Gauss curvature K g = 2(a 1 + a 2 )/ a 1 |z| 2 + a 2 |w| 2 3 .

Remark 4.2.
The reader may easily verify that in the case a 1 = a 2 = 1, we recover the usual Fubini-Study metric on CP 1 . For this reason we refer to g as the Fubini-Study metric of CP(a 1 ,a 2 ).

Constructing the twistor space
We will next construct the twistor space J + (O) in the case of the weighted projective line CP(a 1 ,a 2 ) and where [∇] is the projective equivalence class of the Levi-Civita connection of the Fubini-Study metric on CP(a 1 ,a 2 ) constructed in Lemma 4.1. As we will see, in this special case the twistor space J + (CP(a 1 ,a 2 )) can indeed be embedded into the weighted projective plane CP (a 1 ,(a 1 + a 2 )/2,a 2 ). We will henceforth also write J + for the twistor space whenever the underlying orbifold is clear from the context. We refer the reader to [20] for additional details.
In the case of the orbifold CP(a 1 ,a 2 ), the orbifold canonical bundle with respect to the Riemann surface structure induced by the orientation and the Fubini-Study metric g described in Lemma 4.1 can be defined as a suitable quotient of SU(2) × C.
Likewise, K −1 CP(a1,a2) arises from acting with spin −(a 1 + a 2 ), and K CP(a1,a2) arises from the complex conjugate of the spin (a 1 + a 2 ) action -that is, also from the action with spin − (a 1 + a 2 ). Therefore, we obtain B = SU(2) × S 1 C, where now S 1 acts with spin −2(a 1 + a 2 ) on C. The Hermitian bundle metric h on B arises from the usual Hermitian inner product on C, and hence we obtain where D ⊂ C denotes the open unit disk and S 1 acts with spin −2(a 1 + a 2 ) on D.
We now define an almost complex structure J on J + (CP(a 1 ,a 2 )). On SU(2) × D we consider the complex-valued 1-forms where μ denotes the standard coordinate on D. Abusing notation and writing T e iϕ for the combined S 1 -action on SU(2) × D, we obtain Furthermore, by construction, the forms ξ 1 and ξ 2 are semibasic for the projection SU(2)× D → J + (CP(a 1 ,a 2 )). It follows that there exists a unique almost complex structure J on J + (CP(a 1 ,a 2 )) whose (1,0)-forms pull back to SU(2) × D to become linear combinations of ξ 1 and ξ 2 . Finally, in [34, §4.2] it is shown that the almost complex structure so constructed agrees with the complex structure on the twistor space associated to (CP(p,q),[ g ∇]), where g ∇ denotes the Levi-Civita connection of g.

Remark 4.3.
More precisely, in [34, §4.2] only the case of smooth surfaces is considered, but the construction carries over to the orbifold setting without difficulty.

Embedding the twistor space
Recall that the twistor space of the 2-sphere J + S 2 equipped with the complex structure coming from the projective structure of the standard metric maps biholomorphically onto CP 2 \ RP 2 . The map arises as follows. Consider S 2 as the unit sphere in R 3 and identify the tangent space T e S 2 to an element e ∈ S 2 with the orthogonal complement {e} ⊥ ⊂ R 3 . Then an orientation-compatible complex structure J on T e S 2 is mapped to the element where v ∈ T e S 2 is any nonzero tangent vector. With respect to our present model of J + S 2 as an associated bundle, this map takes the explicit form after applying a linear coordinate change (see Section 4.6). In this new coordinate system, the real projective space RP 2 sits inside CP 2 as the image of the unit sphere in C × R under the map j = π •ĵ : C × R → CP 2 , where j : C × R → C 3 is defined as j(z,t) = (z,it,z) and where π : C 3 \{0} → CP 2 is the quotient projection. Note that for μ = 0, the map Ξ restricts to the Veronese embedding of CP 1 into CP 2 . We observe that in the weighted case, the very same map Ξ also defines a smooth map of orbifolds. In fact, we are going to show the following statement in a sequence of lemmas: is a biholomorphism onto CP (a 1 ,(a 1 + a 2 )/2,a 2 ) \j S 2 , where j = π •ĵ as before. Moreover, j S 2 is a real projective plane RP 2 ((a 1 + a 2 )/2) with a cyclic orbifold singularity of order (a 1 + a 2 )/2.

Remark 4.5.
More precisely, by biholomorphism we mean a diffeomorphism Ξ which is holomorphic in the sense that it is (J,J 0 )-linear. By this we mean that it satisfies J 0 • Ξ = Ξ • J, where J denotes the almost complex structure defined on J + (CP(a 1 ,a 2 )) in Section 4.3 and J 0 the standard complex structure on the weighted projective space CP (a 1 ,(a 1 + a 2 )/2,a 2 ).
In the following we also describe SU(2) × S 1 D as a quotient of C 2 \{0} × C by the respective weighted C * -action. Here λ ∈ C * acts as λ/|λ| on the second factor. Then the map Ξ is covered by the C * -equivariant map Since we already know that the map Ξ is an immersion in the unweighted case, and since the C * -actions on C 2 \{0} × D and on C 3 \{0} do not have fixed points, it follows that the mapΞ, and hence also the map Ξ in the weighted case, is an immersion as well. Alternatively, the same conclusion can be drawn from an explicit computation, which shows that the determinant of the Jacobian of the mapΞ is given by det(J(z,w,μ)) = 4 1 − |μ| 2 |z| 2 + |w| 2 4 . There are different ways to continue the proof of Proposition 4.4. For instance, one can show that the map Ξ extends to a smooth orbifold immersion of a certain compactification of J + onto CP (a 1 ,(a 1 + a 2 )/2,a 2 ), so that the complement of J + is mapped onto j S 2 . Compactness and the fact that CP (a 1 ,(a 1 + a 2 )/2,a 2 ) is simply connected as an orbifold then imply that this map is a diffeomorphism. Since we do not need such a compactification otherwise at the moment, we instead prove the proposition by hand, which, in total, is less work. More precisely, we proceed by proving the following three lemmas. Proof. Suppose we have [z : w : μ] ∈ J + with Ξ([z : w : μ]) ∈ j S 2 . We can assume that |z| 2 + |w| 2 = 1. Then there exist some λ ∈ C * such that λ 2a1 z 2 − w 2 μ = λ 2a2 w 2 − z 2 μ , λ a1+a2 (zw + zwμ) ∈ iR and If zw = 0, the last two conditions imply that |μ| = 1, a contradiction. Let us assume that z = 0. Then w = 0, and so the first condition implies that −λ 2a1 μ = λ 2a2 . Together with 1 = −λ 2(a1+a2) μ, this also implies |μ| = 1. The same conclusion follows analogously in the case z = 0. Hence, in any case we obtain a contradiction, and so the lemma is proven. Again we can assume that |z| 2 + |w| 2 = 1 and |z | 2 + |w | 2 = 1. There exists some λ ∈ C * such that Computing the expression z 2 2 − z 1 z 3 on both sides implies μ = λ 2(a1+a2) μ . We set z = λ a1 z , w = λ a2 w , μ = μ = λ 2(a1+a2) μ and obtain Computing the expressions z 1 + μz 3 and z 3 + μz 1 on both sides yields Because |μ| < 1, it follows that z 2 = λ 2a1 z 2 and w 2 = λ 2a2 w 2 , and hence z = ε z λ a1 z = ε z z and w = ε w λ a2 w = ε w w for some ε z ,ε w ∈ {±1}. Plugging this into the third component of equation (4.9), we get zw + zwμ = ε z ε w (zw + zwμ).
If z = 0 = w, then the expression on the left-hand side is nontrivial because |μ| < 1, and then ε z and ε w have the same sign. In this case it follows, perhaps after replacing λ by −λ, that z = λ a1 z , w = λ a2 w and hence [z : w : μ] = [z : w : μ ]. Otherwise we can draw the same conclusion, again perhaps after replacing λ by −λ.
We have shown that Ξ is a bijective immersion onto the complement of j S 2 in CP (a 1 ,(a 1 + a 2 )/2,a 2 ). It follows from the local structure of such maps that the inverse is smooth as well. Hence the map Ξ is a diffeomorphism onto the complement of j S 2 .

Projective transformations
Let O be an orbifold equipped with a torsion-free connection ∇ on its tangent bundle. A projective transformation for (O,∇) is a diffeomorphism Ψ : O → O which sends geodesics of ∇ to geodesics of ∇ up to parametrisation. In the case where O is a smooth manifold, the group of projective transformations of ∇ is known to be a Lie group (see, e.g., [23]). In our setting, the projective transformations of the Besse orbifold metric on CP(a 1 ,a 2 ) also form a Lie group, since the automorphisms of the associated generalised path geometry form a Lie group (see [21] for details). Moreover, a vector field is called projective if its (local) flow consists of projective transformations. Clearly, if ∇ is a Levi-Civita connection for some Riemannian metric g, then every Killing vector field for g is a projective vector field. The set of vector fields for ∇ form a Lie algebra given by the solutions of a linear second-order partial differential equation system of finite type. In the case of two dimensions and writing a projective vector field as W = W 1 (x,y) ∂ ∂x + W 2 (x,y) ∂ ∂y for local coordinates (x,y) : U → R 2 and real-valued functions W i on U , the partial differential equation system is [12] where R 0 = −Γ 2 11 , and where Γ i jk denote the Christoffel symbols of ∇ with respect to (x,y). In order to show that the deformations we are going to construct in Section 4.6 are nontrivial, we need to know that the identity component of the group of projective transformations of (CP(a 1 ,a 2 ),g) consists solely of isometries. Up to rescaling, any rotationally symmetric Besse metric on CP(a 1 ,a 2 ) is isometric to the metric completion of one of the following examples (see [ where (r,φ) ∈ (0,π) × ([0,2π]/0 ∼ 2π). Proof. Since a 1 > a 2 ≥ 1, every projective transformation τ fixes the singular point of order a 1 , the north pole (r = 0), and hence also its antipodal point of order a 2 , the south pole (r = π). Moreover, it leaves the unique exceptional geodesic, the equator (r = π/2), invariant. After composition with an isometry, we can assume that τ fixes a point x 0 on the equator. Then τ also leaves invariant the minimising geodesic between x 0 and the north pole. Because a 1 > 2, it follows that the differential of τ at the north pole is a homothety -that is, it scales by some factor λ > 0. Therefore, τ in fact leaves invariant all geodesics starting at the north pole and consequently fixes the equator pointwise. In particular, the derivative of τ in the east-west direction along the equator is the identity. We write our Besse orbifold metric in coordinates as in equation (4.13). Let x be some point on the equator. We can assume that it has coordinates (r,φ) = (π/2,0). We look at regular unit-speed geodesics γ(s) = (r(s),φ(s)) with φ (0) > 0 that start at x and do not pass the singular points. Let r m be the maximal (or minimal) latitude attained by such a geodesic. By [4,Theorem 4.11] this latitude is attained at a unique value of s during one period. By symmetry and continuity, the corresponding φ-coordinate φ m is constant as long as r (0) does not change its sign. According to Clairaut's relation we have sin 2 (r)φ (s) = sin(r m ) along γ, and the geodesic oscillates between the parallels r = r m and r = π − r m [4, p. 101]. Letγ(s) = r(s),φ(s) be the unit-speed parametrisation of the geodesic τ (γ). Suppose the differential of τ at x scales by a factor of λ > 0 in the north-south direction. Then we havẽ and a corresponding relation between sin (r m ) and sin(r m ) by Clairaut's relation. Therefore, τ maps the curve c : [0,π/2] t → (t,φ m ) to the curvẽ c : [0,π/2] t → arcsin sin(t) λ 2 + (1 − λ 2 ) sin(t) 2 ,φ m , withc (0) = (1/λ ,0). Hence, we have λ = 1/λ . In particular, in our coordinates the differential of τ looks the same at every point of the equator. It follows that τ = τ λ maps