Odd-dimensional orbifolds with all geodesics closed are covered by manifolds

Manifolds all of whose geodesics are closed have been studied a lot, but there are only few examples known. The situation is different if one allows in addition for orbifold singularities. We show, nevertheless, that the abundance of new examples is restricted to even dimensions. As one key ingredient we provide a characterization of orientable manifolds among orientable orbifolds in terms of characteristic classes.


Introduction
It is an essential question in Riemannian geometry to understand the behaviour of closed geodesics and to relate it to properties of the ambient space.
A vast literature has been devoted to studying the existence, and number of closed geodesics in compact Riemannian manifolds. Often results in this respect, or their proofs, depend on the topology of the space. Striking examples are that every metric on a closed manifold has at least one closed geodesic due to Lyusternik-Fet [26], and that, by the combined works of Gromoll-Meyer [17] and Sullivan-Vigué-Poirrier [40], every closed Riemannian manifold admits infinitely many closed geodesics as long as its rational cohomology is not generated by one element.
Another, somewhat opposite direction of research, has been to understand the Riemannian manifolds all of whose geodesics are closed [3], here called Besse manifolds. The compact rank one symmetric spaces (CROSSes) S n , CP n , HP n , CaP 2 are the canonical examples. These are in fact the only simply connected spaces known to support a Besse metric. Moreover, by the collective work of Bott, Samelson and McCleary, their cohomology rings are the only ones that can show up in the simply connected setting [7,36,28]: with X a CROSS as above.
In particular, if n := dimM is odd, thenM is homeomorphic to S n (using the Poincaré Conjecture).
Apart from this result, very little is known: for example, it is not known if exotic spheres admit Besse metrics. Furthermore, except a couple of constructions giving rise to infinite families of Besse metrics on spheres, no other Besse metrics are known. In particular, it is not known if there are non-canonical Besse metrics on the projective spaces. As a matter of fact, Lin-Schmidt proved that the only Besse metric on a real projective space of dimension at least 4 is the round one [25].
The study of closed geodesics can be extended to the Riemannian orbifold setting (cf. Section 2.1). In particular, it makes sense to search for closed geodesics on Riemannian orbifolds [6,19,12] and to investigate Besse orbifolds, that is, Riemannian orbifolds all of whose geodesics are closed [18,23]. Allowing for orbifold singularities gives rise to interesting new phenomena and questions. For instance, while Guruprasad and Haefliger proved the GM-SVP result for Riemannian orbifolds and the Lyusternik-Fet result for non-developable orbifolds [19], it is in general not known if every closed Riemannian orbifold has a closed geodesic. Moreover, it provides a much wider class of examples: one immediately gets infinitely many examples of Besse orbifolds in every even dimension, given by weighted projective spaces (see Section 2.1) with interesting properties; e.g. their geodesic length spectra are much more complicated than in the manifold case [18].
With this in mind, it may come as a surprise that in odd dimensions even the orbifold setting is very rigid, i.e. not richer than the manifold category, which is exactly expressed by our main result.
Theorem A. Any odd-dimensional Besse orbifold is of the form (M, g)/Γ for some Riemannian manifold M homeomorphic to a sphere, a Besse metric g, and some finite group Γ of isometries of (M, g). Moreover, if n = 3 and Γ contains a fixed-point-free involution, then (M, g) is isometric to a round sphere and Γ ⊂ O(n + 1).
Notice that this theorem, in particular, extends Theorem 1.1 for orbifolds in the odd dimensional case. For even dimensions, we still think that Theorem 1.1 can be generalized as follows: Conjecture 1.2. The integral (orbifold) cohomology ring of an even-dimensional, simply connected Besse orbifold is generated by one element.
For further motivation to this conjecture and for a more refined version of it we refer the reader to Section 8.
1.1. Idea of the proof, and structure of the article. The first step toward the proof is to find a way to determine when an orbifold is actually a manifold, based on its orbifold cohomology. In this respect, we prove: orb (O; Z p ) (for every p odd prime) are nilpotent. We then focus on a Besse orbifold O. First, an orbifold version of Wadsley's Theorem gives, as a byproduct, that the universal cover of a Besse orbifold is again a Besse orbifold. This reduces our work to the simply connected case. We then consider a loop space Ω orb p,q O which somehow keeps track of the orbifold structure of O. Such a space has already been introduced in [19]. Here, we give a different, though equivalent definition in terms of the frame bundle, instead of using the language of groupoids. Then we prove, via Morse theoretic arguments, that the Betti numbers of Ω orb p,q O are uniformly bounded. Secondly, we show that Ω orb p,q O is an H-space, and we use this structure to prove that the cohomology H * (Ω orb p,q O, F ), with F = Q and Z p , is extremely simple (i.e. isomorphic as a graded vector space to a polynomial algebra F [x]). Finally, this is used to prove that, in the simply connected case, the orbifold cohomology of O satisfies the conditions of Theorem 1.3 and thus O is a (Besse) manifold.
The paper is structured as follows. In Section 2 we recall basic notions and results about (Riemannian) orbifolds and Hopf algebras that are required for our presentation. In particular, in Section 2.2 we describe the class of examples of Besse orbifolds given by weighted projective spaces in more detail. In Section 3 we provide a short and selfcontained proof of Theorem 1.3. In Section 4 we introduce loop spaces for orbifolds and apply Morse theory to obtain the universal bounds on the Betti numbers of Ω orb p,q O for a Besse orbifold O. In Section 5 we apply rational homotopy theory to deduce that for an odd-dimensional simply connected Besse orbifold the rational cohomology ring of the loop space is a polynomial ring. In Section 6 we build on ideas of McCleary and Browder to determine the corresponding cohomology ring with coefficients in a finite field of prime order, at least as a vector space. In Section 7 we show via an application of the Leray-Serre spectral sequence that this information suffices to guarantee the assumptions of Theorem 1.3 and therewith prove our main result. Finally, in Section 8 we provide more background on our Conjecture 1.2. Moreover, there is an appendix that contains complete proofs for several auxiliary statements on Hopf algebras (see Appendix A) and for an orbifold version of Wadsley's theorem (see Appendix B). tion to the starting question of this paper and the work [35] of Quillen.
The first named author was supported both by a Heisenberg grant and another research grant of the German Research Foundation; he is moreover associated to the DFG Priority Programme 2026. The second named author is partially supported by the DFG funded project SFB/TRR 191. The third named author is partially supported by the NSF grant DMS-1810913.

Riemannian orbifolds.
A length space is a metric space in which the distance between any two points can be realized as the infimum of the lengths of all rectifiable paths connecting these points [11]. An n-dimensional Riemannian orbifold O n is a length space such that for each point x ∈ O there exists a neighborhood 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. In this case we call M a manifold chart and U a local chart of O around x. We call a chart good if Γ fixes a preimage of x in M . Behind this definition lies the fact that an isometric action of a finite group on a simply connected Riemannian manifold can be recovered from the corresponding metric quotient [24,Lem. 2.1]. By this fact every Riemannian orbifold admits a canonical smooth structure. Roughly speaking, this means that there exist equivariant, smooth transition maps between the manifolds charts (cf. [5]). Conversely, every smooth orbifold admits a Riemannian metric and in this sense the two notions are equivalent. In the following we sometimes omit the prefix "Riemannian" if a certain property only depends on the smooth structure. For a point x on an orbifold the linearized 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 otherwise singular.
An (orbifold) geodesic on a Riemannian orbifold is a continuous path that can locally be lifted to a geodesic in a manifold chart. A closed geodesic is a continuous loop that is a geodesic on each subinterval. A prime geodesic is a closed geodesic that is not a concatenation of nontrivial closed geodesics. See also [23] for a more detailed discussion on orbifold geodesics. We call a Riemannian metric on an orbifold as well as a connected Riemannian orbifold Besse, if all its geodesics "are closed", i.e. if they factor through closed geodesics. A Besse orbifold is geodesically complete and hence complete by the Hopf-Rinow theorem for length spaces [11,Thm. 2.5.28].

2.2.
Examples of Besse orbifolds. Consider the action of the unit circle S 1 ⊂ C 1 on the unit sphere S 2n+1 ⊂ C n+1 defined by z(z 0 , . . . , z n ) = (z a 0 z 0 , . . . , z an z n ) for some weights a i ∈ Z\{0}. Set a = (a 0 , . . . , a n ). This action is almost free, i.e. its isotropy groups are finite, and the quotient space CP n a = S 2n+1 /S 1 with its quotient metric is a 2n-dimensional Riemannian orbifold. It is referred to as a weighted complex projective space. The geodesics on CP n a are the images of horizontal geodesics on S 2n+1 , i.e. of those geodesics that are perpendicular to the S 1 -orbits [18]. In particular, all geodesics of CP n a are closed, i.e. CP n a is a Besse orbifold. However, not all of them have the same length [18]. For n = 1 the orbifold CP n a is also known as a football or spindle orbifold. In [23, Sect. 2.2] an infinite-dimensional family of Besse metrics on spindle orbifolds is described.
Similarly, one can show that if ρ is a representation of SU(2) on a complex vector space C n such that all irreducible components are complex even-dimensional, then the induced action of SU(2) on the unit sphere in C n is almost free and the corresponding quotient, which is also referred to as a weighted quaternionic projective space [15], is a Besse orbifold [16, p. 154]. Indeed, the only infinite, closed subgroups of SU(2) are its maximal, one-dimensional tori, all of which are conjugated, and SU(2) itself, and the restriction of an irreducible representation of SU(2) on C n to its maximal torus S 1 has weights (n − 1) − 2k, k = 0, . . . , n − 1, so that infinite isotropy groups can only occur for odd n.
All the weighted projective spaces we have discussed in this section are even-dimensional and, by the long exact sequence in homotopy, simply-connected as orbifolds, i.e. they do not admit non-trivial orbifold coverings (see Sections 2.3 and 2.5).

Orbifold coverings.
A covering of complete n-dimensional Riemannian orbifolds O ′ and O is a submetry p : O ′ → O, i.e. a map satisfying p(B r (x)) = B r (p(x)) for all x ∈ O ′ and all r > 0. In this case each point x ∈ O has a neighborhood U isometric to some M/Γ for which each connected component U i of p −1 (U ) is isometric to M/Γ i for some subgroup Γ i < Γ such that the isometries are compatible with the natural projections M/Γ i → M/Γ [24]. In fact, for smooth orbifolds the concept was originally defined by Thurston using the latter property [39]. According to him an orbifold is called good (or developable) if it is covered by a manifold. Otherwise it is called bad. Note that a finite covering of a Besse orbifold is itself Besse. Thurston also showed 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 as the deck transformation group of the universal covering. The orbifold fundamental group also admits interpretations in terms of loops. To explain the one outlined in [24] we first introduce the frame bundle.  [33,34] so that the O(n)-action is isometric, and the projection π : F O → O is an orbifold Riemannian submersion, in the sense that for every point z in F O, there is a neighbourhood V of z such that U = π(V ) is a good chart, and π| V factors as Vπ −→Ũ An orbifold is called orientable if its frame bundle is disconnected. By passing to the orientable double cover [24,Lem. 4.8] we can assume that all our Besse orbifolds are orientable. For an orientable orbifold we also sometimes simply work with the bundle of oriented orthonormal frames, on which SO(n) acts isometrically with quotient O, and still denote this bundle as F O.
Similar to the construction of the frame bundle one can construct a tangent bundle T O which is an orbifold and which carries a natural metric as well corresponding to the Sasaki-metric in the manifold case [37,38]. Furthermore, We define the orbifold Pontryagin classes While the orbifold cohomology with rational coefficients coincides with the usual cohomology of the underlying topological space, the orbifold cohomology with Z p coefficients may have infinite cohomological dimension. For instance, for a finite subgroup G < O(n) the orbifold cohomology of R n /G with Z p coefficients coincides with the group cohomology of G with Z p coefficients which is always infinite dimensional.
Nevertheless, we have the following: Let O be a compact, orientable orbifold. Then H * orb (O; Z p ) is a finitely generated module over H := H * (BO(n); Z p ).
Proof. Since O is orientable, choose an orientation and take F O to be the bundle of oriented orthonormal frames of O. Nothing changes from before, except that now the map F O is a principal SO(n)-bundle, and the projection F O×ESO(n) → BO is classified by a map BO → BSO(n). Consider the Serre spectral sequence with Z p -coefficients (p a prime) of the homotopy fibration is a compact manifold, and thus H q (F O; Z p ) is a vector space over Z p with finite dimension (say m). Moreover, E * , * 2 is isomorphic, as an H := H * (BSO(n); Z p )-module, to H p (BSO(n); Z ⊕m p ) ∼ = H ⊕m and, in particular, it is a finitely generated module over H.
We claim, by induction, that every page (E r , d r ) is a finitely generated differential module over H: in fact, this is true for (E 2 , d 2 ) since H embeds into E * ,0 2 , and by the product structure in E 2 . Moreover, if this is true for (E r−1 , d r−1 ), then H commutes with d r−1 and induces an H-module structure on E r . Moreover, there is a ring homomorphism φ r : H → E * ,0 r such that the action of h ∈ H on x ∈ E r is given by h · x = φ r (h) ∪ x. In particular, since d r is a derivation and the H-action is via multiplication by elements in E * ,0 r , it follows that d r commutes with H. Moreover, as a finitely generated module over the Noetherian ring H, E r−1 is a Noetherian H-module and thus any H-submodule is finitely generated. In particular, ker(d r−1 ) is finitely generated as an H-module, and therefore the quotient E r = ker(d r−1 )/d r−1 (E r−1 ) is finitely generated as well.
Since F O is finite dimensional, the spectral sequence degenerates at the (r 0 = dim F O)page, thus E * , * ∞ = E * , * r 0 is a finitely generated H-module. Since E * , * ∞ ≃ H * (BO; Z p ) as H-modules, the result follows.
2.7. Hopf algebras. Through the paper, we will work with the following definitions.
3. An algebra is a graded vector space A = i A i over a field K, together with a graded linear map Φ : A ⊗ A → A, and an element 1 As usual, we will refer to Φ(x ⊗ y) simply as x · y or, if there is no source of confusion, simply as xy.
Definition 2.5. A Hopf algebra is a graded vector space A = i A i over a field K, together with the structure of an algebra Φ : A ⊗ A → A, 1 ∈ A 0 ; and of a coalgebra Given any algebra A, the dual space A * = Hom(A, K) inherits the structure of a coalgebra, by defining Ψ : for anȳ x ∈ A * and y, z ∈ A. In particular, if A is a Hopf algebra, then A * is a Hopf algebra as well.
The requirement that A 0 = K is not always made, and Hopf algebras satisfying this condition are usually called connected. Throughout this paper, we will only deal with connected Hopf algebras so we will not make this distinction. Furthermore, we will only work with K = Q or K = Z p , which are examples of perfect fields. In this case, the following structure theorem applies (cf. [31, Thm 7.11]): Hopf algebra over the perfect field K, the multiplication in A is (graded) commutative, and dim A i < ∞ for all i then, as an algebra, A is isomorphic with a tensor product i∈I A i of Hopf algebras A i , where A i is a Hopf algebra with a single generator x i .
We will need the following definitions: Definition 2.7. Given a Hopf algebra A, an element x ∈Ā is called: • decomposable if it can be written as a finite sum x = i y i z i for some y i , z i ∈Ā, and indecomposable otherwise.

Definition 2.8. A Hopf algebra A is called:
• primitive if there exists a set of primitive generators for A.
• biprimitive if it is both primitive and coprimitive.
Given a Hopf algebra A, the algebra and coalgebra structures define filtrations F i A and G i A, by: Given such filtrations, one can define the spaces which are both isomorphic to A as graded vector spaces. Moreover, both spaces inherit a Hopf algebra structure from A. In particular, one can define the Hopf algebra 0 EE 0 (A), and the following holds: , Corollary 2.6). If A is an associative Hopf algebra, then 0 EE 0 (A) is a biprimitive Hopf algebra.
Biprimitive Hopf algebras will play a role in the context of differential Hopf algebras, that is, Hopf algebras endowed with a differential d : A → A of degree 1, such that the maps Φ : A ⊗ A → A and Ψ : A → A ⊗ A become morphisms of differential algebras (where the differential on A ⊗ A is given by d(x ⊗ y) = (dx) ⊗ y + (−1) deg(x) x ⊗ (dy). We have the following: Theorem 2.10 ([9], Theorem 3.9). Let (A, d) be a biprimitive differential Hopf algebra over Z p . Then, in the category of differential Hopf algebras, (A, d) is isomorphic to a tensor product • Q is primitively generated, and d(Q) = 0.

A topological condition for an orbifold to be a manifold
The goal of this section is to provide an algebro-topological condition to determine when an orbifold is in fact a manifold. Most of the argument below (see Proposition 3.3) also follows from a more general result [35, Theorem 7.7, p. 568] due to Quillen. As a service to the reader, we provide a short and self-contained reasoning relevant to our purposes.
We shall draw on the following lemma which will help to identify a special system of local coefficients used in an E 2 -page of a Leray-Serre spectral sequence in the proof of Proposition 3.3.
Lemma 3.1. Given an orientable n-dimensional orbifold O and a subgroup Z p ⊆ SO(n), in the fibration the monodromy action of the fundamental group of the base, π 1 (BO), on the cohomology of the fibre, H * (SO(n)/Z p ; Z p ), is trivial.
Proof. We prove the result in slightly larger generality replacing SO(n) by a connected, compact Lie group G, Z p by a discrete group H ⊆ G and F O by a connected G-CWcomplex X. In this case we have the fibration We claim that for any element [γ] ∈ π 1 (X G ), its action on G/H is homotopic to the identity, and thus its action on Letγ : I → X × EG denote a lift of γ, and letγ(0) = (x 0 , v 0 ),γ(1) = g −1 0 ·γ(0). Consider now the bundle G/H → X H → X G . Let Γ : G/H × I → X G be the map Γ(gH, t) = γ(t), and take the liftΓ 0 : G/H → X H of Γ at t = 0 given byΓ 0 (gH) = [g −1 ·γ(0)] H (which identifies G/H with the fiber at γ(0)). This map can be extended tô It is easy to check that this map is well defined, is a lift of Γ, and that at t = 1 it is given byΓ , we can thus rephrase the statement of the proposition in terms of the frame bundle: is finite for every prime p. Suppose now that the SO(n) action is not free. Then there is some (compact) isotropy group H, which contains some finite cyclic subgroup Z p , for some prime p. Thus, there exists some Z p ⊆ SO(n) with nonempty fixed point set, the cohomology of the total space of the bundle The existence of a fixed point x ∈ F O of the Z p -action yields a section of this bundle given by s(b) = [x, b] G , where we use the notation of Lemma 3.1. Hence in this case the induced morphism on cohomology Next, the natural map is trivial, and thus the Leray-Serre spectral sequence satisfies and converges to H p+q Zp (F O; Z p ). As noted above, H * Zp (F O; Z p ) has infinite cohomological dimension. We claim that this implies that H * SO(n) (F O; Z p ) has infinite dimension as well (thus proving the proposition). We prove the claim by contradiction: if H * SO(n) (F O; Z p ) had finite cohomological dimension, then the E 2 -page of the spectral sequence would have finite dimension, and therefore the same would be true for the E ∞ -page. This is a contradiction, since the spectral sequence converges to H * Zp (F O; Z p ) for which we showed that it is infinite dimensional.

Orbifold loop space, and Morse theory
The goal of this section is to reduce the proof of Theorem A to the simply connected case, and to prove the following: From now on, for sake of notation we will denote Ω(BO) by ΩBO.

4.1.
The orbifold loop space. Let us start by defining a loop space of O, which incorporates the information about the orbifold structure. An equivalent definition in the language of groupoids is given in [19].
Let O be an orbifold with frame bundle π : F O → O. Given two points p, q ∈ O, with p regular, and fixing z ∈ π −1 (q), define the orbifold loop space as . In particular, there is a homeomorphism Ω orb p,z ′ O ≃ Ω orb p,z O, and we call either of them simply by Ω orb p,q O. An alternative definition for the orbifold loop space is The two definitions are different, however it is a standard fact that the inclusion Ω orb p,q O → Ω orb p,q O is a homotopy equivalence. This is proved, for example, with great detail in [29], Theorem 17.1.
Moreover,Ω orb p,q O has the homotopy type of a CW-complex (cf. [30]). Standard Morse Theory gives a more precise description of the CW-complex structure: let us equip F O with a Riemannian metric g F , such that O(n) acts by isometries, and the projection π : F O → O is a Riemannian submersion as discussed in Section 2.4. Moreover, let E : Ω orb p,q O → R be the energy functional taking γ : Standard variation arguments show that the critical points of E are geodesics γ from π −1 (p) to z, with γ ′ (0) perpendicular to π −1 (p). Moreover, for any such critical point, γ, its index ind I (γ) (number of negative eigenvalues of the Hessian of E at γ) coincides with the number of focal points to π −1 (p) along γ in (0, 1) counted with multiplicity, and the nullity (i.e. dimension of the kernel of the Hessian of E at γ) coincides with the multiplicity of γ(1) as a focal point of π −1 (p) along γ. For z generic, the nullity of γ is zero for every critical point γ of E, and standard Morse Theory ( [29], Thm. 17.3) shows that Ω orb p,q O is homotopy equivalent to a CW-complex, with a d-dimensional cell attached for every critical point γ of index d.
The first result is to show that the information about the critical points of E and their index can be read off from the orbifold geometry of O.
Recall that on Riemannian orbifolds one can still define geodesics and Jacobi fields: a curve c : Similarly, an orbifold Jacobi field along an orbifold geodesic c is a section J of c * T O such that, whenever c| [a,b] is contained in a good open set U ⊂ O and is covered by a geodesicc : I →Ũ , there is a Jacobi fieldJ : [a, b] →c * TŨ such that (π U ) * J = J. Equivalently, orbifold Jacobi fields can be described by variations of orbifold geodesics (although one needs more care with this definition).
Just as in the manifold case, given an orbifold geodesic c :  I (c). Proof. As mentioned above, if γ : I → F O is a critical point for E, then it is a geodesic in F O starting perpendicularly to the fiber π −1 (p). Since π is a Riemannian submersion, the geodesic γ is a horizontal geodesic, and thus its projection c = π • γ is an (orbifold) geodesic in O. Moreover, just as in the manifold case, any orbifold geodesic from p to q can be lifted horizontally to a horizontal geodesic, ending at z ∈ π −1 (q), and everywhere perpendicular to the fibers of π (in particular, starting orthogonal to π −1 (p)). Clearly the two maps are inverses of one another, and this gives the bijection.
Let Λ p be the space of Jacobi fields along γ, given as variations through horizontal geodesics from π −1 (p). The zeroes of Jacobi fields in Λ p count precisely the focal points of π −1 (p), thus ind I (γ) equals the number of Jacobi fields in Λ p vanishing at points in I, counted with multiplicity. To prove the equivalence between the indices, it is thus enough to show that π : F O → O induces, for every t 0 ∈ [0, 1], a one-to-one correspondence between: • Jacobi fields J ∈ Λ p with J(t 0 ) = 0, and • Orbifold Jacobi fields J along c with J(0) = J(t 0 ) = 0. For each t ∈ [0, 1] there is a small time segment I = (t − ǫ, t + ǫ) such that γ(I) is entirely contained in an open subset V ⊂ F O for which U := π(V ) is a good local chart, and such that π : V → U factors as Vπ −→Ũ π U −→ U , whereŨ is the manifold chart of U andπ is a standard Riemannian submersion.
Working in such open sets shows that Jacobi fields along γ project to orbifold Jacobi fields along c and, conversely, that orbifold Jacobi fields along c can be lifted to Jacobi fields along γ. Since horizontal geodesics from π −1 (p) project viaπ to geodesics inŨ fromp =c(0), any Jacobi field J ∈ Λ p projects viaπ * to an orbifold Jacobi field alongc withπ * J(0) = 0. Moreover, we see that orbifold Jacobi fields along c vanishing at t 0 correspond to the Jacobi fields in Λ p vanishing at the same point.
The space Λ p is a vector space of dimension dim F O −1, and it splits into 2 summands: • The space Λ v , spanned by (the restriction to γ of) the action fields X * , X ∈ so(n). This subspace has dimension = dim π −1 (p), and consists of nowherevanishing Jacobi fields. • The space Λ h , spanned by Jacobi fields J normal to γ ′ (t) such that J(0) = 0 and J ′ (0) ∈ ν γ(0) (π −1 (p)). This space has dimension dim F O − dim(π −1 (p)) − 1 = dim O − 1, andπ * defines an isomorphism between Λ h and the space of Jacobi fields alongc vanishing at t = 0. Suppose now that J 1 , . . . , J m ∈ Λ p are linearly independent Jacobi fields such that J i (t 0 ) = 0. Then under the splitting J i = X * i + K i , with X * i ∈ Λ v and K i ∈ Λ h , the Jacobi fields K i must be linearly independent as well: in fact, if we had a i K i = 0 for some nonzero coefficients a i , then The Jacobi fieldsπ * J 1 , . . .π * J m alongc satisfyπ * J i (t 0 ) = 0. Moreover, sinceπ * J i = π * K i and the K i are linearly independent, it follows thatπ * J 1 , . . . ,π * J m are linearly independent as well, and this proves that ind I (c) ≥ ind I (γ).
To prove the opposite inequality, take linearly independent Jacobi fields J 1 , . . . , J m alongc such that J i (0) = J i (t 0 ) = 0, and let K 1 , . . . K m ∈ Λ h denote the Jacobi fields along γ such thatπ * (K i ) = J i . Then the K i are linearly independent with K i (0) = 0, and let v i := K i (t 0 ) ∈ T γ(t 0 ) π −1 (γ(t 0 )) . Letting X * i ∈ Λ v such that X * i (t 0 ) = v i , we get linearly independent Jacobi fields K i − X * i ∈ Λ p vanishing at t 0 . This proves that ind I (γ) ≥ ind I (c), as we wanted.
This theorem, together with the Morse theory arguments, implies the following:  Let O be an orbifold, and p, q ∈ O. If p ∈ O is chosen to be a regular point, then Ω orb p,q O is weakly homotopy equivalent to ΩBO. Proof. Consider the space P p = {γ ∈ C 0 (I, F O) | γ(0) ∈ π −1 (p)}. Since p is a regular point, the group O(n)-action on P p by (g ·γ)(t) = g ·(γ(t)) is free. In particular, the Borel construction (P p ) O(n) = (P p × ESO(n))/O(n) is homotopy equivalent to the standard quotient P p /SO(n). Moreover, the map i : O(n) → P p sending g to the constant path γ(t) ≡ g · z is an inclusion, and in fact an O(n)-equivariant homotopy equivalence. This induces a homotopy equivalence between the quotients P p /O(n) ≃ O(n)/O(n) = {pt} and thus (P p ) O(n) is contractible.
The map ev 1 : P p → F O sending a curve γ to γ(1) is a fibration, with fiber (ev 1 ) −1 (q) = Ω orb p,q O. The map ev 1 is also O(n)-equivariant, and thus it induces a fibration between the Borel spaces (  The proof follows closely the original proof of Wadsley for manifolds, with minor modifications. For sake of completeness, we included a proof of the orbifold version of this theorem in Appendix B. One immediate corollary is the following: Let L be the minimal common period of all closed geodesics on O. We claim that a geodesic c based at p of length L represents the same element in π orb 1 (O, x) as its inverse. Let γ be a lift of c to F O and let g ∈ O(n) such that gγ(1) = γ(0). Continuously deforming the initial vector from v to −v through horizontal vectors defines a homotopy through horizontal geodesics between γ and (gγ) −1 = g(γ) −1 . In particular, we have [γ] = [γ] −1 in π orb 1 (O, x) as claimed. Consequently, any element in π orb 1 (O, p) is represented by a horizontal geodesic of length < 2L. LetÕ → O be the universal covering and let FÕ → F O be the induced covering [24,Lem. 4.6]. It follows that the fibers of FÕ → F O have diameter ≤ 2(2L + diam(π −1 (p)). Since FÕ is complete and connected and the fibers are discrete, they are finite and hence so is π orb 1 (O, p). The corollary above reduces the proof of Theorem A to the simply connected case. Furthermore, we can use Wadsley's Theorem and Corollary 4.4 to prove the following: Proof. The proof is virtually identical to the one in the manifold case, cf. Sections 7.41 and 7.42, p. 193 in [3]. For the sake of completeness, and because this result is central in proving our Main Theorem, we recall the arguments here.
In the following we fix a regular point p in O and choose some point q that is not conjugate to p along any geodesic (this can be done by a straightforward consequence of Sard's theorem applied to the normal exponential map from the normal bundle of π −1 (p) to F O). By Proposition 4.5 it is sufficient to show that the Betti numbers of Ω orb p,q O are uniformly bounded. Moreover, by Corollary 4.4, it is enough to prove that for every d, Notice that any closed orbifold geodesic from p with length L must have the same index, and nullity n − 1, the maximum possible. In fact, on the one hand, every orbifold Jacobi field J along c with J(0) = 0 and J ′ (0) ⊥ c ′ (0) defines a variation through geodesics from p, which by Wadsley's Theorem must return to p at time L, and thus J(L) = 0, thus proving the second statement. On the other hand, letting c 0 , c 1 be any two geodesics from p of length L, and letting {c s } s∈[0,1] be a smoothly-varying family of orbifold geodesics c s : I → O of length L from c 0 to c 1 , it is well-known that the only discontinuities of the function s → ind orb I (c s ) occur at times s i in which the nullity of c s i jumps up. However, since the nullity of any such c s is n − 1, it follows that ind orb I (c s ) is in fact constant. In particular, any two orbifold geodesics of length L have the same orbifold index, call it C.
Since O is compact, and p and q are not conjugate to each other, for any e ∈ R there are only finitely many geodesics between p and q with energy ≤ e. Let {c 1 , . . . c k } be the geodesics from p to q with length ≤ L.
By Wadsley's Theorem, the orbifold geodesics with length in (mL, (m + 1)L) must be {c i (ℓ(c i )) = q). In particular, for any integer m there are exactly k geodesics with length in (mL, (m + 1)L).
Since any geodesic longer than L picks up n − 1 conjugate points at times multiple of L, we obtain that ind orb I (c In this section and the next, O will denote an odd-dimensional, simply connected Besse orbifold. In particular, O is orientable and we only work with the bundle of oriented orthonormal frames. From the previous section, the loop space ΩBO has uniformly bounded Betti numbers. Using the extra information that O is odd dimensional, and the fact that ΩBO is an H-space, we will show that, in fact, H * (ΩBO; K) ≃ K[x] is a polynomial algebra in one generator of even degree, whenever K = Q or Z p . We will deal with two cases separately: in this section we will treat the case K = Q; and in the next section, we will look at the case K = Z p .
The main structure that we will exploit is the fact that ΩBO (here thought as the space of continuous pointed loops at some fixed point z ∈ BO) is an H-space, with respect to the map ΩB × ΩB → ΩB given by concatenation of paths. In particular, the cohomology H * (BO; K) (with respect to any field of coefficients K) is a commutative, associative Hopf algebra.
When dealing with rational coefficients, we use Sullivan's theory of minimal models. The methods and results of this section are fairly standard.
Recall that to any simply connected topological space X, one can associate a differential graded algebra (ΛV X , d): that is, a free graded commutative algebra over Q where V X = V odd X ⊕ V even X is a vector space over Q, together with a differential d on ΛV X , of degree 1. This differential graded algebra, called minimal model of X, is defined by the properties that: Among the properties of the minimal model, one has that for simply connected X, the degree-d part V d X of V X satisfies V d X ≃ Hom(π d (X), Q). Although we are mainly concerned with the odd-dimensional case, let us formulate the following result also for even dimensions.
Proof. First of all, recall that the natural map induced by projection on the first factor is a rational homotopy equivalence (it induces isomorphisms in rational cohomology, and thus in rational homotopy as well). Therefore Since V d ≃ π d (O) ⊗ Q and the homotopy groups of ΩO are just the ones of O scaled by one, it follows that the minimal model of ΩO can be also written as It only remains to remark that the loop space cohomology in both cases obviously has universally bounded Betti numbers. 6. Computing H * (ΩBO; Z p ) (p prime), for simply-connected Besse orbifolds In this section we consider the cohomology of ΩBO with coefficients in Z p , p a prime. The methods of this sections follow closely the arguments of McCleary in [28]. The whole section is devoted to proving the following: so we need to prove that d r = 0 for all r. We prove this by showing that (B r ) odd = 0 for all r, so that the result follows by "lacunary principle".
Recall that by Browder (cf. [8]), for every B k there is a spectral sequence of spectral sequences s EE r (B k ), with the following properties: Hopf algebra (cf. definitions in Section 2.7) isomorphic to B k as a graded vector space. (ii) For every r, k fixed, there are differentials d s such that ( s EE r (B k ), d s ) is a spectral sequence, converging to 0 EE r+1 (B k ). (iii) For every k fixed, there are differentials d r such that ( 0 EE r (B k ), d r ) is a spectral sequence converging to 0 EE 0 (B k+1 ). Since each s EE r (B k ) is biprimitive by [9,Lem. 3.2], by Theorem 2.10 it is isomorphic, as a differential Hopf algebra, to a product where y, z are primitive, |y| odd and d s y = z.
where y ′ , z ′ are primitive, with |y ′ | odd and d s z ′ = y ′ .
• Q is primitively generated, and d s Q = 0. Recall that the set of primitive elements in a Hopf algebra A is a vector space. We will see that, if A is biprimitive, then any homogeneous basis of this subspace forms a set of generators of A in the tensor product description above.
We now prove that (B k ) odd = 0 for all k, by contradiction. Supposing that some B k contains elements of odd degree, we want to produce a subalgebra of B 1 = H * (ΩBO, Z p ) whose dimension grows unbounded with respect to the degree, contradicting Theorem 4.8. We do this by applying the following: Lemma 6.2. Suppose A is a (associative and commutative) biprimitive Hopf algebra over Z p , and suppose that we have two linearly independent sequences a = {a 1 , a 2 , . . .}, b = {b 1 , b 2 , . . .} of homogeneous primitive elements of A, such that the subalgebras A a , A b generated by a and b respectively, are nonzero in degrees forming arithmetic sequences. Then the dimension of A grows unbounded in the degree.

Proof. Since A is biprimitive, it has the form
where the x i 's are primitive elements and d i is either 2 if deg(x i ) is odd, or p if deg(x i ) is even, cf. for example the discussion in [9, Corollary 2.2]. Let P ⊂Ā be the set of primitive elements, and let P → F 1 A/F 2 A ≃ span{x i } be the projection map. This is both surjective (since A is primitive) and injective (A is coprimitive) and therefore P = span{x i }. We claim that for any basis of P consisting of homogeneous primitive elements {u 1 , u 2 , . . .}, we can write It is enough to prove it for a given degree: given {u 1 , . . . , u m } and {x 1 , . . . , x m } bases of P d (degree d summand of P ), we need to show that the algebra generated by is of the same shape as above. Also, this is clear in odd degrees, due to the fact that the product is skew commutative. In even degrees, this is due to the fact that, with Z p coefficients, the map A → A, x → x p is an algebra homomorphism: the kernel of the natural map φ : Z p [x 1 , . . . , x m ] → A is generated by x p i and thus, letting u i = ij α ij x j for some A = (α ij ) ∈ GL(m, Z p ) and letting B = (β ij ) the inverse of A, we have that the kernel of φ is the ideal generated by j (β ij u j ) p = j β ij u p j , which is the same as the ideal generated by the u p j . Therefore, given the linearly independent set S = {a 1 , b 1 , a 2 , b 2 , . . .} of homogeneous elements of P , this can be extended to a basis {u 1 , u 2 , . . .} of homogeneous elements of P , such that A can be written as in (3). In particular, the algebras A a , A b are of this form as well, and the algebra generated by S is isomorphic to A a ⊗ A b . Since A a , A b have elements in degrees forming arithmetic sequences, it is easy to check that the dimensions of the degree components of A a ⊗ A b (and thus the ones of A) grow unbounded.
One sequence, satisfying the properties of the lemma above is readily available. Notice in fact that, since B ∞ is a polynomial ring, its biprimitive form is The elements z i can be lifted to a set of primitive elements which is nonzero in degrees multiples of (n − 1)p and thus satisfies the condition of the lemma.
To finish the proof, we thus need to produce the second set b. We will consider two cases.
6.1. Case I: the spectral sequence s EE r (B k ) does not contain factors of type L j . Recall that we are assuming, by contradiction argument, that there exists some element of odd degree in some B r , and then also in 0 EE 0 (B r ). In this case the following lemma applies for (s 0 , r 0 , k 0 ) = (0, 0, r). Lemma 6.3. Suppose that the biprimitive spectral sequence s EE r (B k ) only contains factors of type K i and Q. If there is an element y ∈ s 0 EE r 0 (B k 0 ) of odd degree, then there exists a sequence b = {x 1 , x 2 , . . .} of primitive elements of degree |x i | = (|y| + 1)p i in algebra Z p [w], |w| = |y| + 1. Moreover, every x i gets killed at some stage in the spectral sequence s EE r (B k ).
Proof. We can assume that y is primitive. Since B ∞ does not have elements of odd degree, the element y must be killed at some stage (s 1 , r 1 , k 1 ). Since there are no factors of type L j , the only way this can happen is that there is some x 1 ∈ s 1 EE r 1 (B r 1 ) such that dy = x 1 . Since the differential commutes with the coproduct, the element x 1 is primitive as a differential of a primitive element. Moreover, |x 1 | = |y| + 1, and in s 1 +1 EE r 1 (B r 1 ) there is a new element of odd degree being created, namely y 2 = [yx p−1 1 ]. By Lemma A.1 below, y 2 is primitive, and |y 2 | = |y| + (p − 1)(|y| + 1) = p(|y| + 1) − 1. By the same reason as before, at some later time (s 2 , r 2 , k 2 ) we can find x 2 ∈ s 2 EE r 2 (B r 2 ) such that dy 2 = x 2 . Again, x 2 is primitive and |x 2 | = |y 2 | + 1 = p(|y| + 1). The lemma now follows by inductively repeating the same steps above and by lifting the x i to 0 EE 0 (B 1 ) via an application of Lemma A.3.
Because the elements in the sequence b in Lemma 6.3 above do not survive to 0 EE 0 (B ∞ ), they are linearly independent of the elements in the sequence a in (4). Lemma 6.2 can then be applied, contradicting the uniform boundedness of 0 EE 0 (B 1 ). This proves Proposition 6.1 in this first case.
6.2. Case II: the spectral sequence s EE r (B k ) contains a factor of type L j . We can now assume that at some stage in the biprimitive spectral sequence s EE r (B k ), we have a factor of type L j . In other words, there is an element x s,r ∈ s EE r (B k ) of even degree (call it D), such that dx s,r = 0. By the explicit form (2) of each stage s EE r (B k ), we can choose x s,r to be primitive. According to Lemma A.3 the element x s,r is in fact represented by a primitive element x 0,0 ∈ 0 EE 0 (B k ). Using the explicit shape of the biprimitive form 0 EE 0 (B k ) shows that the primitive element x 0,0 can be expressed as a linear combination of indecomposable generators ξ i of 0 EE 0 (B k ) as in [9], Page 158. We can assume that x 0,0 is represented by one such generator ξ 0 (still guaranteeing our assumption dx s,r = 0). By [9, Theorem 2.7] ξ 0 is represented by an iterated p-th power x p m 0 of an indecomposable generator x 0 of E 0 (B k ). Since B k ≃ E 0 (B k ) as algebras, ξ 0 = x p m 0 is represented by an iterated p-th power ξ = x p m of some indecomposable element x ∈ B k . However, if m = 0, we would have dξ = 0 in (B k , d) and, by standard facts about the spectral sequence of a filtered complex, it would follow that dx s,r = 0 for all x s,r ∈ s EE r (B k ) represented by ξ. This gives a contradiction, and hence x s,r is represented by the indecomposable element x. We claim that dx cannot be written as dw for some decomposable element w: in fact, if this was the case, then d(x − w) = 0 would imply that every element in s EE r (B k ) represented by x − w has zero differential. On the other hand, since w ∈ F 2 (B k ) (second subset in the filtration of B k , which coincides with the set of decomposable elements), then , and this implies that x s,r would also be represented by x − w ∈ F 1 (B k ), a contradiction since dx s,r = 0.
Recall (cf. [28]) that the homology H * (ΩBO; Z p ) is a Hopf algebra as well, and that there is a homology Bockstein spectral sequence (B k , d k ) with B 1 = H * (ΩBO; Z p ) and such that each B k is a Hopf Algebra, which is identified with the dual of B k . In particular, since dx ∈ B k is linearly independent from every dw, w decomposable, it then follows that there exists an elementȳ ∈ B k such thatȳ(dx) = 0 andȳ(dw) = 0 for every w decomposable. Then: • dȳ(w) = 0 for all decomposable w. Hence, dȳ is primitive by Lemma A.2.
• dȳ(x) = 0, which implies thatx := dȳ = 0. From this, it follows from Theorem 3 of [28] thatx produces ∞-implications. That is, there exists a sequence of primitive elementsx 1 ,x 2 , . . . in homology, withx i ∈ B r i for some r i , and |x i | = Dp i . Moreover, it follows from the proof of Theorem 3 of [28] that: In particular, they represent primitive elements in E 0 B r i by [9, Proposition 1.3, (i)], and hence primitive elements in 0 EE 0 (B r i ) by [9, Proposition 1.2, (iv)], which, by Lemma A.3, can be lifted to primitive elements in 0 EE 0 (B r ), again called x i . Moreover, since |x i | = Dp i , it follows by dimensional reasons that the algebra that is, the algebra generated by the x i , is isomorphic as a vector space to a polynomial algebra in one generator of degree D.
Since the elements in the sequence b, i.e. the elements x i , do not survive to 0 EE 0 (B ∞ ), they are linearly independent of the elements in the sequence a in (4). Once again Lemma 6.2 can then be applied, contradicting the uniform boundedness of 0 EE 0 (B 1 ).
This proves Proposition 6.1 in the second and final case.

Proof of the Main Theorem
Let O be an odd-dimensional Besse orbifold. By Corollary 4.7, the universal coverÕ is a Besse orbifold as well. We first prove thatÕ is in fact a manifold.
Consider the Leray-Serre spectral sequence of the path-fibration ΩBÕ → P BÕ → BÕ with Z p coefficients. The second page is E r,s 2 = H r (BÕ; Z p )⊗H s (ΩBÕ; Z p ). From Proposition 6.1, we know that H * (ΩBÕ; Z p ) ∼ = Z p [e n−1 ] as a graded vector space. Since the total space of the fibration is contractible, the spectral sequence converges to the cohomology of a point. In particular, the lowest non-zero degree in which BO has nontrivial Z p cohomology is n and the dimension of this cohomology group is 1. Hence, for r ≤ n we have E r,s 2 = Λ(e n ) ⊗ Z p [e n−1 ] as graded vector spaces, and e n · e n−1 = 0 in E p,q 2 . Since x = e n has even degree we find using the Leibniz rule that It follows inductively that for k < p we have x k = 0 and d n (x k ) = 0. This implies that H i (BÕ; Z p ) = 0 for all 0 < i < p(n − 1) + 1, i = n.
In the case p = 2, since H <n (BÕ; Z p ) = 0, all the orbifold Stiefel Whitney classes are trivial with the possible exception of w n (Õ). However, it follows from Wu's formula that the Stiefel Whitney classes are generated over the Steenrod algebra by those in degrees that are a power of two. Since n > 2 is not a power of two, w n (Õ) is trivial as well.
From Proposition 3.3, it follows thatÕ is a Besse manifold. SinceÕ is odd dimensional, it follows from Theorem 1.1 thatÕ is a (simply connected) homology sphere, hence a homotopy sphere and, by the Poincaré Conjecture, a topological sphere. In particular, . Furthermore, if n = 3 and Γ contains an involution g acting freely onÕ, then g generates a copy of Z 2 acting freely onÕ. By [25,Theorem 2], it follows thatÕ/ g (and henceÕ) has constant sectional curvature.

A conjecture
Recall the construction of weighted projective spaces CP n a , the quotients S 2n+1 /S 1 by an almost free action, with a = (a 0 , . . . , a n ) and with a i ∈ Z\{0} from Section 2.1. These constitute simply connected Besse orbifolds in even dimensions. Consequently, our main result cannot be generalized to even dimensions.
As a service to the reader let us quickly recall the argument. We identify H * orb (CP n a ; Z) ∼ = H * S 1 (S 2n+1 ; Z) and consider the long exact sequence of the pair (C n+1 , S 2n+1 ).
Considering (C n+1 , S 2n+1 ) as a pair of trivial vector bundles over the point the Thom , and α is multiplication with the equivariant Euler class e S 1 (C n+1 ) = a 0 ·. . .·a n ·u n+1 (computed using the splitting principle). Consequently, the morphism α is injective, and the long exact sequence above splits to yield the result. Alternatively, the result can be obtained by an application of the Gysin sequence to the fibration The same reasoning shows that the integral cohomology ring of an 4n-dimensional weighted quaternionic projective space is of the form Z[u]/(p·u n+1 ) with u of degree 4 and some integer p, which, by [35,Theorem 7.7,p. 568], is 1 if and only if the corresponding action of SU(2) on S 4n−1 is free.
Recall further that due to 1.1 the best known result in the manifold case is a cohomological classification which, in particular, implies that the integral cohomology algebra is generated by one element only.
Taken together, these observations motivate the following conjecture.
Conjecture 8.1. The integral cohomology algebra of a simply-connected even-dimensioevendimensionalnal Besse orbifold O of dimension d is generated by one element u of even degree k such that d = k · n and k ∈ {2, 4, 8}, n ∈ N, such that . Note that by [35,Theorem 7.7,p. 568] in this case O has the structure of a manifold if and only if |l| = 1.
Remark 8.2. The reader may observe that we can support this conjecture further: Indeed, our Morse theoretic arguments (with rational coefficients) together with the corresponding arguments from rational homotopy theory presented in Proposition 5.1 show that also in the case when dim O is even its rational (orbifold) cohomology algebra is a truncated polynomial algebra generated by one element of even degree.
Lemma A.2. Let A be a Hopf algebra, andȳ ∈ A * (the dual of A). Thenȳ(w) = 0 holds for every w decomposable, if and only ifȳ is primitive.
The next Lemma, applies to the specific case of Browder's spectral sequence s EE r (B k ) of biprimitive Hopf algebras (see Section 6). Notice that, given two triplets of integers (s, r, k), (s ′ , r ′ , k ′ ), with (s, r, k) < (s ′ , r ′ , k ′ ) in the lexicographic order, it follows that the page s ′ EE r ′ (B k ′ ) occurs after the page s EE r (B k ). In this situation, we say that a subalgebra E ′ ⊆ s ′ EE r ′ (B k ′ ) lifts to a subalgebra E ⊂ s EE r (B k ), if there is an algebra isomorphism φ : E → E ′ such that every x ∈ E represents φ(x). Lemma A.3. Given two triplets of integers (s, r, k) < (s ′ , r ′ , k ′ ) in the lexicographic order, the algebra s ′ EE r ′ (B k ′ ) lifts to a subalgebra E ⊂ s EE r (B k ), such that primitive elements of even degree lift to primitive elements.
Proof. Let E s ′ ,r ′ ,k ′ (m) ⊆ s ′ EE r ′ (B k ′ ) be the subalgebra generated by primitive elements of degree ≤ m. We will show that for any m, there are lifts of E s ′ ,r ′ ,k ′ (m) to subalgebras E s,r,k (m) of s EE r (B k ), such that E s,r,k (m) ⊆ E s,r,k (m + 1) and so that the lifts are compatible with the inclusions.
We will prove this by induction: By the explicit shape of these algebras, it is clear how to lift s ′ EE r ′ (B k ′ ) to s ′ −1 EE r ′ (B k ), and in particular any E s ′ ,r ′ ,k ′ (m). Notice that primitive elements of even degree must live in the Q factor, hence their lift is again primitive.
Then, clearly, E 0,r ′ ,k ′ (m) lifts to a subalgebra of s ′′ EE r ′ −1 (B k ′ ), and, by iterating the point before, it lifts to 0 EE r ′ −1 (B k ′ ). • If s ′ = 0 and r ′ = 0 then, again since As before, E 0,0,k ′ (m) lifts to a subalgebra of 0 EE r ′′ (B k ′ −1 ) and, by iterating the point before, it lifts to 0 EE 0 (B k ′ −1 ). Iterating these steps, one can lift E s,r,k (m) to any earlier page. Taking the direct limit as m → ∞, one gets the result.
Appendix B. Wadsley's Theorem for orbifolds B.1. Smooth actions on orbifolds. In order to prove Proposition 4.6 on the existence of a minimal common period for the geodesics of a Besse orbifold we first need to explain some properties of smooth actions on orbifolds. We assume that the reader is familiar with the notion of a smooth orbifold (see e.g. [5] [14,Lem. 2.11]. The latter moreover shows that G/G x embeds into O as a full suborbifold [5] in the sense that the images of all lifts of i : G/G x → O to good manifold charts are invariant, and hence point-wise fixed, under the actions of local groups (see proof of [14,Lem. 2.11]). In particular, if Γ x is the local group of O n at x, then the tangent space of O at x splits as R k /Γ x × T x Gx, where k = n − dim(G/G x ). This splitting is preserved under the action of G and so the normal bundle ν(Gx) of Gx in O is a R k /Γ x -bundle over G/G x . B.2. Proof of Wadsley's theorem. The geodesic flow induces a smooth R-action on the unit sphere bundle T 1 O of a Besse orbifold O which is itself an orbifold. The orbits of this action are geodesics with respect to the natural (Sasaki) metric on T 1 O, they project to the geodesics on O with the same arc-length parametrization and the same period, and all the geodesics on O can be obtained in this way (cf. [3, Ch. 1.K]). In the manifold case a theorem by Wadsley says that an R-action with these properties factors through an S 1 -action [41], see also [3,Appendix A]. We are going to argue that the same statement and its conclusion holds in the present setting of orbifolds. In the manifold case the proof involves a local analysis of Poincaré return maps. In order to have the same tool available in the orbifold case we first prove the following statement. Proof. According to our discussion in Section B.1 the normal bundle ν(L) of L in O is an R n−1 /Γ x -bundle over S 1 where Γ x is the local group of a point on L. Using the exponential map we find a small neighborhood W of L in O diffeomorphic to ν(L). We identify W and ν(L) via this diffeomorphism. LetW be the universal covering of W as a topological space with a smooth orbifold structure induced form W so thatW → W is also a covering of orbifolds. Since W is homotopy equivalent to S 1 the deck transformation group of the coveringW → W is isomorphic to Z and by construction it acts transitively on the fibers, i.e. the covering is normal. The spaceW is an R n−1 /Γ x -bundle over R and, in fact, it as a trivial bundle. In this case this can for instance be seen by lifting the induced R-action dφ : R × ν(L) → ν(L) toW . In particular, its orbifold fundamental group is isomorphic to Γ x . Moreover, it shows that the universal (orbifold) coveringW of W is R n−1 × R. Let Γ be the deck transformation group of the coveringW → W and let N be its subgroup isomorphic to Γ x corresponding to the coveringW →W . Since the coveringW → W is normal, the subgroup N is normal in Γ and the quotient Γ/N is isomorphic to Z, the deck transformation group of the coveringW → W . This implies that Γ is a semidirect product N ⋊ H of N with an infinite cyclic subgroup H of Γ. Since H does not have finite subgroups, it acts freely onW and so M =W /H is a manifold. By construction the group N leaves the preimage of L inW point-wise fixed. It follows that the induced covering p : M → W restricts to a homeomorphism on its preimage of L. This completes the proof of the lemma.
In the situation of the preceding lemma the vector field on W induced by the R-action lifts to a vector field on M whose local flow covers the flow on W . A small disk D transversal to the periodic orbit L in W at a point x is covered by a small diskD in M transversal to the homeomorphic liftL of L in M . In M we can find a smaller diskÊ ⊂D centered onL for which a smooth Poincaré mapθ :Ê →D with smooth return timê τ :Ê → R is defined, cf. [3, A.14]. Setting E = p(Ê) we then obtain a smooth Poincaré map θ : E → D covered byθ whose first return time τ : E → R satisfies τ •p =τ and so is smooth as well. Let i > 0 and suppose thatÊ is sufficiently small so thatθ i|Γx| :Ê →D and θ i : E → D are defined. Then a preimage inÊ of a fixed point y ∈ E of θ i is a fixed point ofθ i|Γx| . Now we explain the proof of the orbifold version of Wadsley's result which implies our Proposition 4.6.
Proposition B.2. Let ϕ : R × O → O be a smooth action on an orbifold such that all orbits are periodic geodesics parametrized by arclength by the action. Then the R-action factors through an S 1 -action, i.e. all orbits have a minimal common period.
Proof. We only explain those parts of the proof in detail that require modifications compared to the proof in the manifold case given in [3]. Otherwise we refer to [3] for the details. We follow the notation from [3]. Let ̺ : O → R be the function that assigns to a point x ∈ O the length of the orbit through x. The function ̺ is not necessarily locally bounded. However, if the orbit of a point x 0 ∈ E (notation as in the preceding paragraph) defined by x i = θ i (x 0 ), i > 0, remains in E for every value of i, then we have the following relation. Since the periodic orbit through x meets E only a finite number of times, the θ-orbit of x is finite, say of order n(x), we have τ (x i ).
Since the local analysis using Poincaré maps is available in the orbifold setting, the following statement can be proven verbatim as in the manifold case, see [3,Prop. A.17].
Proposition B.3. Let L be a fixed orbit of length λ. Let k > 0 and ε > 0 be given. Then there is a neighborhood U of L such that if x ∈ U and ̺(x) ≤ k, then (i) for some integer i with 0 ≤ i ≤ ⌊k/λ⌋ + 1 < ε we have |̺(x) − iλ| < ǫ and (ii) the orbit through x lies in an ε-neighborhood of L.
Using this proposition the following corollary is proven in the same way as in [3,Cor. A.18].
Corollary B.4. The function ̺ is lower semicontinuous. If N is any compact subset of O, then the set V of points of continuity of ̺ |N is an open subset of N . If N is locally compact, then V is dense in N . Now suppose that ̺ is bounded by some k > 0 in a neighborhood of an orbit L through a point x. Then as in [3, A.19] one can apply Proposition B.3 to find a small Poincaré disk E ′ centered at x with θ(E ′ ) = E ′ . Since ̺ is bounded by k on E ′ the orders of the θ-orbits are bounded by some m. For such an m the composition θ m! is the identity on E ′ . The map θ : E ′ → E ′ is covered by a diffeomorphismθ :Ê ′ →Ê ′ of finite order of a diskÊ ′ in M . Using this cover the argument from [3, A.20] shows that the set of points in E ′ whose θ-orbit has the same order as θ, say r, is open and dense in E ′ . Therefore, equation (5) shows that the function α(x) = lim sup y→x ̺(y) satisfies (6) α(x) = r−1 i=0 τ (θ i x) for x ∈ E ′ . Since α is invariant under the flow, this shows that α is smooth in a neighborhood of L. Moreover, if X denotes the vector field defined by the R-action on O, then the flow of the vector field X/α in a neighborhood of L factors through an S 1 -action. Hence, in order to prove the proposition, it suffices to show that α is constant on O. Set B 1 = {x ∈ O | ̺ is unbounded in every neighborhood of x} and write ϕ t (x) for ϕ(t, x). By continuity we have ϕ α(x) (x) = x for all x ∈ O\B 1 . Let γ : (−1, 1) → O be a smooth path with γ(0) = x and consider the one-parameter family of geodesics defined by (s, t) −→ ϕ tα(γ(s)) (γ(s)) for t ∈ [0, 1]. For fixed s the length of this geodesic is α(γ(s)). On the other hand, the first variation formula, which also works for orbifolds, implies that this length is independent of s. This implies that α is locally constant on M \B 1 . Hence, in order to prove the proposition, it suffices to show that B 1 is empty.
The proof of B 1 = ∅ is an application of a theorem by Newman and works by contradiction. Since Newman's theorem only applies to manifolds the proof in the orbifold case has to be slightly refined as follows compared to the proof in [3, p. 220]. Let B 2 = {x ∈ B 1 | ̺ |B 1 is not continuous at x}. Corollary B.4 implies that B 1 is closed and has void interior in O and that B 2 is closed with void interior in B 1 . Let x ∈ B 1 \B 2 and let U be a connected open neighborhood of x which is a quotient of a manifold chart by the local group Γ x , which is disjoint from B 2 and such that (7) |̺(u) − ̺(v)| < ̺(w)/8 for u, v, w ∈ U ∩ B 1 .
Such an U exists since ̺ is lower semicontinuous on O and its restriction to B 1 is continuous on B 1 \B 2 . Let V be a component of O\B 1 which meets U and such that By continuity the restriction of the time c flow map ϕ c toV is the identity, and by definition of V we have ∂V ⊂ B 1 . Moreover, we have Since U is connected and meets B 1 , there is a point y ∈ U ∩ ∂V . By standard orbifold theory the order of Γ y divides the order of Γ x . Since ϕ c (y) = y, there is an integer k such that c = k̺(y). By (7) and (8), k ≥ 2|Γ y |. Let p : M → W be given by Lemma B.1 applied to the periodic orbit L through y. Set V ′ = p −1 (V ∩ W ) and letŷ ∈ M be a preimage of y. LetD be a small open disk of codimension one with p(D) ⊂ U , transverse to the orbit throughŷ and with centerŷ. LetÊ be a smaller concentric disk on whichθ i is defined for 1 ≤ i ≤ k!|Γ y |, with eacĥ θ iÊ ⊂D. By lower semicontinuity of ̺ and the fact that ϕ c is the identity onV we can assume that the restriction of θ k! to p(Ê) ∩V is the identity. Then also the restriction of θ k!|Γy| toÊ ∩V ′ is the identity. It follows as in [3] that there exists an open, connected and Γ y -invariant neighborhood J ofŷ in E witĥ θ(J ∩V ′ ) = J ∩V ′ .
We define ψ : J → J by ψ(z) =θ |Γy| (z) for z ∈ J ∩V ′ and ψ(z) = z otherwise. By (7) the restriction of θ to p(Ê) ∩ ∂V is the identity. Hence, also the restriction ofθ |Γy| toÊ ∩ ∂V ′ is the identity and so the map ψ is continuous. Since ψ k! = id, ψ is a homeomorphism of finite order. Since k ≥ 2|Γ y |, ψ is not the identity on J ∩ V ′ . By Newman's theorem J ∩ V ′ is dense in J. Since p is open, it follows that V is dense in a neighborhood of y, and hence ϕ c is the identity in a neighborhood of y. This contradicts the fact that any neighborhood of y contains points from B 1 , because of y ∈ ∂V and (9). The proof is complete.