2 Preliminaries

2.1 Background on orbifolds

For a detailed account on different perspectives on orbifold we refer the reader to e.g. [1, 5, 25, 36]. Here we only quickly recall some basic notions which are relevant for our purpose. An \(n\)-dimensional Riemannian orbifold \(\mathcal{O}^n\) can be defined as a length space such that for each point \(x \in \mathcal{O}\) there exists a neighbourhood \(U\) of \(x\) in \(\mathcal{O}\), an \(n\)-dimensional Riemannian manifold \(M\) and a finite group \(\Gamma\) acting by isometries on \(M\) such that \(U\) and \(M/\Gamma\) are isometric [25]. In this case we call \(M\) a manifold chart for \(\mathcal{O}\). Every Riemannian orbifold admits a canonical smooth structure, i.e., roughly speaking, there exist equivariant, smooth transition maps between manifolds charts. Conversely, every smooth orbifold is “metrizable” in the above sense. 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 \(\Gamma_x\) and is called the local group of \(\mathcal{O}\) at \(x\). A point \(x\in \mathcal{O}\) is called regular if its local group is trivial and otherwise singular.

For example, the metric quotient \(\mathcal{O}_{a}\), \(a=(a_1,a_2)\), of the unit sphere \(S^{3}\subset \mathbb{C}^{2}\) by the isometric action of \(S^1\subset \mathbb{C}\) defined by \[z (z_1,z_2)=(z^{a_1}z_1,z^{a_2}z_2)\] for co-prime numbers \(a_1\geqslant a_2\) 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 a \((a_1,a_2)\)-spindle orbifold. The quotient map \(\pi\) from \(S^3\) to \(\mathcal{O}_{a}\) 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 \(M/\Gamma=U=\pi(V)\) is a chart, and \(\pi|_V\) factors as \(V\stackrel{\tilde{\pi}}{\longrightarrow} M{\longrightarrow} M/\Gamma=U\), where \(\tilde{\pi}\) 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 above \(S^1\)-action and induces an isometric \(S^1\)-action on \(\mathcal{O}_{a}\). Let \(\Gamma_k\) be a cyclic subgroup of the anti-Hopf \(S^1\)-action. The quotient \(S^3/\Gamma_k\) is a lens space of type \(L(k,1)\). By moding out such \(\Gamma_k\)-actions on \(\mathcal{O}_{a}\) we obtain spindle orbifolds \(S^2(a_1,a_2)\) with arbitrary \(a_1\) and \(a_2\) as quotients. These spaces fit in the following commutative diagram \[\begin{xy} \xymatrix { S^3 \ar[r] \ar[d] & \mathcal{O}_{a} \cong S^2(a_1,a_2) \ar[d] \\ S^3/\Gamma_k \cong L(k,1) \ar[r] & \mathcal{O}_{a}/\Gamma_k \cong S^2(k'a_1,k'a_2) } \end{xy}\] for some \(k'|k\). Here the left vertical map is an example of a (Riemannian) orbifold covering \(p:\mathcal{O}\rightarrow\mathcal{O}'\), i.e. each point \(x\in \mathcal{O}'\) has a neighbourhood \(U\) isomorphic to some \(M/\Gamma\) for which each connected component \(U_i\) of \(p^{-1}(U)\) is isomorphic to \(M/\Gamma_i\) for some subgroup \(\Gamma_i<\Gamma\) such that the isomorphisms are compatible with the natural projections \(M/\Gamma_i \rightarrow M/\Gamma\) (see [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 \(\pi_1^{orb}(\mathcal{O})\) of a connected orbifold \(\mathcal{O}\) 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] to be \[\tag{2.1} k'=\frac{k}{\gcd(k,a_1-a_2)}.\]

More generally, in his fundamental monograph [38] Seifert studies foliations of \(3\)-manifolds 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] and the references stated therein). In particular, in [38] 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 is described in detail in [18]. Let us record the following special statement which will be needed later.

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 them to 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 to demand certain lifting conditions. For instance, a map between orbifolds is called smooth if it locally lifts to smooth maps between manifolds 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]. 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 a \(S^2(a_1,a_2)\) spindle orbifold is an \(L(a_1+a_2,1)\) lens space [24]. 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 \(\mathrm{SU}(2)\times_{S^1} V\) 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.

2.2 Besse orbifolds

The Riemannian spindle orbifolds \(\mathcal{O}_a\cong S^2(a_1,a_2)\) constructed in the preceding section have the additional property that all their geodesics are closed, i.e. 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].

2.3 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 only consider oriented Finsler surfaces and use definitions for Finsler structures from [9]:

A Finsler structure on an oriented surface \(M\) is a smooth hypersurface \(\Sigma \subset TM\) for which the basepoint projection \(\pi : \Sigma \to M\) is a surjective submersion which has the property that for each \(p \in M\) the fibre \(\Sigma_p=\pi^{-1}(p)=\Sigma \cap T_pM\) is a closed, strictly convex curve enclosing the origin \(0\in T_pM.\) A smooth curve \(\gamma : [a,b] \to M\) is said to be a \(\Sigma\)-curve if its velocity \(\dot{\gamma}(t)\) lies in \(\Sigma\) for every time \(t \in [a,b]\). For every immersed curve \(\gamma : [a,b] \to M\) there exists a unique orientation preserving diffeomorphism \(\Phi : [0,\mathscr{L}] \to [a,b]\) such that \(\phi:=\gamma \circ \Phi\) is a \(\Sigma\)-curve. The number \(\mathscr{L} \in \mathbb{R}^{+}\) is the length of \(\gamma\) and the curve \(\dot{\phi} : [a,b] \to \Sigma\) is called the tangential lift of \(\gamma\). Note that in general the length may depend on the orientation of the curve.

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 [7]. Let \(\Sigma \subset TM\) be a Finsler structure. Then there exists a unique coframing \(\mathscr{P}=(\chi,\eta,\nu)\) of \(\Sigma\) with dual vector fields \((X,H,V)\) which satisfies the structure equations \[\tag{2.2} \begin{align} \mathrm{d}\chi=&-\eta\wedge\nu,\\ \mathrm{d}\eta=&-\nu\wedge(\chi-I\eta),\\ \mathrm{d}\nu=&-(K\chi-J\nu)\wedge\eta, \end{align}\] for some smooth functions \(I,J,K :\Sigma \to \mathbb{R}\). Moreover the \(\pi\)-pullback of any positive volume form on \(M\) is a positive multiple of \(\chi\wedge\eta\) and the tangential lift of any \(\Sigma\)-curve \(\gamma\) satisfies \[\dot{\gamma}^*\eta=0 \quad \text{and} \quad \dot{\gamma}^*\chi=\mathrm{d}t.\] A \(\Sigma\)-curve \(\gamma\) is a \(\Sigma\)-geodesic, that is, a critical point of the length functional, if and only if its tangential lift satisfies \(\dot{\gamma}^*\nu=0\). The integral curves of \(X\) therefore project to \(\Sigma\)-geodesics on \(M\) and hence the flow of \(X\) is called the geodesic flow of \(\Sigma\).

For a Riemannian Finsler structure the functions \(I,J\) vanish identically, as a result of which \(K\) is constant on the fibres of \(\pi : \Sigma\to M\) and therefore the \(\pi\)-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 \(\pi : \Sigma \to M\).

Let \(\Sigma \subset TM\) and \(\hat{\Sigma}\subset T\hat{M}\) be two Finsler structures on oriented surfaces with coframings \(\mathscr{P}\) and \(\hat{\mathscr{P}}\). An orientation preserving diffeomorphism \(\Phi : M \to \hat{M}\) with \(\Phi^{\prime}(\Sigma)=\hat{\Sigma}\) is called a Finsler isometry. It follows that for a Finsler isometry \((\Phi^{\prime}\vert_{\Sigma})^*\hat{\mathscr{P}}=\mathscr{P}\) and conversely any diffeomorphism \(\Xi : \Sigma \to \hat{\Sigma}\) which pulls-back \(\hat{\mathscr{P}}\) to \(\mathscr{P}\) is of the form \(\Xi=\Phi^{\prime}\) for some Finsler isometry \(\Phi : M \to \hat{M}\).

Following [9], we use the following definition:

As in the case of a Finsler structure we denote the dual vector fields of \((\chi,\eta,\nu)\) by \((X,H,V)\). Note that a generalized Finsler structure naturally defines an oriented generalized 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.

2.4 Weyl structures and connections

A Weyl connection on an orbifold \(\mathcal{O}\) is an affine torsion-free connection on \(\mathcal{O}\) preserving some conformal structure \([g]\) on \(\mathcal{O}\) in the sense that its parallel transport maps are angle preserving with respect to \([g]\). An affine torsion-free connection \(\nabla\) is a Weyl connection with respect to the conformal structure \([g]\) on \(\mathcal{O}\) if for some (and hence any) conformal metric \(g \in [g]\) there exists a \(1\)-form \(\theta \in \Omega^1(\mathcal{O})\) such that \[\tag{2.3} \nabla g=2\theta\otimes g.\] Conversely, it follows from Koszul’s identity that for every pair \((g,\theta)\) consisting of a Riemannian metric \(g\) and \(1\)-form \(\theta\) on \(\mathcal{O}\) the connection \[\tag{2.4} {}^{(g,\theta)}\nabla_XY={}^g\nabla_XY+g(X,Y)\theta^{\sharp}-\theta(X)Y-\theta(Y)X, \quad X,Y \in \Gamma(T\mathcal{O})\] is the unique affine torsion-free connection satisfying (2.3). Here \({}^g\nabla\) denotes the Levi-Civita connection of \(g\) and \(\theta^{\sharp}\) is the vector field dual to \(\theta\) with respect to \(g\). Notice that for \(u \in C^{\infty}(\mathcal{O})\) we have the formula \[{}^{\exp(2u)g}\nabla_XY={}^g\nabla_XY-g(X,Y)(\mathrm{d}u)^{\sharp}+\mathrm{d}u(X)Y+\mathrm{d}u(Y)X, \quad X,Y \in \Gamma(T\mathcal{O}).\] From which one easily computes the identity \[{}^{(\exp(2u)g,\theta+\mathrm{d}u)}\nabla={}^{(g,\theta)}\nabla.\] Consequently, we define a Weyl structure to be an equivalence class \([(g,\theta)]\) subject to the equivalence relation \[(\hat{g},\hat{\theta})\sim (g,\theta) \quad \iff \quad \hat{g}=\mathrm{e}^{2u}g\;\; \text{and} \;\; \hat{\theta}=\theta+\mathrm{d}u, \quad u \in C^{\infty}(\mathcal{O}).\] Clearly, the mapping which assigns to a Weyl structure \([(g,\theta)]\) its Weyl connection \({}^{(g,\theta)}\nabla\) is a one-to-one correspondence between the set of Weyl structures – and the set of Weyl connections on \(\mathcal{O}\).

The Ricci curvature of a Weyl connection \({}^{(g,\theta)}\nabla\) on \(\mathcal{O}\) is \[\mathrm{Ric}\left({}^{(g,\theta)}\nabla\right)=\left(K_g-\delta_g\theta\right)g+2\mathrm{d}\theta\] where \(\delta_g\) denotes the co-differential with respect to \(g\).

In the case where \(\mathcal{O}\) is oriented we may equivalently say the Weyl structure \([(g,\theta)]\) is positive if the \(2\)-form \((K_g-\delta_g\theta)d\sigma_g\) – which only depends on the orientation and given Weyl structure – is an orientation compatible volume form on \(\mathcal{O}\). 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 geometric significance of the form \(\chi\) 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 [34].

We conclude this section with a definition:

Note that the Levi-Civita connection of any (orientable) Besse orbifold \(\mathcal{O}\) (see 2.2) gives rise to a Besse–Weyl structure on \(S \mathcal{O}\).