4 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.

4.1 The twistor space

Inspired by the twistorial construction of holomorphic projective structures by Hitchin [19] and LeBrun [26], it was shown in [14, 35] how to construct a “twistor space” for smooth projective structures. Here we restrict our description to the case of an oriented surface \(M\). Let \(\mathsf{J}_+(M) \to M\) denote the fibre bundle whose fibre at \(x \in M\) consists of the orientation compatible linear complex structures on \(T_xM\). By definition, the sections of \(\mathsf{J}_+(M) \to M\) are in bijective correspondence with the (almost) complex structures on \(M\) that induce the given orientation.

The choice of a torsion-free connection \(\nabla\) on \(TM\) allows to define an integrable almost complex structure on \(\mathsf{J}_+(M)\) which depends only on the projective equivalence class \([\nabla]\) of \(\nabla\). The projective equivalence class \([\nabla]\) of \(\nabla\) consists of all torsion-free connections on \(TM\) having the same unparametrised geodesics as \(\nabla\). We refer to the resulting complex surface \(\mathsf{J}_+(M)\) as the twistor space of \((M,[\nabla])\). In [31], it is shown that a Weyl connection in the projective equivalence class \([\nabla]\) corresponds to a section of \(\mathsf{J}_+(M) \to M\) 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, \(\mathbb{CP}^1\) equipped with the projective structure arising from the Levi-Civita connection of the Fubini–Study metric – the twistor space \(\mathsf{J}_+(S^2)\) is biholomorphic to \(\mathbb{CP}^2\setminus\mathbb{RP}^2\). Here we think of \(\mathbb{RP}^2\) as sitting inside \(\mathbb{CP}^2\) 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 \(\mathbb{CP}^2\setminus\mathbb{RP}^2\), see [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 [9].

The construction of the twistor space can still be carried out for the case of a projective structure \([\nabla]\) on an oriented orbifold \(\mathcal{O}\). Again, sections of \(\mathsf{J}_+(\mathcal{O})\to \mathcal{O}\) having holomorphic image correspond to Weyl connections in \([\nabla]\). Since the spindle orbifold \(S^2(a_1,a_2)\) may also be thought of as the weighted projective line \(\mathbb{CP}(a_1,a_2)\) with weights \((a_1,a_2)\) (see 4.2), one would expect that \(\mathsf{J}_+(\mathbb{CP}(a_1,a_2))\) can be embedded holomorphically into the weighted projective plane, where we equip \(\mathbb{CP}(a_1,a_2)\) 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 4.4. However, a difficulty that arises is that there is more than one natural candidate for the Fubini–Study metric on \(\mathbb{CP}(a_1,a_2)\). We will next identify the correct metric for our purposes.

4.2 The Fubini–Study metric on the weighted projective line

The (complex) weighted projective space is the quotient of \(\mathbb{C}^n\setminus\{0\}\) by \(\mathbb{C}^*\), where \(\mathbb{C}^*\) acts with weights \((a_1,\ldots,a_n) \in \mathbb{N}^n\), that is, by the rule \[z \cdot (z_1,\ldots,z_n)=(z^{a_1}z_1,\ldots,z^{a_n}z_n)\] for all \(z \in \mathbb{C}^*\) and \((z_1,\ldots,z_n) \in \mathbb{C}^n\setminus\{0\}\). It inherits a natural quotient complex structure from \(\mathbb{C}^n\). We denote the projective space with weights \((a_1,\ldots,a_n)\) by \(\mathbb{CP}(a_1,\ldots,a_n)\). Clearly, taking all weights equal to one gives 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 co-prime with \(a_1\geqslant a_2\) and both numbers odd.

For what follows we would like to have an explicit Besse orbifold metric on \(\mathbb{CP}(a_1,a_2)\) which induces the quotient complex structure of \(\mathbb{CP}(a_1,a_2)\). The quotient Besse orbifold metric on \(S^2(a_1,a_2)\) described in 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]). In fact, since the biholomorphism group of \(\mathbb{CP}(a_1,a_2)\) for \((a_1,a_2)\neq (1,1)\) 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 \(\mathbb{CP}(a_1,a_2)\) as a quotient of \(\mathrm{SU}(2)\). In particular, we will identify \(\mathrm{SU}(2)\) as an \((a_1+a_2)\)-fold cover of the unit tangent bundle of \(\mathbb{CP}(a_1,a_2)\).

We consider \[\mathrm{SU}(2):=\left\{\begin{pmatrix} z& -\overline w\\ w& \overline z\end{pmatrix} : (z,w) \in \mathbb{C}^2, |z|^2+|w|^2=1\right\}\] and think of \(\mathrm{U}(1)\) as the subgroup consisting of matrices of the form \[\mathrm{e}^{\mathrm{i}\vartheta}\simeq \begin{pmatrix} \mathrm{e}^{-\mathrm{i}\vartheta} & 0 \\ 0 & \mathrm{e}^{\mathrm{i}\vartheta}\end{pmatrix}\] for \(\vartheta \in \mathbb{R}\). Consider the smooth \(S^1\)-action \[\tag{4.1} T_{\mathrm{e}^{\mathrm{i}\vartheta}}:=L_{\mathrm{e}^{\mathrm{i}(a_1-a_2)\vartheta/2}}\circ R_{\mathrm{e}^{\mathrm{i}(a_1+a_2)\vartheta/2}} : \mathrm{SU}(2)\to \mathrm{SU}(2)\] for \(\vartheta \in \mathbb{R}\) and where \(L_g\) and \(R_g\) denote left – and right multiplication by the group element \(g \in \mathrm{SU}(2)\). Explicitly, we have \[T_{\mathrm{e}^{\mathrm{i}\vartheta}}\left(\begin{pmatrix} z& -\overline w\\ w& \overline z\end{pmatrix}\right)=\begin{pmatrix} \mathrm{e}^{-\mathrm{i}a_1\vartheta}z& -\mathrm{e}^{\mathrm{i}a_2 \vartheta} \overline w\\\mathrm{e}^{-\mathrm{i}a_2\vartheta} w& \mathrm{e}^{\mathrm{i}a_1\vartheta}\overline z\end{pmatrix},\] and hence the corresponding quotient can be identified with the weighted projective line \(\mathbb{CP}(a_1,a_2)\).

Recall that the Maurer–Cartan form \(\varrho\) is defined as \(\varrho_g(v):=\left(L_{g^{-1}}\right)^{\prime}_g(v)\) where \(v \in T_g\mathrm{SU}(2)\). Writing the Maurer–Cartan form \(\varrho\) of \(\mathrm{SU}(2)\) as \[\varrho=\begin{pmatrix} -\mathrm{i}\kappa & -\overline\varphi \\ \varphi & \mathrm{i}\kappa\end{pmatrix}\] for a real-valued \(1\)-form \(\kappa\) and a complex-valued \(1\)-form \(\varphi\) on \(\mathrm{SU}(2)\), the structure equation \(\mathrm{d}\varrho+\varrho\wedge\varrho=0\) is equivalent to \[\mathrm{d}\varphi=-2\mathrm{i}\kappa\wedge\varphi\quad \text{and} \quad \mathrm{d}\kappa=-\mathrm{i}\varphi\wedge\overline{\varphi}\] Since \[\varrho=\begin{pmatrix} z& -\overline w\\ w& \overline z\end{pmatrix}^{-1}\mathrm{d}\begin{pmatrix} z& -\overline w\\ w& \overline z\end{pmatrix}\] we also obtain \[\mathrm{d}z=-\overline{w}\varphi-\mathrm{i}z \kappa\quad \text{and}\quad \mathrm{d}w= \overline{z}\varphi-\mathrm{i}w \kappa\] as well as \[\tag{4.2} \varphi=z\mathrm{d}w-w \mathrm{d}z\quad \text{and} \quad \kappa=\mathrm{i}\left(\overline{z}\mathrm{d}z+\overline{w}\mathrm{d}w\right).\]

In order to compute a basis for the \(1\)-forms that are semi-basic for the projection \(\pi_{a_1,a_2} : \mathrm{SU}(2)\to \mathbb{CP}(a_1,a_2)\), we evaluate the Maurer–Cartan form on the infinitesimal generator \(Z:=\left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{t=0}T_{\mathrm{e}^{\mathrm{i}t}}\) of the \(S^1\)-action. We obtain \[\begin{aligned} \varrho(Z)&=\begin{pmatrix} z& -\overline w\\ w& \overline z\end{pmatrix}^{-1}\left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{t=0}\begin{pmatrix} \mathrm{e}^{-\mathrm{i}a_1 t}z& -\mathrm{e}^{\mathrm{i}a_2 t} \overline w\\\mathrm{e}^{-\mathrm{i}a_2 t} w& \mathrm{e}^{\mathrm{i}a_1 t}\overline z\end{pmatrix}\\ &=\begin{pmatrix} -\mathrm{i}(a_1|z|^2+a_2|w|^2) & \mathrm{i}(a_1-a_2)\overline{zw} \\ \mathrm{i}(a_1-a_2) zw& \mathrm{i}(a_1|z|^2+a_2|w|^2) \end{pmatrix},\end{aligned}\] so that \(\kappa(Z)=\mathscr{U}\) and \(\varphi(Z)=\mathscr{V}\), where \[\mathscr{U}=a_1|z|^2+a_2|w|^2\quad\text{and} \quad \mathscr{V}=\mathrm{i}(a_1-a_2)zw.\] Consequently, we see that the complex-valued \(1\)-form \[\tag{4.3} \omega=\mathscr{U}\varphi-\mathscr{V}\kappa\] satisfies \(\omega(Z)=0\) and hence – by definition – is semibasic for the projection \(\pi_{a_1,a_2} : \mathrm{SU}(2)\to \mathbb{CP}(a_1,a_2)\). Because of the left-invariance of \(\varrho\) we have \(T_{\mathrm{e}^{\mathrm{i}\vartheta}}^*\varrho=(R_{\mathrm{e}^{\mathrm{i}(a_1+a_2)\vartheta/2}})^*\varrho\) and hence \[\begin{aligned} (T_{\mathrm{e}^{\mathrm{i}\vartheta}})^*\varrho&=%(R_{\e^{\i(a_1+a_2)\vartheta/2}})^*\varrho\\ \begin{pmatrix} -\mathrm{i}\kappa & \mathrm{e}^{\mathrm{i}(a_1+a_2)\vartheta}\overline\varphi \\ \mathrm{e}^{-\mathrm{i}(a_1+a_2)\vartheta}\varphi & \mathrm{i}\kappa \end{pmatrix},\end{aligned}\] where we have used the equivariance property \(R_g^*\varrho=g^{-1}\varrho g\) which holds for all \(g \in \mathrm{SU}(2)\). Since \((T_{\mathrm{e}^{\mathrm{i}\vartheta}})^*\mathscr{U}=\mathscr{U}\) and \((T_{\mathrm{e}^{\mathrm{i}\vartheta}})^*\mathscr{V}=\mathrm{e}^{-\mathrm{i}(a_1+a_2)\vartheta}\mathscr{V}\), we obtain \[\tag{4.4} \begin{align} (T_{\mathrm{e}^{\mathrm{i}\vartheta}})^*\omega&=\mathrm{e}^{-\mathrm{i}(a_1+a_2)\vartheta}\omega\\ (T_{\mathrm{e}^{\mathrm{i}\vartheta}})^*\zeta&=\zeta, \end{align}\] where \[\tag{4.5} \zeta=\kappa/\mathscr{U}.\] Infinitesimally we obtain \[\mathrm{d}\omega=-\mathrm{i}(a_1+a_2)\zeta\wedge\omega\quad \text{and}\quad \mathrm{d}\zeta=-\frac{\mathrm{i}K_g}{2(a_1+a_2)}\omega\wedge\overline{\omega}\] with \(K_g=2(a_1+a_2)/\mathscr{U}^3\). For later usage we also record the identities \[\tag{4.6} \begin{align} \mathrm{d}z&=-(\overline{w}/\mathscr{U})\omega-\mathrm{i}a_1 z\zeta,\\ \mathrm{d}w&=(\overline{z}/\mathscr{U})\omega-\mathrm{i}a_2 w\zeta. \end{align}\] Observe that if we write \(\omega=\alpha+\mathrm{i}\beta\) for real-valued \(1\)-forms \(\alpha,\beta\) on \(\mathrm{SU}(2)\), then we obtain the structure equations \[\begin{aligned} \mathrm{d}\alpha=-(a_1+a_2)\beta\wedge\zeta, \quad \mathrm{d}\beta&=-(a_1+a_2)\zeta\wedge\alpha, \quad \mathrm{d}\zeta=-\frac{K_g}{(a_1+a_2)}\alpha\wedge\beta. \end{aligned}\] Now since \((a_1,a_2)\) are co-prime, the cyclic group \(\mathbb{Z}_{a_1+a_2}\subset S^1\) of order \(a_1+a_2\) acts freely on \(\mathrm{SU}(2)\). Therefore, the quotient \(\mathrm{SU}(2)/\mathbb{Z}_{a_1+a_2}\) is a smooth manifold equipped with a smooth action of \(S^1/\mathbb{Z}_{a_1+a_2}\simeq S^1\) which we denote by \(\underline{T}_{\mathrm{e}^{\mathrm{i}\vartheta}}\). Writing \(\upsilon : \mathrm{SU}(2)\to \mathrm{SU}(2)/\mathbb{Z}_{a_1+a_2}\) for the quotient projection, we have the equivariance property \[\upsilon \circ T_{\mathrm{e}^{\mathrm{i}\vartheta}}=\underline{T}_{\mathrm{e}^{\mathrm{i}(a_1+a_2)\vartheta}}\circ \upsilon\] for all \(\mathrm{e}^{\mathrm{i}\vartheta} \in S^1\). Denoting the infinitesimal generator of the \(S^1\) action \(\underline{T}_{\mathrm{e}^{\mathrm{i}\vartheta}}\) by \(\underline{Z}\) we thus obtain \[\upsilon^{\prime}(Z)=(a_1+a_2)\underline{Z}.\] Likewise, denoting the framing of \(\mathrm{SU}(2)\) that is dual to \((\alpha,\beta,\zeta)\) by \((A,B,Z)\), the equivariance properties (4.4) imply that we obtain unique well defined vector fields \(\underline{A},\underline{B}\) on \(\mathrm{SU}(2)/\mathbb{Z}_{a_1+a_2}\) so that \[\upsilon^{\prime}(A)=\underline{A}\quad\text{and}\quad \upsilon^{\prime}(B)=\underline{B}.\] In particular, the structure equations for \((\alpha,\beta,\zeta)\) imply the commutator relation \[[\underline{Z},\underline{A}]=\underline{B}\quad\text{and}\quad [\underline{Z},\underline{B}]=-\underline{A} \quad \text{and} \quad[\underline{A},\underline{B}]=K_g\underline{Z},\] where, by abuse of notation, we here think of \(K_g\) as a function on the quotient \(\mathrm{SU}(2)/\mathbb{Z}_{a_1+a_2}\). These commutator relations in turn imply that the coframing \((\underline{\alpha},\underline{\beta},\underline{\zeta})\) of \(\mathrm{SU}(2)/\mathbb{Z}_{a_1+a_2}\) that is dual to \((\underline{A},\underline{B},\underline{Z})\) defines a generalized 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 \(\mathbb{CP}(a_1,a_2)\), so that \(\pi^*g=\underline{\alpha}\otimes\underline{\alpha}+\underline{\beta}\otimes\underline{\beta}\) and so that the area form of \(g\) satisfies \(\pi^*d\sigma_g=\underline{\alpha}\wedge\underline{\beta}\). Here \(\pi : \mathrm{SU}(2)/\mathbb{Z}_{a_1+a_2} \to \mathbb{CP}(a_1,a_2)\) denotes the quotient projection with respect to the \(S^1\) action \(\underline{T}_{\mathrm{e}^{\mathrm{i}\vartheta}}\). Moreover, the map \[\Phi : \mathrm{SU}(2)/\mathbb{Z}_{a_1+a_2} \to T\mathbb{CP}(a_1,a_2), \quad u\mapsto \pi^{\prime}_u(\underline{A}(u))\] is a diffeomorphism onto the unit tangent bundle \(S\mathbb{CP}(a_1,a_2)\) of \(g\) which has the property that the pullback of the canonical coframing on \(S\mathbb{CP}(a_1,a_2)\) yields \((\underline{\alpha},\underline{\beta},\underline{\zeta})\). Thus, we will henceforth identify the unit tangent bundle \(S\mathbb{CP}(a_1,a_2)\) of \((\mathbb{CP}(a_1,a_2),g)\) with \(\mathrm{SU}(2)/\mathbb{Z}_{a_1+a_2}\).

We will next show that \(g\) is a Besse orbifold metric. For an element \(y\) in the Lie algebra \(\mathfrak{su}(2)\) of \(\mathrm{SU}(2)\) we let \(Y_y\) denote the vector field on \(\mathrm{SU}(2)\) generated by the flow \(R_{\exp(ty)}\). Recall that the Maurer–Cartan form \(\varrho\) satisfies \(\varrho(Y_y)=y\) for all \(y \in \mathfrak{su}(2)\). It follows that the basis \[e_1=\begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix}\quad \text{and}\quad e_2=\begin{pmatrix}0 & \mathrm{i}\\ \mathrm{i}& 0\end{pmatrix}\quad \text{and} \quad e_3=\begin{pmatrix}-\mathrm{i}& 0 \\ 0 & \mathrm{i}\end{pmatrix}\] of \(\mathfrak{su}(2)\) yields a framing \((Y_{e_1},Y_{e_2},Y_{e_3})\) of \(\mathrm{SU}(2)\) which is dual to the coframing \((\mathrm{Re}(\varphi),\mathrm{Im}(\varphi),\kappa)\). Therefore, using (4.3), (4.5) and the definition of \(\alpha,\beta\), we obtain \(A=Y_{e_1}/\mathscr{U}\). The flow of \(Y_{e_1}\) is given by \(R_{\exp(te_1)}\) and hence periodic with period \(2\pi\). Recall that the geodesic flow of the metric \(g\) on \(\mathbb{CP}(a_1,a_2)\) is \(\underline{A}\). Thinking of \(\mathscr{U}\) as a function on \(S\mathbb{CP}(a_1,a_2)\), we have \[\underline{A}=\upsilon^{\prime}(A)=\upsilon^{\prime}(Y_{e_1})/\mathscr{U}\] and hence \(g\) is a Besse orbifold metric.

In complex notation, we have \((\pi_{a_1,a_2})^*g=\omega\circ\overline{\omega}\), where \(\circ\) denotes the symmetric tensor product and \(\omega=\alpha+\mathrm{i}\beta\). The complex structure on \(\mathbb{CP}(a_1,a_2)\) defined by \(g\) and the orientation is thus characterized by the property that its \((1,\! 0)\)-forms pull-back to \(\mathrm{SU}(2)\) to become complex multiples of \(\omega\). In particular, this complex structure coincides with the natural quotient complex structure of \(\mathbb{CP}(a_1,a_2)\) since \(\omega\) is a linear combination of \(\mathrm{d}z\) and \(\mathrm{d}w\), see (4.2).

Finally, observe that \(K_g\) is strictly positive. We have thus shown:

4.3 Constructing the twistor space

We will next construct the twistor space \(\mathsf{J}_+(\mathcal{O})\) in the case of the weighted projective line \(\mathbb{CP}(a_1,a_2)\) and where \([\nabla]\) is the projective equivalence class of the Levi-Civita connection of the Fubini–Study metric on \(\mathbb{CP}(a_1,a_2)\) constructed in Lemma 4.1. As we will see, in this special case the twistor space \(\mathsf{J}_+(\mathbb{CP}(a_1,a_2))\) can indeed be embedded into the weighted projective plane \(\mathbb{CP}(a_1,(a_1+a_2)/2,a_2)\). We will henceforth also write \(\mathsf{J}_+\) for the twistor space, whenever the underlying orbifold is clear from the context.

In the case of a Riemann surface \((M,J)\) the orientation inducing complex structures on \(M\) are in one-to-one correspondence with the Beltrami differentials on \((M,J)\). A Beltrami differential \(\mu\) on \(M\) is a section of \(B=K_{M}^{-1}\otimes \overline{K_M}\), where \(K_M=(T^{*}_{\mathbb{C}}M)^{1,0}\) denotes the canonical bundle of \((M,J)\) with inverse \(K_M^{-1}\) and where \(\overline{K_M}\) denotes its complex conjugate bundle. The line bundle \(B\) carries a natural Hermitian bundle metric \(h\) and Beltrami differentials are precisely those sections which satisfy the condition \(h(\mu,\mu)<1\) at each point of \(M\). This identifies \(\mathsf{J}_+(M)\) with the open unit disk bundle in \(B\). We refer the reader to [20] for additional details.

In the case of the orbifold \(\mathbb{CP}(a_1,a_2)\), 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 \(\mathrm{SU}(2)\times \mathbb{C}\).

Recall that the metric \(g\) on \(\mathbb{CP}(a_1,a_2)\) satisfies \((\pi_{a_1,a_2})^*g=\omega\circ\overline{\omega}\) and \((\pi_{a_1,a_2})^*d\sigma_g=\frac{\mathrm{i}}{2}\omega\wedge\overline{\omega}\), where \(\omega=\alpha+\mathrm{i}\beta\). In particular, the complex structure on \(\mathbb{CP}(a_1,a_2)\) induced by \(g\) and the orientation has the property that its \((1,\! 0)\)-forms pull-back to \(\mathrm{SU}(2)\) to become complex multiples of \(\omega\).

Moreover, recall that from (4.4) that we have \((T_{\mathrm{e}^{\mathrm{i}\vartheta}})^*\omega=\mathrm{e}^{-\mathrm{i}(a_1+a_2)\vartheta}\omega\). We thus define \[K_{\mathbb{CP}(a_1,a_2)}=\mathrm{SU}(2)\times_{S^1}\mathbb{C},\] where \(S^1\) acts on \(\mathrm{SU}(2)\) by (4.1) and on \(\mathbb{C}\) with spin \((a_1+a_2)\), that is, by the rule \[\mathrm{e}^{\mathrm{i}\vartheta}\cdot z=\mathrm{e}^{\mathrm{i}(a_1+a_2)\vartheta}z.\] Likewise, \(K^{-1}_{\mathbb{CP}(a_1,a_2)}\) arises from acting with spin \(-(a_1+a_2)\) and \(\overline{K_{\mathbb{CP}(a_1,a_2)}}\) 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 \(B=\mathrm{SU}(2)\times_{S^1} \mathbb{C}\), where now \(S^1\) acts with spin \(-2(a_1+a_2)\) on \(\mathbb{C}\). The Hermitian bundle metric \(h\) on \(B\) arises from the usual Hermitian inner product on \(\mathbb{C}\) and hence we obtain \[\mathsf{J}_+(\mathbb{CP}(a_1,a_2))=\mathrm{SU}(2)\times_{S^1}\mathbb{D},\] where \(\mathbb{D}\subset \mathbb{C}\) denotes the open unit disk and \(S^1\) acts with spin \(-2(a_1+a_2)\) on \(\mathbb{D}\).

We now define an almost complex structure \(\mathfrak{J}\) on \(\mathsf{J}_+(\mathbb{CP}(a_1,a_2))\). On \(\mathrm{SU}(2)\times \mathbb{D}\) we consider the complex-valued \(1\)-forms \[\xi_1=\omega+\mu\overline{\omega}\quad\text{and}\quad \xi_2=\mathrm{d}\mu+2(a_1+a_2)\mathrm{i}\mu\zeta,\] where \(\mu\) denotes the standard coordinate on \(\mathbb{D}\). Abusing notation and writing \(T_{\mathrm{e}^{\mathrm{i}\varphi}}\) for the combined \(S^1\)-action on \(\mathrm{SU}(2)\times \mathbb{D}\), we obtain \[(T_{\mathrm{e}^{\mathrm{i}\varphi}})^*\xi_1=\mathrm{e}^{-\mathrm{i}(a_1+a_2)\vartheta}\xi_1\quad \text{and}\quad \left(T_{\mathrm{e}^{\mathrm{i}\varphi}}\right)^*\xi_2=\mathrm{e}^{-2\mathrm{i}(a_1+a_2)\vartheta}\xi_2.\] Furthermore, by construction, the forms \(\xi_1\) and \(\xi_2\) are semi-basic for the projection \(\mathrm{SU}(2)\times \mathbb{D} \to \mathsf{J}_+(\mathbb{CP}(a_1,a_2))\). It follows that there exists a unique almost complex structure \(\mathfrak{J}\) on \(\mathsf{J}_+(\mathbb{CP}(a_1,a_2))\) whose \((1,\! 0)\)-forms pull-back to \(\mathrm{SU}(2)\times \mathbb{D}\) to become linear combinations of \(\xi_1\) and \(\xi_2\). Finally, in [34] it is shown that the so constructed almost complex structure agrees with the complex structure on the twistor space associated to \((\mathbb{CP}(p,q),[{}^g\nabla])\) where \({}^g\nabla\) denotes the Levi-Civita connection of \(g\).

4.4 Embedding the twistor space

Recall that the twistor space of the \(2\)-sphere \(\mathsf{J}_+(S^2)\) equipped with the complex structure coming from the projective structure of the standard metric maps biholomorphically onto \(\mathbb{CP}^2\setminus \mathbb{RP}^2\). The map arises as follows. Consider \(S^2\) as the unit sphere in \(\mathbb{R}^3\) and identify the tangent space \(T_eS^2\) to an element \(e \in S^2\) with the orthogonal complement \(\{e\}^{\perp}\subset \mathbb{R}^3\). Then an orientation compatible complex structure \(J\) on \(T_eS^2\) is mapped to the element \([v+iJv]\in \mathbb{CP}^2\setminus \mathbb{RP}^2\) where \(v\in T_e S^2\) is any non-zero tangent vector. With respect to our present model of \(\mathsf{J}_+(S^2)\) as an associated bundle this map takes the following explicit form \[\begin{array}{cccl} \Xi: &\mathsf{J}_+(S^2)=\mathrm{SU}(2)\times_{S^1} \mathbb{D} & \rightarrow& \mathbb{C}\mathbb{P}^2 \\ & [z:w:\mu] & \mapsto& [z^2-\mu\overline{w}^2:zw+\overline{z}\overline{w}\mu:w^2-\mu\overline{z}^2] \end{array}\] after applying a linear coordinate change (see Appendix A). In this new coordinate system the real projective space \(\mathbb{RP}^2\) sits inside \(\mathbb{CP}^2\) as the image of the unit sphere in \(\mathbb{C}\times \mathbb{R}\) under the map \(j=\pi\circ\hat j: \mathbb{C}\times \mathbb{R}\rightarrow\mathbb{CP}^2\) where \(j: \mathbb{C}\times \mathbb{R}\rightarrow\mathbb{C}^3\) is defined as \(j(z,t)= (z,it,\overline{z})\) and where \(\pi:\mathbb{C}^3\backslash \{0\} \rightarrow\mathbb{CP}^2\) is the quotient projection. Note that for \(\mu=0\) the map \(\Xi\) restricts to the Veronese embedding of \(\mathbb{CP}^1\) into \(\mathbb{CP}^2\).

We observe that in the weighted case the very same map \(\Xi\) also defines a smooth map of orbifolds. In fact, we are going to show the following statement in a sequence of lemmas.

In the following we also describe \(\mathrm{SU}(2)\times_{S^1} \mathbb{D}\) as a quotient of \(\mathbb{C}^2\backslash\{0\} \times \mathbb{C}\) by the respective weighted \(\mathbb{C}^*\)-action. Here \(\lambda \in \mathbb{C}^*\) acts as \(\lambda/|\lambda|\) on the second factor. Then the map \(\Xi\) is covered by the \(\mathbb{C}^*\)-equivariant map \[\begin{array}{cccl} \hat \Xi: &\mathbb{C}^2\backslash\{0\} \times \mathbb{D} & \rightarrow& \mathbb{C}^3 \backslash\{0\}\\ & (z,w,\mu) & \mapsto& (z^2-\mu\overline{w}^2,zw+\overline{z}\overline{w}\mu,w^2-\mu\overline{z}^2). \end{array}\] Since we already know that the map \(\Xi\) is an immersion in the unweighted case, and since the \(\mathbb{C}^*\)-actions on \(\mathbb{C}^2\backslash\{0\} \times \mathbb{D}\) and on \(\mathbb{C}^3 \backslash\{0\}\) do not have fixed points, it follows that the map \(\hat \Xi\), and hence also the map \(\Xi\) 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 map \(\hat \Xi\) is given by \(\mathrm{det}(J(z,w,\mu))=4(1-|\mu|^2)(|z|^2+|w|^2)^4\).

There are different ways to continue the proof of Proposition 4.4. For instance, one can show that the map \(\Xi\) extends to a smooth orbifold immersion of a certain compactification of \(\mathsf{J}_+\) onto \(\mathbb{C}\mathbb{P}(a_1,(a_1+a_2)/2,a_2)\) so that the complement of \(\mathsf{J}_+\) is mapped onto \(j(S^2)\). Compactness and the fact that \(\mathbb{C}\mathbb{P}(a_1,(a_1+a_2)/2,a_2)\) 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.

We have shown that \(\Xi\) is a bijective immersion onto the complement of \(j(S^2)\) in \(\mathbb{C}\mathbb{P}(a_1,(a_1+a_2)/2,a_2)\). It follows from the local structure of such maps that the inverse is smooth as well. Hence, the map \(\Xi\) is a diffeomorphism onto the complement of \(j(S^2)\).

In order to complete the proof of Proposition 4.4 it remains to verify that the map \(\Xi\) is holomorphic. The map \(\Xi\) is holomorphic if and only if it pulls back \((1,0)\)-forms to \((1,0)\)-forms. By definition the \((1,0)\)-forms of \(\mathsf{J}_+\) pull back to linear combinations of \(\xi_1\) and \(\xi_2\) on \(\mathrm{SU}(2)\times \mathbb{D}\). On the other hand, the \((1,0)\)-forms on \(\mathbb{C}\mathbb{P}(a_1,(a_1+a_2)/2,a_2)\) pull back to the \((1,0)\)-forms on \(\mathbb{C}^3\backslash \{0\}\) which vanish on the infinitesimal generator of the defining \(\mathbb{C}^*\)-action on \(\mathbb{C}^3\backslash \{0\}\). The latter are linear combinations of the complex valued \(1\)-forms \[\Pi_1=a_2z_3\mathrm{d}z_1-a_1z_1\mathrm{d}z_3 \quad \text{and} \quad \Pi_2=\left(\frac{a_1+a_2}{2}\right)z_2\mathrm{d}z_1-a_1z_1\mathrm{d}z_2.\] Hence, we need to show that \(\Psi=\hat{\Xi}|_{\mathrm{SU}(2)\times \mathbb{D}} : \mathrm{SU}(2)\times \mathbb{D}\to \mathbb{C}^{3}\setminus\{0\}\) satisfies \[\tag{4.9} \xi_1\wedge \xi_2 \wedge \Psi^*\Pi_1 = 0 \quad \text{and} \quad \xi_1\wedge \xi_2 \wedge \Psi^*\Pi_2 = 0.\] Recall the identities (4.6) \[\begin{aligned} \mathrm{d}z&=-(\overline{w}/\mathscr{U})\omega-\mathrm{i}a_1 z\zeta,\\ \mathrm{d}w&=(z/\mathscr{U})\omega-\mathrm{i}a_2 w\zeta. \end{aligned}\] Using these identities a tedious – but straightforward – calculation gives \[\begin{gathered} \Psi^*\Pi_1=-\frac{2}{\mathscr{U}}\left(a_1(z^2-\mu\overline{w}^2)\overline{z}w+a_2(w^2-\mu\overline{z}^2)z\overline{w}\right)\xi_1\\ +\left(a_1(z^2-\mu\overline{w}^2)\overline{z}^2-a_2(w^2-\mu\overline{z}^2)\overline{w}^2\right)\xi_2\end{gathered}\] and \[\begin{gathered} \Psi^*\Pi_2=-\frac{1}{\mathscr{U}}\left(a_1(z^3\overline{z}+\mu w\overline{w}^3)+a_2(zw+\mu\overline{zw})z\overline{w}\right)\xi_1\\ +\frac{1}{2}\left(a_1(\mu\overline{z}\overline{w}^2-|w|^2z-2|z|^2z)\overline{w}-a_2(\mu\overline{zw}+zw)\overline{w}^2\right)\xi_2,\end{gathered}\] thus (4.9) is satisfied and \(\Xi\) is a biholomorphism. This finishes the proof of Proposition 4.4.

4.5 Projective transformations

Let \(\mathcal{O}\) be an orbifold equipped with a torsion-free connection \(\nabla\) on its tangent bundle. A projective transformation for \((\mathcal{O},\nabla)\) is a diffeomorphism \(\Psi : \mathcal{O} \to \mathcal{O}\) which sends geodesics of \(\nabla\) to geodesics of \(\nabla\) up to parametrisation. In the case where \(\mathcal{O}\) is a smooth manifold the group of projective transformations of \(\nabla\) is known to be a Lie group (see for instance [23]). In our setting, the projective transformations of the Besse orbifold metric on \(\mathbb{CP}(a_1,a_2)\) also form a Lie group, since the automorphisms of the associated generalized 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 \(\nabla\) 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 \(\nabla\) form a Lie algebra given by the solutions of a linear second order PDE system of finite type. In the case of two dimensions and writing a projective vector field as \(W=W^1(x,y)\frac{\partial}{\partial x}+W^2(x,y)\frac{\partial}{\partial y}\) for local coordinates \((x,y) : U \to \mathbb{R}^2\) and real-valued functions \(W^i\) on \(U\), the PDE system is [12] \[\tag{4.10} \begin{align} 0&=W^2_{xx}-2R^0W^1_x-R^1W^2_x+R^0W^2_y-R^0_xW^1-R^0_yW^2,\\ 0&=-W^1_{xx}+ 2W^2_{xy}-R^1W^1_x-3R^0W^1_y-2R^2W^2_x-R^1_xW^1-R^1_yW^2,\\ 0&=-2W^1_{xy}+W^2_{yy}-2R^1W^1_y-3R^3W^2_x-R^2W^2_y-R^2_xW^1-R^2_yW^2,\\ 0&=-W^1_{yy}+R^3W^1_x-R^2W^1_y-2R^3W^2_y-R^3_xW^1-R^3_yW^2, \end{align}\] where \[R^0=-\Gamma^2_{11},\quad R^1=\Gamma^1_{11}-2\Gamma^2_{12}, \quad R^2=2\Gamma^1_{12}-\Gamma^2_{22},\quad R^3=\Gamma^1_{22}\] and where \(\Gamma^i_{jk}\) denote the Christoffel symbols of \(\nabla\) with respect to \((x,y)\).

In order to show that the deformations we are going to construct in 4.6 are nontrivial, we need to know that the identity component of the group of projective transformations of \((\mathbb{CP}(a_1,a_2),g)\) consists solely of isometries. Up to rescaling, any rotationally symmetric Besse metric on \(\mathbb{CP}(a_1,a_2)\) is isometric to the metric completion of one of the following examples (see [24] and [4]): let \(h: [-1,1] \rightarrow(- \frac{a_1+a_2}{2},\frac{a_1+a_2}{2})\) be a smooth, odd function with \(h(1)=\frac{a_1-a_2}{2}=-h(-1)\) and let a Riemannian metric on \((0,\pi)\times ([0,2\pi]/0 \sim 2\pi) \ni (r,\phi)\) be defined by \[\tag{4.11} g_h=\left( \frac{a_1+a_2}{2}+h(\cos(r))\right)^2 d\theta^2 + \sin^2(r) d\phi^2.\] Our specific Besse orbifold metric \(g\) on \(\mathbb{CP}(a_1,a_2)\) takes the form \(g_h/4\) with \(h(x)=\frac{1}{2}(a_1-a_2)x\) with respect to the parametrization \[\tag{4.12} [z:w]=\left[\cos(r/2)\mathrm{e}^{-\mathrm{i}\phi/(a_1+a_2)}:\sin(r/2)\mathrm{e}^{\mathrm{i}\phi/(a_1+a_2)}\right]\] where \((r,\phi) \in (0,\pi)\times([0,2\pi]/0 \sim 2\pi)\).

4.6 Deformations of Finsler metrics and the Veronese embedding

Recall from 4.1 that sections of \(\mathsf{J}_+\to \mathbb{CP}(a_1,a_2)\) with holomorphic image correspond to Weyl connections in \([\nabla]\). Moreover, a projective transformation gives rise to a biholomorphism of \(\mathsf{J}_+\), and it pulls-back a Weyl connection \(\nabla_1\in [\nabla]\) to \(\nabla_2 \in [\nabla]\) if and only if the corresponding holomorphic curves are mapped onto each other.

Let us identify \(\mathsf{J}_+\) with \(\mathbb{CP}(a_1,(a_1+a_2)/2,a_2)\setminus j(S^2)\) via Proposition 4.4 in the case where \(\nabla=\nabla^g\) for the Besse orbifold metric \(g\) from Lemma 4.1. In this case, the complex structure on \(\mathbb{CP}(a_1,a_2)\) arising from the chosen orientation and the metric \(g\) corresponds to the Veronese embedding \[\begin{array}{cccl} \Theta=\Xi_{|\mathbb{C}\mathbb{P}(a_1,a_2)}: &\mathbb{C}\mathbb{P}(a_1,a_2) & \rightarrow& \mathbb{C}\mathbb{P}(a_1,(a_1+a_2)/2,a_2) \\ & [z:w] & \mapsto& [z^2:zw:w^2]. \end{array}\] We would like to construct deformations of Finsler metrics via deformations of this embedding. Note that the image of the Veronese embedding is defined by the equation \(y^2_2=y_1y_3\), where we use \((y_1,y_2,y_3)\) as coordinates on \(\mathbb{CP}(a_1,(a_1+a_2)/2,a_2)\). An explicit complex one-dimensional family of deformations is given by the equation \(y_2^2=\lambda y_1y_3\) for some \(\lambda \in \mathbb{C}^*\). Choosing \(\lambda\) sufficiently close to \(1\) will cut out a holomorphic curve which continues to be a section of \(\mathsf{J}_+ \to \mathbb{CP}(a_1,a_2)\) and hence corresponds to a positive Weyl connection since the metric \(g\) has strictly positive Gauss curvature. Therefore, according to our Theorem A, small deformations of the Veronese embedding through holomorphic curves give rise to deformations of the Finsler metric dual to \(g\) through Finsler metrics of constant curvature \(1\) and all geodesics closed.

It remains to show that the so obtained Finsler metrics are not all isometric. Again, according to our Theorem A, this amounts to showing that the resulting Weyl structures do not coincide up to an orientation preserving diffeomorphism. Let \(\mathscr{W}_{\lambda_i}\) for \(i=1,2\) denote the Weyl structures corresponding to the deformations by \(\lambda_1\neq\lambda_2\) sufficiently close to \(1\). Suppose \(\Psi : \mathbb{CP}(a_1,a_2) \to \mathbb{CP}(a_1,a_2)\) is an orientation preserving diffeomorphism which identifies \(\mathscr{W}_{\lambda_1}\) with \(\mathscr{W}_{\lambda_2}\). By construction, the Weyl structures \(\mathscr{W}_{\lambda_i}\) have Weyl connections whose geodesics agree with the geodesics of the Besse orbifold metric \(g\) up to parametrisation. Therefore, \(\Psi\) is a projective transformation for the Levi-Civita connection of \(g\) and hence by Lemma 4.9 an isometry for \(g\) up to possibly applying a transformation from a discrete set of non-isometric projective transformations.

Every orientation preserving diffeomorphism of \(\mathbb{CP}(a_1,a_2)\) naturally lifts to a diffeomorphism of \(\mathsf{J}_+\) and in the case of an orientation preserving isometry \(\Upsilon : \mathbb{CP}(a_1,a_2) \to \mathbb{CP}(a_1,a_2)\) the lift \(\mathsf{J}_+ \to \mathsf{J}_+\) is covered by a map \(\mathrm{SU}(2)\times \mathbb{D} \to \mathrm{SU}(2)\times \mathbb{D}\) which is the product of the identity on the \(\mathbb{D}\) factor and the natural lift of \(\Upsilon\) to \(\mathrm{SU}(2)\) on the first factor. With respect to our coordinates (4.12) the isometries generated by the Killing vector field \(\frac{\partial}{\partial \phi}\) lift to \(\mathrm{SU}(2)\) to become left-multiplication by the element \(\mathrm{e}^{\mathrm{i}\vartheta}\). Thus, under our biholomorphism \(\Xi : \mathsf{J}_+ \to \mathbb{CP}(a_1,(a_1+a_2)/2,a_2)\setminus j(S^2)\) lifts of orientation preserving isometries to \(\mathsf{J}_+\) take the form \[[y_1:y_2:y_3] \mapsto [\mathrm{e}^{-2\mathrm{i}\vartheta}y_1:y_2:\mathrm{e}^{2\mathrm{i}\vartheta}y_3]\] for \(\vartheta \in \mathbb{R}\). Observe that each such transformation leaves each member of the family \(y_2^2=\lambda y_1y_3\) invariant. In particular, the deformed Besse–Weyl structures are rotationally symmetric as well. Since \(\Psi\) identifies the two Weyl structures, its lift \(\tilde{\Psi} : \mathsf{J}_+ \to \mathsf{J}_+\) must map the holomorphic curves cut out by \(y_2^2=\lambda_iy_1y_3\) for \(i=1,2\) onto each other and hence \(\Psi\) must be a member of the discrete set of non-isometric projective transformations. Since we have a real two-dimensional family of deformations of the Veronese embedding, we conclude that we have a corresponding real two-dimensional family of non-isometric, rotationally symmetric Finsler metrics of constant curvature \(K=1\) on \(S^2\) and with all geodesics closed.