On metrics on 2-orbifolds all of whose geodesics are closed

  • 1 Mathematisches Institut, Universität zu Köln, Weyertal 86–90, 50931, Köln, Germany
Christian LangeORCID iD: https://orcid.org/0000-0002-7338-5584

Abstract

We show that the periods and the topology of the space of closed geodesics on a Riemannian 2-orbifold all of whose geodesics are closed depend, up to scaling, only on the orbifold topology and compute it. In the manifold case we recover the fact proved by Gromoll, Grove and Pries that all prime geodesics have the same length, without referring to the existence of simple geodesics. We partly strengthen our result in terms of conjugacy of contact forms and explain how to deduce rigidity on the projective plane based on a systolic inequality due to Pu.

1 Introduction

Riemannian manifolds all of whose geodesics are closed have been studied since the beginning of the twentieth century, when the first nontrivial examples were constructed by Tannery and Zoll. The famous book of Besse [3] still describes the state of knowledge of the subject to a large extent. Some notable exceptions are concerned with relations between the lengths of geodesics on a Riemannian manifold all of whose geodesics are closed, henceforth called Besse manifold, and its topology. For instance a conjecture of Berger, stating that on a simply connected Besse manifold all prime geodesics have the same length, was proved by Gromoll and Grove for 2-spheres [14] and recently by Radeschi and Wilking for all topological spheres of dimension at least 4 [26]. Apart from spheres, which admit many Besse metrics, i.e. Riemannian metrics all of whose geodesics are closed, the only known Besse manifolds are the other compact rank one symmetric spaces. Moreover, it was shown by Pries that the conclusion of Berger’s conjecture also holds for the real projective plane [24], i.e. that all prime geodesics of a Besse metric have the same length.

Only little is known in the more general setting of Riemannian orbifolds. We define a Besse metric on an orbifold as a Riemannian orbifold metric all of whose orbifold geodesics are closed, and a Besse orbifold as an orbifold endowed with a Besse metric (cf. Section 2.1). On Besse orbifolds new phenomena occur that are not present in the manifold case. For instance, Berger’s conjecture does not hold for so-called spindle orbifolds [16], which admit many Besse metrics (see Section 2.2). However, it turns out that there is still a relation between the periods of geodesics on a Besse 2-orbifold and its topology. In fact, we generalize the results of Gromoll, Grove and Pries mentioned above in a unifying approach to the setting of Riemannian 2-orbifolds. We prove the following result.

Theorem A.

The geodesic periods of a Besse 2-orbifold are determined up to scaling by the orbifold topology. In the manifold case all prime geodesics have the same length.

The geodesic periods of a Besse 2-orbifold can be thought of as the set of lengths of prime (orbifold) geodesics counted with multiplicity. Note, however, that we prove the theorem for a slightly more general notion of geodesic periods; see Definition 2.17. By the orbifold topology we mean the orbifold diffeomorphism type, which can, in the case of 2-orbifolds, be encoded by finitely many numerical invariants (see Section 2.1).

In the manifold case the proofs by Gromoll, Grove and Pries hinge on the existence of at least three simple closed geodesics, i.e. closed geodesics without self-intersections (cf. Remark 3.2). By using a connectedness argument, they moreover show that all prime geodesics are simple. This observation combined with the Blaschke conjecture for S2 proved by Green [13] shows that a Besse metric on the real projective plane has constant curvature [24]. Our proof is independent of the existence of simple closed geodesics. Moreover, we show how rigidity on the real projective plane can be deduced from our result based on a systolic inequality due to Pu and the fact that the geodesic flows of any two Besse metrics on the real projective plane are conjugated by a contactomorphism (see Section A.1). The conjugacy of geodesic flows of Besse metrics is shown to hold on any 2-orbifold with isolated singularities (see Section 4), generalizing the case of the 2-sphere considered in [1].

The paper is structured as follows. After reviewing some preliminaries and examples, we first prove that a 2-orbifold admits a Besse metric if and only if it is either bad or spherical, in other words, if and only if its orbifold Euler characteristic is positive (see Proposition 2.7). Moreover, in Section 2.2 we explain that in many cases there exists an abundance of Besse metrics. The space of oriented prime geodesics on a Besse 2-orbifold 𝒪 has a natural orbifold structure 𝒪g and admits a natural involution i coming from time reversal. We call 𝒪g the orbifold of oriented geodesics and 𝒪g/i the orbifold of non-oriented geodesics, and prove the following rigidity result.

Theorem B.

For a Besse 2-orbifold O the orbifolds of oriented and non-oriented geodesics are determined by the orbifold topology of O. More precisely, the following cases can occur.

  1. (i)𝒪S2/G, 𝒪gS2/G× and 𝒪g/iS2/G* as orbifolds, where G<O(3) is a finite subgroup, G×={det(g)g:gG}<SO(3) and G*=G,-1<O(3).
  2. (ii)pq odd, 𝒪S2(p,q), 𝒪gS2(p+q2,p+q2) and 𝒪g/i2(p+q2).
  3. (iii)pq even, 𝒪S2(p,q), 𝒪gS2(p+qκ,p+qκ) and 𝒪g/iD2(p+qκ;) with κ being 1 or 2 depending on whether p+q is odd or even.
  4. (iv)pq odd, 𝒪D2(;p,q), 𝒪gS2(2,2,p+q2) and 𝒪g/iD2(2;p+q2).
  5. (v)pq even, 𝒪D2(;p,q), 𝒪gS2(2,2,p+qκ) and 𝒪g/iD2(;2,2,p+qκ) with κ being 1 or 2 depending on whether p+q is odd or even.

For explanations on the notations we refer to Section 2. The covering 𝒪g𝒪g/i encodes information on the geodesic periods of 𝒪 (see Section 2.7). In this way we will apply Theorem B in the proof of Theorem A. However, in Section 2.7 we show by example that the geodesic periods and the orbifolds of geodesics in general do not determine each other. For the proof of Theorem A in the case of orbifolds with non-isolated singularities additional geometric arguments are required (see Section 3.5). The proof of Theorem B relies on the following ideas. The unit tangent bundle M=T1𝒪 of a Besse 2-orbifold 𝒪 with isolated singularities is a manifold and the geodesic flow on it is periodic due to a result of Wadsley [32]. We obtain two transversal Seifert fiberings on M, one from the geodesic flow and another one from the projection T1𝒪𝒪. Properties of these Seifert fiberings and their interplay imply the claim in many cases. For Besse 2-orbifolds with codimension one singularities the result is obtained by considering the metric double cover. An explicit list of all 2-orbifolds admitting a Besse metric together with their orbifolds of geodesics and their geodesic periods can be found in the appendix, Table 1.

Since our approach does not rely on the existence of simple closed geodesics, it also works in more general Hamiltonian settings [10]. In [10] a Hamiltonian version of the result of Gromoll and Grove is proven, which could not have been obtained along the lines of the original proof. However, note that in general our result is a Riemannian phenomenon that cannot be seen from the Hamiltonian point of view. For instance, the real projective plane and the teardrop S2(3) have the same unit tangent bundle (see Lemma 3.1 and Section 3.2), but the geodesic periods of Besse metrics on them differ (see appendix, Table 1).

2 Preliminaries

2.1 Orbifolds

For a definition of a (smooth) orbifold we refer to [6, 9]. A Riemannian orbifold can be defined as follows.

Definition 2.1.

An n-dimensional Riemannian orbifold𝒪 is a length space such that for each point x𝒪, there exists a neighborhood U of x in 𝒪, an n-dimensional Riemannian manifold M and a finite group Γ that acts by isometries on M such that U with the restricted metric and M/Γ with the quotient metric are isometric.

A length space is a metric space in which the distance of any two points can be realized as the infimum of the lengths of all rectifiable paths connecting these points [8]. Behind the above definition lies the fact that an effective isometric action of a finite group on a simply connected Riemannian manifold can be recovered from the corresponding metric quotient. In the case of spheres this is proven in [30]; the general case can be deduced from it (see [21]). In particular, a Riemannian orbifold in the above sense admits a smooth orbifold structure and a compatible Riemannian structure that in turn induces the metric structure. For a point x on a Riemannian orbifold, the isotropy group of a preimage of x in a Riemannian manifold chart is uniquely determined up to conjugation. Its conjugacy class in O(n) is denoted as Γx and is called the local group of 𝒪 at x. Riemannian orbifolds are stratified by manifolds. The k-dimensional stratum consists of those points x𝒪 for which dim(Fix(Γx))=k.

The underlying topological space |𝒪| of a 2-orbifold 𝒪 is a manifold with boundary. In this case the orbifold 𝒪 is orientable if and only if |𝒪| is an orientable surface without boundary. A 2-orbifold can have three types of singularities. Isolated singularities in the interior of |𝒪| whose local groups are cyclic and orientation preserving, mirror singularities on the boundary of |𝒪| whose local groups are generated by a reflection, and corner-reflector singularities on the boundary of |𝒪| whose local groups are dihedral groups generated by two distinct reflections. The closure of the 1-dimensional stratum is the boundary of |𝒪| and consists of the mirror- and the corner-reflector singularities. We denote a 2-orbifold 𝒪 with l isolated singularities in the interior of |𝒪| (whose local group are) of order n1,,nl and k isolated corner-reflector singularities on the boundary of |𝒪| whose local groups are dihedral of order 2m1,,2mk by 𝒪=|𝒪|(n1,,nl;m1,,mk). If the boundary of |𝒪| is empty, we simply write 𝒪=|𝒪|(n1,,nl). We will use the following conventions. If two Riemannian orbifolds 𝒪 and 𝒪 are isometric, we write 𝒪=𝒪. If two (smooth) orbifolds are isomorphic, we write 𝒪𝒪. For 2-dimensional orbifolds 𝒪 and 𝒪 we have 𝒪𝒪 if and only if the underlying topological spaces of 𝒪 and 𝒪 are homeomorphic and the numerical invariants associated to 𝒪 and 𝒪 above coincide.

The metric quotient of a Riemannian 2-orbifold by a finite group of isometries is again a Riemannian 2-orbifold [21]. If 𝒪 is a Riemannian 2-orbifold with non-empty 1-dimensional stratum, then its metric double𝒪^ along the closure of this stratum (see [8, Definition 3.1.12]) is a Riemannian orbifold with isolated singularities. Moreover, in this case the natural involution of 𝒪^ is an isometry with quotient 𝒪 [21].

We are interested in (orbifold) geodesics in the following sense.

Definition 2.2.

A geodesic on a Riemannian orbifold is a continuous path that can locally be lifted to a geodesic in a Riemannian 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.

In particular, we are interested in Riemannian orbifold metrics all of whose geodesics “are closed”, i.e. factor through closed geodesics. We call such metrics Besse. We call a Riemannian orbifold whose metric is Besse a Besse orbifold.

Definition 2.3.

By the period of a closed geodesic we mean its length as a parametrized curve.

We stress this definition since the length of a geodesic on a Riemannian orbifold as a parametrized curve may differ from the length of its geometric image. To illustrate this and to provide some familiarity with geodesics on a Riemannian 2-orbifold 𝒪, let us summarize some of their properties. In the regular part geodesics behave like ordinary geodesics in a Riemannian manifold. A geodesic hitting an isolated singularity is either reflected or goes straight through it depending on whether the order of the singularity is even or odd. In particular, a closed geodesic that hits an isolated singularity of even order passes its trajectory twice during a single period. Moreover, we see that an orbifold geodesic is in general not locally length minimizing. However, on the other hand, a locally length minimizing path, which has to lie completely in either the 1- or the 2-dimensional stratum, is locally covered by length minimizing paths in Riemannian manifold charts and hence an orbifold geodesic. Suppose the 1-dimensional stratum of 𝒪 is non-empty and let 𝒪^ be the metric double of 𝒪 along the closure of this stratum. Then a geodesic hitting the topological boundary of 𝒪 continues as the projection to 𝒪 of its continuation in 𝒪^. In particular, a geodesic hitting the 1-dimensional stratum is reflected according to the usual reflection law. A geodesic that remains in the closure of the 1-dimensional stratum can never leave it.

We need the following concept.

Definition 2.4.

A covering orbifold of a Riemannian orbifold 𝒪 is a Riemannian orbifold 𝒪 together with a surjective map φ:𝒪𝒪 such that each point x𝒪 has a neighborhood U isometric to some M/G for which each connected component Ui of φ-1(U) is isometric to M/Gi for some subgroup Gi<G such that the isometries respect the projections.

For instance, if a finite group G acts isometrically on a Riemannian 2-orbifold 𝒪, then the projection 𝒪𝒪/G is a covering of Riemannian orbifolds. Thurston showed that the theory of covering spaces (and fundamental groups) works for orbifolds analogously as for manifolds [9, Section 1.2.2], see [31] and e.g. [4, Section 2.2].

Note that a finite covering orbifold of a Besse orbifold is itself Besse. In particular, the metric double cover of a Besse 2-orbifold with mirror singularities is a Besse 2-orbifold with isolated singularities. An orbifold is called good (or developable) if it is covered by a manifold, and otherwise it is called bad [31]. The only bad 2-orbifolds are depicted in Figure 1 (see [28, Theorem 2.5]).

For orbifolds an Euler characteristic can be defined that is multiplicative under coverings and coincides with the usual Euler characteristic in the manifold case [9]. A 2-orbifold has positive Euler characteristic if and only if it is either bad or spherical, i.e. a quotient of S2 by a finite subgroup of O(3). All spherical 2-orbifolds are listed in Table 1 for p=q (cf. [9]). A detailed description of the corresponding finite subgroups of O(3) can for instance be found in [19]. Spherical 2-orbifolds inherit a standard Besse metric from S2. In Proposition 2.7 we will see that a 2-orbifold admits a Besse metric if and only if its orbifold Euler characteristic is positive. The 2-orbifolds with positive Euler characteristic are listed in Table 1.

Figure 1
Figure 1

Left: A (p,q)-spindle orbifold S2(p,q). Right: A (p,q)-half-spindle orbifold D2(;p,q) (no assumptions on p and q). (p,1)-spindle orbifolds are also known as teardrops. The orbifolds in the picture are bad if and only if pq.

Citation: Journal für die reine und angewandte Mathematik 2020, 758; 10.1515/crelle-2017-0050

2.2 Besse metrics on 2-orbifolds

In [16] a Besse (p,q)-spindle orbifold (cf. Figure 1) is constructed for integers p and q with gcd(p,q)=1 and 1<p<q as follows. Consider the action

S1×S3S3,(z,(z1,z2))(zpz1,zqz2).

The quotient S3/S1 is a Besse (p,q)-spindle orbifold. The geodesics on S3/S1 are precisely the projections of horizontal geodesics on S3, i.e. geodesics that are orthogonal to the S1-orbits. In [16, Theorem 3.6] the lengths of the geodesic’s trajectories on the quotient S3/S1 are computed. The difference to our result in the case that p+q is odd is due to the fact that the length of a geodesic’s trajectory may differ from its length as a parametrized curve as explained in the preceding section.

Another construction of Besse metrics on (p,q)-spindle orbifolds for arbitrary p and q is similar to the construction of non-standard Zoll metrics on S2 due to Tannery and Zoll [3, 35]. Let h:[-1,1](-p+q2,p+q2) be a smooth, odd function with

h(1)=p-q2=-h(-1)

and let a Riemannian metric on X=(0,π)×([0,2π]/02π)(θ,ϕ) be defined by

g=(p+q2+h(cos(θ)))2dθ2+sin2(θ)dϕ2.

Then the metric completion of (X,g) is a Besse (p,q)-spindle orbifold; see [3, Theorem 4.13] and note that p and q are defined differently therein. Since the metric is invariant under a reflection in ϕ([0,2π]/02π), it descends to a Besse metric on the corresponding quotient. Hence, we have

Proposition 2.5.

Every bad 2-orbifold and every spherical 2-orbifold, that is, every 2-orbifold with positive Euler characteristic, admits a Besse metric.

Another method to construct Besse metrics on 2-spheres is due to Guillemin [15]. He shows that for any odd function σ on the standard round sphere (S2,g0) there exists a one-parameter family of smooth functions ρt on S2 such that ρ0=1, dρtdt=σ at t=0 and such that exp(ρt)g0 is a Besse metric for small t. Note that if G<O(3) does not contain minus the identity, then σ can be chosen to be nontrivial and G-equivariant.

On spherical orbifolds with more than two isolated singular points only the round Besse metrics seem to be known. However, in view of the result in [15] we believe in the following:

Conjecture.

Let G<O(3). If S2/G is not covered by the real projective plane, i.e. if -1G, then the moduli space of Besse metrics on S2/G is infinite-dimensional.

2.3 2-orbifolds that admit Besse metrics

In [32] Wadsley proves the following result (cf. [3, Theorem 7.12]).

Theorem (Wadsley).

If the orbits of a flow on a Riemannian manifold are periodic geodesics parametrized by arc-length, then the flow itself is periodic, so that the orbits have a common period.

Using Wadsley’s theorem, we can prove the following property of Besse 2-orbifolds.

Proposition 2.6.

A Besse 2-orbifold is compact.

Proof.

First suppose that 𝒪 is a Besse 2-orbifold with isolated singularities. Then the unit tangent bundle M=T1𝒪 is an orientable manifold [2]. It inherits an orientation and a natural Riemannian metric (Sasaki metric) from 𝒪 (see [3, Chapter 1.K]). With respect to this metric the integral curves of the geodesic field on M are geodesics that project to the geodesics on 𝒪 with the same arc-length parametrization and the same period. All the geodesics on 𝒪 can be obtained in this way (see [3, Chapter 1.K]). It follows from Wadsley’s result that these integral curves on M have a common period, say l, and thus so have the geodesics on 𝒪.

Since 𝒪 is geodesically complete as an orbifold by the Besse condition, every length minimizing path γ:[0,a)𝒪, which is also an orbifold geodesic, can be extended to a continuous path γ¯:[0,a]𝒪. Therefore 𝒪 is complete by the Hopf–Rinow theorem [8, Theorem 2.5.28] and every pair of points on 𝒪 can be connected by a (minimizing and hence orbifold) geodesic [8, Theorem 2.5.23]. It follows that diam(𝒪)l and thus that 𝒪 is compact [8, Theorem 2.5.28 (ii)]. For a Besse 2-orbifold 𝒪 whose singular points are not isolated the same argument applies to its metric double, which is a Besse 2-orbifold with isolated singularities. Hence, in this case 𝒪 is compact as the continuous image of its double. ∎

Now we can prove the following characterization of 2-orbifolds that admit Besse metrics.

Proposition 2.7.

A 2-orbifold admits a Besse metric if and only if it is either bad or spherical, i.e. if and only if its orbifold Euler characteristic is positive.

Proof.

By Proposition 2.5 it remains to prove the only if direction. So let 𝒪 be a Besse 2-orbifold. By Proposition 2.6, 𝒪 is compact. If it is also good, then it is in fact finitely covered by a Besse manifold [28, Theorem 2.5]. Since the fundamental group of a Besse manifold is finite [3, Theorem 7.37], 𝒪 must be spherical in this case. ∎

In the following three sections we recall some facts that will be needed later. The reader may proceed to Section 2.7 on first reading and come back to these sections on demand.

2.4 Almost free circle actions on 3-manifolds and Seifert fiber spaces

Suppose we have a smooth, effective, almost free (i.e. isotropy groups are finite) S1-action on an orientable closed 3-manifold M. Then the orbits are circles and define a decomposition of M into so-called fibers. If some element of S1 fixes a point on a fiber, then it fixes the fiber pointwise. A fiber is called exceptional (or singular) of order k2 if its isotropy subgroup of S1 has order k. Since S1 is compact, there exists an S1-invariant Riemannian metric on M. The metric quotient M/S1 is an orientable Riemannian 2-orbifold with isolated singularities and (the orders of) the exceptional fibers correspond to (the orders of) the singularities of M/S1 (cf. [23]). The manifold M together with a chosen orientation and its decomposition into fibers defines a Seifert fiber space of type o1, or Oo in Seifert’s original notation [29], meaning that both the space and the base are orientable. For the definition of a general Seifert fiber space we refer to [29]. Roughly speaking it is a closed 3-manifold together with a decomposition into S1-fibers which are, however, only locally defined by an S1-action. Conversely, any Seifert fiber space of type o1 can be obtained in the above way [17, Chapter 2].

Remark 2.8.

The fibers of a Seifert fiber space of type o1 can be oriented in a continuous way. When we speak about orientations of the fibers, we always mean such a continuous choice.

A Seifert fiber space (of type o1) is uniquely determined by a finite number of numerical invariants up to orientation- and fiber-preserving diffeomorphism [17, Theorem 1.5]. Two sets of invariants determine the same Seifert fiber space if and only if they are related as described in [17, Theorem 1.5]. Forgetting about the orientation of M and allowing general fiber-preserving diffeomorphisms amounts to enlarging the equivalence relation on the set of numerical invariants [17, Corollary 1.7]. In [27, equation (6.1)] it is shown that two S1-actions on M define the same Seifert fiber space up to orientation, if and only if there exists a diffeomorphism h of M and an automorphism a of S1 such that for all mM and gS1 the relation h(gm)=a(g)h(m) holds. In this case it can be shown that this automorphism can in fact be chosen to be the identity [17, pp. 12–13]. A specific diffeomorphism that conjugates the S1-actions occurring in this paper to their inverse actions is given in Section 2.7 (namely by the map i:T1𝒪T1𝒪). Hence, we have:

Lemma 2.9.

The classification of smooth, almost free S1-actions on M up to conjugation by diffeomorphisms coincides with the classification of Seifert fiberings on M of type o1 up to orientation.

A covering of a Seifert fiber space is a covering that restricts to coverings of fibers on preimages of fibers.

Remark 2.10.

Let M be k-foldly covered by S3 via a map σ and let τk:S1S1 be a k-fold Lie group covering. Then the action of S1 on M via τk can be lifted to an action on S3 since the map S1×S3M, (θ,p)τk(θ)σ(p) is trivial on fundamental groups. In particular, we see that Seifert fiberings on M of type o1 can be lifted to S3. Moreover, since σ preserves the orientations of the fibers induced by the S1-actions, also the group of deck transformations preserves the orientations of the fibers.

Finally, note that the classifications of Seifert fibered spaces (of type o1) in the topological and the smooth category coincide [7, 17, 29].

2.5 Seifert fiberings of lens spaces

Seifert fiberings of lens spaces are described in [12]. Here we remind of some facts. Recall that for coprime integers p,q0 the L(p,q)lens space is defined as a quotient of S3={(z1,z2)2:|z1|2+|z2|2=1} by the free p-action on S3 generated by e2πi/p(z1,z2)=(e2πi/pz1,e2πiq/pz2). Recall that two lens spaces L(p,q) and L(p,q) are diffeomorphic if and only if p=±p and q±q±1 mod p [17, pp. 28–29]. Given a pair of coprime natural numbers α1,α2, a Seifert fibering on S3 can be defined by the action θ(z1,z2)=(eiθα1z1,eiθα2z2). This Seifert fibering descends to a Seifert fibering of L(p,q) and every Seifert fibering on L(p,q) with orientable base can be obtained in this way [12, Theorem 5.1.].

An alternative description of lens spaces and Seifert fiberings on them works as follows. Suppose we have two solid tori T1 and T2 and a diffeomorphism ψ:T1T2. Then the space T1ψT2 obtained by gluing together T1 and T2 via ψ is a lens space. Moreover, if we choose meridiansmi on Ti, i.e. embedded loops in Ti that are null-homotopic in Ti and that generate maximal subgroups of H1(Ti), and longitudesli on Ti, i.e. embedded loops in Ti that generate maximal subgroups of H1(Ti), and if we have ψ(m1)sm2+rl2 in H1(T2), the space T1ψT2 is an L(r,-s) lens space, see [7, Theorem 1.3.4.] and [17, Theorem 4.3]. A standard fibered solid torus is a solid torus T=D2×S1 fibered by the orbits of the almost free S1-action eit(reit0,eit1)=(reiteit0,eikteit1) for some positive integer k. Every smooth Seifert fibering on a solid torus T is fiber-preservingly diffeomorphic to precisely one of the standard fibered solid tori. Suppose that the solid tori T1 and T2 in the situation above are Seifert fibered and that the diffeomorphism ψ:T1T2 preserves fibers. Then we obtain an induced Seifert fibering of the lens space T1ψT2. Moreover, if m1 is a meridian of T1, then the fiber-homeomorphism (and hence fiber-diffeomorphism) type of T1ψT2 is completely determined by the homology class of ψ(m1) in H1(T2) (see [7, Theorem 1.3.4.]).

2.6 Finite group actions on S1 and S2

We will encounter isometric actions of finite groups on Riemannian 2-orbifolds 𝒪 with |𝒪|=S2. Such an action can be smoothed, i.e. there exists a smooth structure on |𝒪| with respect to which the group acts smoothly. Indeed, the orbifold admits an equivariant triangulation and the corresponding simplicial complex can be equivariantly smoothed [22]. Then it follows from the classification of 2-orbifolds with positive Euler characteristic (see [9]) that the action can be conjugated to a linear action on S2 (cf. [34]).

In a similar way one can show that a continuous action of a finite group on a circle can be conjugated by an orientation-preserving homeomorphism to a linear action. Hence, if such an action preserves the orientation of the circle, then it must be cyclic. Moreover, if the order of an orientation-preserving homeomorphism h of the circle S1 is at least 3 but finite, then its linearized action rotates the circle about an angle different from π. In this situation we say that the circle is rotated in a positive or negative direction with respect to a chosen orientation if the angle rotated by the linearized action (obtained through conjugation by an orientation-preserving homeomorphism) measured with respect to the chosen orientation is smaller or greater than π. Observe that h rotates S1 in a positive direction if and only if, following S1 from x in the positive direction, one encountersh(x) before h2(x). In particular, in this way it makes sense to say that a homeomorphism rotates two fibers of a Seifert fiber space of type o1 in the same or in different directions, cf. Remark 2.8. We will need the following statement.

Lemma 2.11.

Let (M,F) be a connected Seifert fiber space of type o1 with only regular fibers. Let h be a homeomorphism of M of finite order n3 that leaves all fibers invariant and preserves their orientation. Suppose that the restriction of h to each fiber has order n. Then h rotates all fibers in the same direction.

Proof.

By the connectedness assumption it suffices to prove the conclusion for a standard fibered solid torus with only regular fibers, i.e. for D2×S1. In this case the conclusion follows from continuity and the “first encounter criterion” above by looking at the orbits of a section S=D2×{*} of the fibered solid torus under h. Indeed, if two fibers were rotated in different directions, then the images h(S) and h2(S) of S would have a nontrivial intersection by the first encounter criterion, resulting in a fixed point of h. This can only happen if h is the identity, in contradiction to our assumption on the order of h. ∎

Remark 2.12.

It can be shown that the assumption in the lemma on the orders of the restrictions of h to the fibers actually follows, too. Moreover, these conclusions still hold for the regular fibers of a general Seifert fiber space of type o1 with a homeomorphism h as in the lemma. However, all our assumptions will be satisfied in our application in Lemma 3.9.

2.7 Orbifolds of geodesics and geodesic periods

In the following a geodesic on a Besse orbifold is supposed to be prime unless stated otherwise. Suppose that 𝒪 is a Besse 2-orbifold with only isolated singularities. This is in particular the case if 𝒪 is orientable. Then M=T1𝒪 is a manifold and the geodesic flow on it is periodic due to Wadsley’s theorem [32] as discussed in Section 2.3. Hence, the flow defines a Seifert fibering g of type o1 on M=T1𝒪 whose fibers inherit a natural orientation from the flow. The quotient 𝒪g=M/g parametrizes the closed orbits of the geodesic flow on M=T1𝒪 and has a natural orbifold structure with isolated singularities, see Section 2.4. A point on 𝒪g of order k corresponds to (an equivalence class of reparametrizations of) a geodesic on 𝒪 whose period is k-times shorter than the period of a generic geodesic on 𝒪. We say that this geodesics is of order k and we call it exceptional if k>1 and regular otherwise. We will see that in this situation 𝒪g is always a quotient of S2 by a finite subgroup of SO(3) and as such a topological sphere [20].

A non-orientable Besse 2-orbifold 𝒪 is a metric quotient of an orientable Besse 2-orbifold 𝒪^ with isolated singularities by an isometric orientation reversing involution s. Since the S1-action on T1𝒪^ defining g is normalized by s, the auxiliary metric on T1𝒪^ can be chosen to be s invariant (cf. Section 2.4) so that s induces an orientation preserving isometry s:𝒪^g𝒪^g. In particular, s either acts trivially on 𝒪^g or rotates 𝒪^g around two points (cf. Section 2.6). We have T1𝒪=T1𝒪^/s (cf. [2]) and set 𝒪g:=𝒪^g/s, which is a Riemannian orbifold. Since 𝒪^g will always be topologically a sphere, so will be 𝒪g. The points on 𝒪g still correspond to (equivalences classes of reparametrizations of) geodesics on 𝒪. However, if the singularities of 𝒪 are not isolated, then T1𝒪 has orbifold singularities that cause singularities on 𝒪g whose orders do not directly correspond to the periods of the respective geodesics on 𝒪. In this situation the following lemma provides information on the periods of geodesics on 𝒪.

Lemma 2.13.

Suppose that O^O is the orientable double cover of a non-orientable Besse 2-orbifold O and that s:O^O^ generates the group of deck transformations. Then a prime geodesic on O^ projects to a geodesic on O. This geodesic on O is not prime if and only if the geodesic on O^ is fixed by s as an element of O^g but not pointwise fixed as a fiber of Fg. In this case the corresponding prime geodesic on O is covered twice by the geodesic on O^.

Proof.

The projection of a geodesic on 𝒪^ to 𝒪 is a geodesic by definition, since a small neighborhood in 𝒪, isometric to some M/G as in Definition 2.1, is covered by a small neighborhood in 𝒪^ isometric to M/H for an index-2 subgroup H of G. A closed geodesic γ (say of period 1) is prime if and only if for each positive integer n and some (and then all) t[0,1-1n] we have γ(t)γ(t+1n) as elements of the unit tangent bundle. A prime geodesic γ on 𝒪^ (of period 1) is invariant under s (i.e. fixed as an element of 𝒪^g) if and only if s(γ(0))=γ(t0) for some t0[0,1] (and then also for some t0{0,12} since s has order 2). It is pointwise fixed by s if and only if for some (and then each) t[0,1] we have s(γ(t))=γ(t). Putting this together proves the second claim. In this case we have s(γ(0))=γ(12)γ(0) and so the last claim follows, too. ∎

Examples for the two possible cases in the lemma are given by a reflection and an inversion of S2. In the first case geodesics in the fixed point set of s are both s-invariant and pointwise fixed by s. In the second case every geodesic is s-invariant but not pointwise fixed by s.

The orbifolds 𝒪g obtained in this way from Besse 2-orbifolds have the following additional symmetry. Consider the involution i:T1𝒪T1𝒪 mapping (x,v) to (x,-v). The involution i is orientation-preserving and interchanges fibers of g representing different orientations of the same geodesic trajectory on 𝒪 while reversing their natural orientation. We can suppose that the auxiliary Riemannian metric on T1𝒪 is also invariant under i (cf. preceding paragraph and Section 2.4). Then i induces an involutive orientation-reversing isometry of 𝒪g with the quotient being a Riemannian orbifold. In particular, it maps singular points to singular points of the same order. We introduce the following concept; cf. [3, Section 2.5] for the manifold analogue.

Definition 2.14.

For a Besse 2-orbifold 𝒪 we define the orbifold of oriented geodesics to be 𝒪g and the orbifold of non-oriented geodesics to be 𝒪g/i.

Recall that we sometimes view geodesics on 𝒪 as points on 𝒪g, that is we forget about the specific reparametrization. We distinguish two kinds of geodesics on 𝒪.

Definition 2.15.

We call a geodesic on 𝒪self-inverse if it is a branch point of the covering 𝒪g𝒪g/i, that is, if it is fixed by i as a point on 𝒪g .

The following statement is a consequence of the discussion after Definition 2.3.

Lemma 2.16.

A geodesic on a Besse 2-orbifold O is self-inverse if and only if it hits an isolated singularity of even order on O or the boundary of |O| perpendicularly (in the sense of centrically at the corner reflector singularities of odd order, cf. Section 2.1). In particular, if Og is topologically a sphere, then the orbifold of non-oriented geodesics Og/i is topologically a disk in the case that O has isolated singularities of even order or a topological boundary, and otherwise a projective plane.

Proof.

By definition a geodesic γ on 𝒪 (say of period 1) is self-inverse if it is fixed by i as an element of 𝒪g. This is precisely the case if -γ(0)=γ(t0) for some t0[0,1]. In this case we have -γ(t02)=γ(t02) in Tγ(t0)𝒪. This happens if and only if γ hits a singular point of 𝒪 at γ(t0) as described in the lemma. For the second claim recall that i reverses the orientation of 𝒪gS2 and is thus either conjugated to a reflection or the inversion by Section 2.6. Hence, if i has a fixed point it is conjugated to a reflection and the corresponding quotient is a disk. Otherwise, it is conjugated to the inversion and the quotient is a projective plane. ∎

By the trajectory of a geodesic we mean its geometric image in 𝒪. The trajectories of geodesics are in one-to-one correspondence with the geodesic’s images in 𝒪g/i. Since the period of a geodesic is i-invariant, we can talk about periods of geodesic trajectories. By the (non)-self-inverse geodesic periods of 𝒪 we mean the set of periods of (non)-self-inverse geodesic trajectories on 𝒪 counted with multiplicity.

Definition 2.17.

By the (labeled) geodesic periods of a Besse 2-orbifold 𝒪 we mean the data encoded in its self-inverse and its non-self-inverse geodesic periods.

In the following we suppose that all Besse 2-orbifolds are normalized such that their maximal geodesic period is one. We will see that for a Besse 2-orbifold 𝒪 almost all geodesic trajectories on 𝒪 have the same regular maximal period and that the periods of the finitely many exceptions are integer factors of this regular period. Suppose that the periods of the exceptional non-self-inverse geodesic trajectories are specified by the integer factors k1,,kl and the periods of the exceptional self-inverse geodesic trajectories are specified by the integer factors k1,,kl. We will also see that there always exists a geodesic of maximal period which is non-self-inverse and that there are either infinitely many or no geodesics of maximal period that are self-inverse. In this case, the geodesic periods of 𝒪 can be recorded symbolically as the labeled unordered tuple (1,k1¯,,kl¯,k1,,kl) or (k1¯,,kl¯,k1,,kl) depending on whether there exist geodesics of maximal period that are self-inverse or not. For ease of parlance we agree upon calling this tuple the geodesic periods of 𝒪.

To summarize, in case of a Besse 2-orbifold with isolated singularities the geodesic periods and the data encoded in the covering 𝒪g𝒪g/i determine each other. In fact, the map i acts as a reflection or an inversion on the topological sphere 𝒪g depending on whether 1 occurs in the geodesic periods or not, the pairs of singularities on 𝒪g interchanged by i correspond to the k1¯,,kl¯, and the singularities on 𝒪g in the fixed point set of i correspond to the k1,,kl. However, in general these data do not determine each other. For instance, S2(2,2) and D2(4;) with some Besse metric have the same geodesic periods, but different orbifolds of oriented and non-oriented geodesics. Also, the orbifolds of oriented and unoriented geodesics of D2(;2) and D2(;4,2), or of S2(2,3,4) and D2(2,3,4) with some Besse metric coincide, but (even) their (“unlabeled”) geodesic periods differ (see Section 3.3). Hence, in order to prove rigidity of the geodesic periods in case of an orbifold with non-isolated singularities, a more detailed analysis involving geometric arguments based on Lemma 2.13 will be necessary; see Section 3.3 and Section 3.5.

3 Proof of the main result

In this section we show our main results on geodesic periods and orbifolds of geodesics of Besse 2-orbifolds. We first treat the case of spindle orbifolds which, together with the case of the real projective plane, forms a central part of our proof.

3.1 Spindle orbifolds

Let 𝒪 be a Besse (p,q)-spindle orbifold. Recall that we do not make assumptions on gcd(p,q). We claim that the orbifold of geodesics 𝒪g and 𝒪g/i and the geodesic periods are given as stated in Table 1 in the appendix. In particular, we show that 𝒪g=S2(p+qκ,p+qκ) where κ is 1 or 2 depending on whether p+q is odd or even. The proof is divided into steps (a)–(f).

Step (a)

Recall from Section 2.7 that the unit tangent bundle M=T1𝒪 is a manifold.

Lemma 3.1.

The unit tangent bundle M=T1O of a (p,q)-spindle orbifold O is a lens space. More precisely, we have ML(p+q,1).

Proof.

To prove the lemma we choose an equator of 𝒪 that separates the two singular points such that 𝒪 decomposes into two closed disks Di contained in open sets Ui, i=1,2, each of which admits an orbifold chart U~i. We denote the cyclic group acting on U~i by Γi and the preimage of Di in U~i by D~i, i=1,2. The space M decomposes accordingly into the preimages Ti of the disks Di that are contained in the open preimages Vi of the Ui. Let V~i be the covering chart of Vi induced by the chart U~i and let T~i be the preimage of Ti in V~i. Then T~i is the restriction of the unit tangent bundle of U~i to the disk D~i and as such a full torus. We choose a smooth and orientation-preserving identification of U~i with an open ball in 2 with respect to which the action of Γi becomes linear, and which restricts to a diffeomorphism D2D~i. This induces a diffeomorphism D2×S1T~i that maps the S1-fibers to the fibers of the foot-point projection π~i:T~iD~i, D2 to a section of π~i, and with respect to which Γi acts diagonally in both factors by rotations. In particular, the Ti are full tori themselves. The space M can be recovered from these full tori by a specification of the gluing homeomorphism ψ:T1T2. The homeomorphism type of T1ψT2 is determined by the homology class of ψ(m1) in T2 for a meridian m1 of T1, i.e. an embedded loop in T1 that is null-homotopic in T1 and that generates a maximal subgroup of H1(T1) (cf. Section 2.5). With respect to the splitting above we define two loops c~1,c~1:S1T~1=S1×S1 by c~1(z)=(z,z) and c~1(z)=(1,z) and two loops c~2,c~2:S1T~2 analogously. Note that the orientations of the fibers are chosen in such a way that the loops c~i are invariant under Γi. We can choose meridians m~i of T~i such that in homology of T~i we have m~i-c~i+c~i. The meridians m~i of T~i project to meridians mi of Ti. Let ci:S1Ti be the projection of c~i and let ci:S1Ti be curves such that in homology c~i projects to rici, where r1=p and r2=q. Then in homology we have mi-rici+ci. Observe that we recover M if the attaching map ψ satisfies ψ(c1)-c2 and ψ(c1)c2, see Figure 2. Picking l2=c2 as a longitude in T2, i.e. an embedded loop in T2 that generates a maximal subgroup of H1(T2) (cf. Section 2.5), we have ψ(m1)pc2+c2=1m2+(p+q)l2. The resulting space T1ψT2 is a lens space of type L(p+q,-1)L(p+q,1) as claimed (cf. Section 2.5). ∎

Figure 2
Figure 2

Gluing construction in Lemma 3.1. Note that the curve c1 is homotopic to the curve c2-1, that is, c2 traversed in the opposite direction.

Citation: Journal für die reine und angewandte Mathematik 2020, 758; 10.1515/crelle-2017-0050

Step (b)

Recall from Section 2.7 that g denotes the Seifert fibering of type o1 on M=T1𝒪 defined by the geodesic flow. As an auxiliary tool we also need the Seifert fibering t on M induced by the projection M=T1𝒪𝒪. Since both M and 𝒪 are orientable, this Seifert fibering is of type o1, too (cf. Section 2.4). We have M/t=𝒪 as a Riemannian orbifold. By Remark 2.10 the Seifert fiberings t and g can be lifted to Seifert fiberings of type o1 of the universal cover M~ of M. We denote these lifts by ~t and ~g, and the corresponding quotients by 𝒪~t:=M~/~t and 𝒪~g:=M~/~g. The fiberings t and g on M as well as their lifts ~t and ~g on M~ are fiberwise transversal.

Step (c)

We have the commutative diagram

article image

where the outer vertical projections are coverings of Riemannian orbifolds. The upper horizontal projections induce surjections π1(M~)π1orb(𝒪t~) and π1(M~)π1orb(𝒪g~) (see [28, Lemma 3.2]). According to the classification of simply connected, compact 2-orbifolds [28, Theorem 2.5] this implies 𝒪~gS2(pg,qg) and 𝒪~tS2(pt,qt) as orbifolds for coprime pg,qg and coprime pt,qt. In other words, Seifert fiberings on S3 are uniquely determined by two coprime positive integers [29] (cf. [12, Proposition 5.2.]). Let Γp+q be the group of deck transformations of the covering M~M. The action of Γ on M~ induces actions on 𝒪~t and 𝒪~g which are not necessarily effective. We denote by Γt and Γg the quotients of Γ by the respective kernels. The groups Γt and Γg are cyclic and we have 𝒪~t/Γt=𝒪 and 𝒪~g/Γg=𝒪g. Since the group Γ preserves the orientations of the fibers of ~t and ~g (cf. Remark 2.10) and the orientation of M~, the groups Γt and Γg preserve the orientation of |𝒪~t|S2 and |𝒪~g|S2, respectively. Therefore, the action of Γt and of Γg can be conjugated to a standard action of a cyclic group on S2 (cf. Section 2.6). Moreover, since Γ acts isometrically on 𝒪~t and 𝒪~g with respect to the orbifold metrics introduced above (cf. Step (b)) and Section 2.4), it fixes the singular points. Both together implies p=|Γt|pt, q=|Γt|qt (up to permutation) and 𝒪g=S2(|Γg|pg,|Γg|qg).

Remark 3.2.

In the case of 𝒪S2 in [14] the theorem of Lusternik–Schnirelmann, which guarantees the existence of three simple closed geodesics, is applied at this point to show that there exists a simple regular geodesic. It is then not difficult to conclude that all geodesics are simple and regular; see [14].

Step (d)

Let i~:M~M~ be a lift of the involution i:MM introduced in Section 2.7.

Lemma 3.3.

The lift i~ preserves the orientations of the fibers of F~t while it reverses the orientations of the fibers of F~g.

Proof.

The claim follows from the respective property of the action of i on the fibers of t and g. ∎

If pq is odd, then i:𝒪g𝒪g has no fixed points by Lemma 2.16. Hence, in this case it follows that |Γg|pg=|Γg|qg, and thus that pg=qg=1 since pg and qg are coprime. For even pq the same conclusion will follow from the subsequent lemma.

Lemma 3.4.

The actions of i~ and Γ on M~S3 commute.

Proof.

Let St1 be a Γ-invariant fiber of ~t and let γΓ be nontrivial. Since i leaves the fibers of t invariant, its lift i~ leaves preimages of t-fibers invariant. In particular, we see that i~ leaves St1 invariant. The orientation of St1 is preserved by γ due to Remark 2.10 and by i~ due to Lemma 3.3. As a lift of i the map i~ normalizes Γ. Therefore the restrictions of γ and i~ to St1 generate a finite group that preserves the orientation of St1. Since the action can be conjugated to a linear (orientation-preserving) action on a circle (cf. Section 2.6), we see that the generated group must be cyclic and thus γ and i~ commute on St1. The fact that their commutator γi~γ-1i~-1:M~M~ is a lift of the identity of M implies that γ and i~ commute everywhere. Hence, the claim follows. ∎

Indeed, now we can show the following:

Lemma 3.5.

The involution i does not fix singular points on Og. In particular, geodesics hitting a singularity on O of even order are regular.

Proof.

By Lemma 2.16 we only need to consider the case that pq is even. Suppose that a singular point on 𝒪g is fixed by i. We have seen in (c) that this singular point has a single preimage in 𝒪~g whose corresponding fiber Sc1 of ~g and its orientation are Γ-invariant. Therefore the map i~ also leaves the fiber Sc1 invariant (cf. proof of Lemma 3.4) but it reverses its orientation by Lemma 3.3. Moreover, as in the proof of Lemma 3.4 we see that Γ and i~ generate a finite group acting on Sc1, and that the action can be assumed to be linear. The fact that the orientation is not preserved by this action implies that the generated group must be a dihedral group. Since Γ and i~ commute by Lemma 3.4, this dihedral group must be abelian. The only abelian dihedral groups have order 2 and 4, respectively. Since the action of Γ on Sc1 is effective (as a restriction of a deck transformation action), it follows that 2p+q=|Γ|2. This contradicts the existence of a singular point on 𝒪 of even order and thus the first claim follows. Now, the second claim is a consequence of Lemma 2.16. ∎

Consequently, in any case we have 𝒪g=S2(|Γg|,|Γg|) and so it remains to determine the order of Γg.

Step (e)

An example of a Seifert fibering on S3 is the Hopf fibration defined by the free S1-action φ+ (or φ-)

φ±:S1×S3S3,(eit,(z1,z2))(eitz1,e±itz2).

Since the actions of φ+ and φ- commute, they induce almost free actions

φ±:S1×L(r,1)L(r,1)

and Seifert fiberings ± on L(r,1), where L(r,1) is the quotient of S3 by the action of φ+ restricted to the r-th roots of unity in S1. The following lemma shows that these are the only actions that can occur in our situation up to conjugation. For a systematic classification of Seifert fiberings of lens spaces we refer the reader to [12]. Given Lemma 2.9, the following lemma is contained therein as a special case [12, Example 4.17 and Section 5]. Here we give a short, self-contained proof.

Lemma 3.6.

Let φ:S1×L(r,1)L(r,1), r>1, be a smooth, effective, almost free action with quotient orbifold of type S2(k,k) for some positive integer k. If there are no exceptional fibers, i.e. if k=1, then φ is smoothly conjugated to φ+. If k>1, then φ is smoothly conjugated to φ-. In the latter case we have r=κk, where κ is 1 or 2 depending on whether r is odd or even. In particular, r is divisible by 4 if k is even.

Remark 3.7.

In the situation of the lemma the quotient orbifold is actually always a spindle orbifold. The only real assumption is that the orders of the singularities coincide.

Proof.

Let be the Seifert fibering defined by φ. By Lemma 2.9 it is sufficient to show that the Seifert invariants of coincide up to orientation with either those of + or -.

Since L(r,1)/ is an S2(k,k) orbifold by assumption, we can obtain the Seifert fiber space (L(r,1),) as T1ψT2, where T1 and T2 are fibered solid tori with an exceptional fiber of order k and where ψ:T1T2 is a fiber-preserving homeomorphism [7, Theorem 1.4.5.]. Choose meridians m1 and m2 on T1 and T2 (cf. Section 2.5) and a longitude l2 on T2 (cf. Section 2.5). In homology we have ψ(m1)sm2+rl2 for some integers s,r. Since T1ψT2 is an L(r,1) lens space with r>1 by assumption, we have r=ε1r and s=ε2+tr0 for some integer t and some ε1,ε2{±1} (cf. Section 2.5). Replacing l2 by tm2+ε1l2 and m2 by ε2m2, we can assume that ψ(m1)m2+rl2. In particular, l1=ψ-1(l2) is a longitude of T2. Let f2 be a regular fiber on T2. Possibly after reversing the orientation of f2, we have f2b2m2+kl2 for some integer b2. The preimage f1=ψ-1(f2) is a regular fiber on T1 and we have f1b1m1+εkl1 for some integer b1 and some ε{±1}. Since f1 and f2 are without self-intersections, we have gcd(b1,k)=gcd(b2,k)=1. Now the conditions ψ(m1)m2+rl2, ψ(l1)l2 and ψ(f1)=f2 imply that b1=b2 and rb1=k(1-ε). Hence, since gcd(b1,k)=1, we are in one of the following three mutually exclusive cases:

  1. (i)ε=1, k=1, b1=b2=0.
  2. (ii)ε=-1, 2k=r even, and b1=b2=1.
  3. (iii)ε=-1, k=r odd, and b1=b2=2.

Since these data completely determine the fiber-homomorphism type of T1ψT2 and since the same argument applies to (L(r,1),±), the claim follows. ∎

More specifically, a computation shows (see [12, Example 4.17]) that the Seifert invariants of the Seifert fibered spaces occurring in the lemma are given as follows:

  1. (i)(L(r,1),)=M(0;(1,r))=(L(r,1),+) with k=1.
  2. (ii)(L(r,1),)=M(0;(k,1),(k,1))=(L(r,1),-) with r=2k even.
  3. (iii)(L(r,1),)=M(0;(k,1+k2),(k,1-k2))=(L(r,1),-) with r=k odd.

Step (f)

In case of a sphere, i.e. when p=q=1, we have |Γg|{1,2} and thus |Γg|=1 and 𝒪g=S2 by the last claim of Lemma 3.6. Suppose that p+q>2. According to Lemma 3.6, (M,g) is either smoothly conjugated to (L(p+q,1),+) or (L(p+q,1),-). Hence, we can assume that ~g is the Hopf fibration on S3 defined by φ+ and that Γp+q acts linearly via φ+ or φ-. Let Sg1 be a Γ-invariant fiber of ~g. By Lemma 3.5 the fibers Sg1 and i~(Sg1) are disjoint. Recall that the fibers of ~g come along with a natural orientation and that the map i~:Sg1i~(Sg1) reverses these orientations. Let γ be a generator of Γ that rotates Sg1 about a minimal angle in positive direction with respect to the natural orientation (cf. Section 2.6 and Figure 3 and note that ord(γ)3). Since Γ and i~ commute by Lemma 3.4, it follows that γ rotates Sg1 and i~(Sg1) in different directions as depicted in Figure 3 in the case p+q=5. Now the fact that a generator of the p+q-action on (S3,) via φ+ rotates all fibers in the same direction implies that g=- in view of Lemma 3.6. In particular, we have 𝒪g=S2(p+qκ,p+qκ) with κ being 1 or 2 depending on whether p+q is odd or even by that lemma. Moreover, by Lemma 2.16, 𝒪g/i is either 2(p+qκ) or D2(p+qκ;) depending on whether pq is odd or even. In particular, it follows that the geodesic periods are given by (p+q2¯) or (1,p+qκ¯) depending on whether pq is odd or even as discussed in Section 2.7.

Figure 3
Figure 3

Illustration of an argument in Section 3.1, Step (f), in the case p+q=5. Note that γ and i~ commute by Lemma 3.4.

Citation: Journal für die reine und angewandte Mathematik 2020, 758; 10.1515/crelle-2017-0050

Remark 3.8.

Observe that in the case p=q we have

𝒪S2/Cp𝒪gand𝒪g/iS2/Cp,-1,

where Cp<SO(2)SO(3) is a cyclic group of order p.

3.2 The real projective plane

In this subsection we apply the above analysis to the real projective plane 𝒪=2 endowed with a Besse metric. By [18], the unit tangent bundle M=T12 of 2 is homeomorphic to the lens space L(4,1). The fiberings g and t on M are defined as in Section 3.1, Step (b). The fibering g lifts to a fibering on the universal covering M~=S3 as in Section 3.1 by Remark 2.10. The main difference to that section is that in the present case the fibering t is not of type o1 since the quotient M/t=2 is not orientable. The fibers of t cannot be oriented in a continuous way. Still, t lifts to the natural fibering of type o1 on T1S2 defined by T1S2S2 and thus to a fibering ~t on M~ by Remark 2.10. As in Section 3.1, Remark 2.10 also shows that Γ=Deck(M~M)4 preserves the orientations of the fibers of ~g. However, this argument does not apply to ~t since t is not of type o1. In fact, the deck transformation of the covering T1S2M reverses the orientations of the fibers of T1S2S2 and thus a generator of Γ reverses the orientations of the fibers of ~t.

We choose a lift i~:M~M~ of the involution i:MM, defined in Section 2.7, that covers the natural lift i:T1S2T1S2. As in Lemma 3.3 the map i~ preserves the orientations of the fibers of ~t while it reverses the orientation of the fibers of ~g. The fact that in this case, compared to the situation in Section 3.1, Γ does not preserve the orientations of the fibers of ~t results in the following statement converse to Lemma 3.4.

Lemma 3.9.

The group Γ is normalized by the map i~ and for a generator γ of Γ we have γi~=i~γ3.

(a) Illustration of an argument in Lemma 3.9. (b) Illustration of an analogous argument. Note that i~Γx=Γi~x since i~ normalizes Γ.

(a)
(a)

Citation: Journal für die reine und angewandte Mathematik 2020, 758; 10.1515/crelle-2017-0050

(b)
(b)

Citation: Journal für die reine und angewandte Mathematik 2020, 758; 10.1515/crelle-2017-0050

Proof.

As a lift of an involution of the unit tangent bundle M the map i~ normalizes the group Γ=Deck(M~M). Since S2 is simply connected as an orbifold, the preimages of the fibers of T1S2S2 under the two-fold covering M~=S3T1S2 are connected and the group Γ0=Deck(M~T1S2)2 leaves the fibers of ~t invariant. Therefore also the lift i~:M~M~ leaves the fibers of ~t invariant (cf. proof of Lemma 3.4). Moreover, we have i~2Γ0 and we know that i~ and Γ0 commute by Lemma 3.4. Since both Γ0 and i~ leave some fiber Sx1 of ~t and its orientation invariant, their restrictions to Sx1 generate a cyclic group (cf. Section 2.6). In a cyclic group at most a single nontrivial element squares to the identity, in this case the generator of Γ0. Therefore, the fact that i~ is not contained in Γ0 implies that i~ has order 4 and that i~2=γ2 for some generator γ of Γ. This shows that the restriction of i~ to each fiber of ~t has order 4 since γ2 is a deck transformation distinct from the identity. Since i~ is not contained in Γ and normalizes Γ, the lift i~ and Γ generate a group Γ~ of order 8. In particular, the cyclic group I of order 4 generated by i~ is normalized by Γ (as an index-2 subgroup of Γ~). Let St1, γSt1 be a pair of fibers of ~t invariant under Γ and pick a point xSt1. By Lemma 2.11 the map i~ rotates St1 and γSt1 in the same direction. Recall that the map γ:St1γSt1 reverses orientations. We can assume that the actions of i~ on St1 and γSt1 are linear (cf. Section 2.6). Because of γIx=Iγx it follows that the situation is as depicted in Figure 4 (a). In particular, γi~x and i~γx differ and hence i~ and Γ do not commute, i.e. the generator i~-1γi~ of Γ is distinct from γ. The only other generator of Γ4 is γ3 and so the second claim follows, too. ∎

We prove that 𝒪g=S2 and thus that all geodesics on 2 have the same period. This was already shown in [24] in a different way. The claim follows analogously as in Section 3.1 from Lemma 3.9: By the same reason as in Section 3.1 the S1-action on M that is induced by the geodesic flow and that defines g satisfies the assumption of Lemma 3.6. More precisely, the arguments in Section 3.1, Step (c), show that S3/~gS2(pg,qg) for coprime pg, qg, that Γ fixes the singularities of S3/~g and preserves its orientation and thus that M/gS2(kpg,kqg) for some k|Γ|. Moreover, the fact that i acts freely on 𝒪g by Lemma 2.16 implies pg=qg=1 as in Section 3.1, Step (d). Hence, by Lemma 3.6, (M,g) is either smoothly conjugated to (L(4,1),+) or to (L(4,1),-), we can assume that ~g is the Hopf fibration on S3 defined by φ+, and that Γ4 acts linearly via φ+ or φ-, respectively. Now let Sg1 be a Γ-invariant fiber of ~g and let γ be a generator of Γ that rotates Sg1 about a minimal angle in positive direction with respect to the natural orientation (cf. Figure 4 (b) and Section 2.6). Since Γ acts freely on Sg1, the Γ-orbit of a point xSg1 has order 4 as depicted in Figure 4 (b). Since 2 has no orbifold singularities, the fibers Sg1 and i~Sg1 are distinct by Lemma 2.16. Now the fact that i~ reverses the orientation of the ~g-fibers (see Lemma 3.3) and Lemma 3.9 imply that the i~-images of Sg1 and Γx look as depicted in Figure 4 (b). In particular, we see that γ rotates Sg1 and i~Sg1 in the same direction. Observe that in case of a 4-action on via φ- the only two Γ-invariant fibers are rotated in different directions (cf. Section 2.6). We conclude that g=+, 𝒪gS2 and 𝒪g/i2 as orbifolds by Lemma 3.6 and Lemma 2.16. In particular, all geodesics have the same length.

In Section A.1 we show how this result can be used to deduce rigidity on the real projective plane. This implies that every Besse orbifold covered by the real projective plane has constant curvature.

3.3 Non-orientable half-spindle orbifolds

In this subsection we assume 𝒪 to be a Besse 2-orbifold of type D2(;p,q). Let 𝒪^ be its orientable double cover of type S2(p,q) and let s be the deck transformation of the covering 𝒪^𝒪. From Section 3.1 we know that 𝒪^gS2(p+qκ,p+qκ)), where κ is 1 or 2 depending on whether p+q is odd or even. Since the action of s on 𝒪^g preserves the orientation and can be linearized (cf. Section 2.6), it is either trivial or fixes precisely two points (cf. Section 2.7). We claim that the latter is always the case. To see this, it suffices to find a geodesic that is not invariant under s. If pq is odd, then there are no self-inverse geodesics 𝒪^ and so any geodesic that hits Fix(s)𝒪^ perpendicularly is not invariant under s. For even pq the geodesics in Fix(s)𝒪^ hit a singular point of even order and are thus regular by Lemma 3.5. Choose a regular geodesic c that lies in Fix(s)𝒪^ and starts at the singularity x𝒪^ of order p in a direction vTx1𝒪^. By continuity there is a neighborhood U of v in Tx1𝒪^ such that any geodesic that starts at x in a direction v~U intersects U at most once. We can assume that U is invariant under s. It follows that any geodesic starting at x in a direction v~U which is different from v is not invariant under s. Hence, in any case s fixes precisely two points on 𝒪^g. Among the fixed points on 𝒪^g are the geodesics contained in Fix(s)𝒪^. As already mentioned, for even pq these geodesics are regular by Lemma 3.5. In the presence of the symmetry s this property holds regardless of the parity of pq:

Lemma 3.10.

The geodesics contained in Fix(s)O^ are regular.

Proof.

Suppose a geodesic c in Fix(s) is exceptional. Then we must have p+q>2, as there would not exist exceptional geodesics otherwise by Theorem B (ii), which has been proven in Section 3.1. In particular, there is some singular point x in Fix(s) hit by c. By Lemma 3.5 we can assume that pq is odd. By our assumptions the fiber Sc1 of g corresponding to c and the fiber Sx1=Tx1𝒪^ of t intersect in a single point. The fiber Sc1 is pointwise fixed by s while the fiber Sx1 is reflected about two points. Since Sc1 and Sx1 are singular by assumption, their preimages under the covering S3T1𝒪^ are connected fibers S~c1 of ~g and S~x1 of ~t invariant under Γ=Deck(S3T1𝒪^) (cf. proof of Lemma 3.4). In particular, both S~c1 and S~x1 are invariant under each lift s~ of s to S3. Since s is the identity on Sc1, we can choose a lift s~ which is the identity on S~c1. Since s~ reverses the orientation of S~x1, it acts as a reflection (see Section 2.6) that fixes precisely two points on S~x1. Both together imply that |S~c1S~x1|2. Because of |S~c1S~x1|=|Γ||Sc1Sx1|=|Γ|=p+q (see Lemma 3.1 for the third equality), we conclude that p+q2, a contradiction. Hence the claim follows. ∎

We apply Lemma 2.13 in order to determine the geodesic periods. Recall that s preserves the orbifold structure of 𝒪^g and from Section 3.1, Step (f), the geodesic periods of S2(p,q).

If p+q is even, there are precisely two geodesics contained in Fix(s). These geodesics are regular by Lemma 3.10 and pointwise fixed by s. Hence, 𝒪g=𝒪^g/sS2(2,2,p+q2) and the geodesic periods of 𝒪 are (1,p+q2). Since 𝒪 has a topological boundary, it follows from Lemma 2.16 that we either have 𝒪g/iD2(;2,2,p+q2) or 𝒪g/iD2(2;p+q2). These cases occur depending on whether pq is even or odd since by Lemma 2.16 these are the conditions for the geodesics in Fix(s), which correspond to the singularities of order 2 on 𝒪g, to be invariant under i or not.

If p+q is odd, there is only one geodesic contained in Fix(s). This geodesic is regular by Lemma 3.10 and pointwise fixed by s. The other s-invariant geodesic must also be regular, because otherwise both singular geodesics would have to be invariant, contradicting the fact that s has only two fixed points on 𝒪^g. Moreover, this geodesic is not pointwise fixed and thus projects to a geodesic of half the period. Hence, in this case we have 𝒪gS2(2,2,(p+q)) and the geodesic periods are (1,2,(p+q)). Since pq is even, it follows as above that 𝒪g/iD2(;2,2,p+q2).

Remark 3.11.

Observe that in the case p=q we have

𝒪S2/Dp𝒪gand𝒪g/iS2/Dp,-1,

where Dp<SO(3) is a dihedral group of order 2p.

3.4 Orientable spherical orbifolds

The orientable spherical orbifolds are the quotients of S2 by the finite subgroups of SO(3). These are cyclic groups Cnn of order n, dihedral groups Dn𝔇n of order 2n and tetrahedral, octahedral and icosahedral groups T, O and I isomorphic to the alternating group 𝔄4, the symmetric group 𝔖4 and the alternating group 𝔄5, respectively. Let us suppose that 𝒪 is a quotient of S2 by one of these groups, call it G, endowed with a Besse metric. In other words, we have a G-invariant Besse metric on S2 and we denote this Besse manifold by 𝒪~. From Theorem B, (ii), in the case p=q=1, which has been proven in Section 3.1, we know that (𝒪~)gS2 as orbifolds. Note that (𝒪~)g is the simply connected orbifold cover of 𝒪g. Indeed, we have an ordinary covering T1𝒪~T1𝒪 of Seifert fibered manifolds, with the fiberings being induced by the geodesic flow lines, and deck transformation group G. After collapsing the fibers this covering induces a covering of orbifolds (𝒪~)g𝒪g with deck transformation group G, i.e. we have 𝒪g(𝒪~)g/G for the induced action of G on (𝒪~)g. Therefore, we simply write 𝒪~g:=(𝒪~)g.

Theorem B (ii) and (iii) in the case p=q moreover show that every rotation in G acts non-trivially as a rotation on 𝒪~g. Since every nontrivial element in G is a rotation, it follows that G acts effectively on 𝒪~g and preserves the orientation. By Lemma 3.4 the actions of G and i on 𝒪~g commute. Therefore these actions can be linearized simultaneously (cf. Section 2.6). Since the faithful representation of G, as an abstract group, in SO(3) is unique, it follows that 𝒪g=𝒪~g/G𝒪 as orbifolds. Since i acts freely on 𝒪~g as an involution, its linearization in SO(3) has to be minus the identity. Therefore, we have 𝒪g/iS2/G* with G*=G,-1. Note that these results are compatible with Section 3.1 and Section 3.3, see Remark 3.8 and Remark 3.11. This finishes the proof of Theorem B in the case of orientable spherical orbifolds, and hence also the proof of Theorem A in this case as discussed in the last paragraph of Section 2.7.

To compute the geodesic periods, recall from Section 2.7 how they are determined by the covering 𝒪gOg/i in the case of a 2-orbifold 𝒪 with only isolated singularities. The only remaining cases in which this determination is not immediately clear from the combinatorial restriction that i preserves the orders of the local groups of 𝒪g correspond to the groups Dn and T. In these cases we give a more geometric description which will be needed later.

Case 2, 3

If G=Dn, where n2, then 𝒪,𝒪gS2(2,2,n) have double covers 𝒪^,𝒪^gS2(n,n). For even n there exist two geodesics on 𝒪^ connecting the two singular points on 𝒪^ of order n and passing a branch point of the covering 𝒪^𝒪. In fact, we can choose minimizing segments between one of these singularities and the two branch points. Since the covering is two-fold, these segments extend to trajectories of the desired geodesics. These two geodesics are regular by Lemma 3.5. Moreover, by construction they project to exceptional geodesics on 𝒪 of half the (regular) period that are fixed by i. Hence, we have 𝒪g/iD2(;2,2,n) and the geodesic periods are (1,2,2,n).

For odd n geodesics on 𝒪^ passing a branch point of the covering 𝒪^𝒪 project to geodesics on 𝒪 of the same period. In particular, the geodesics of order 2 on 𝒪 do not hit singularities of even order. Therefore, by Lemma 2.16 we have 𝒪g/iD2(2;n) and the geodesic periods are (1,2¯,n). We record the following lemma.

Lemma 3.12.

An exceptional geodesic on O^S2(n,n) hits one of the branch points of O^OS2(2,2,n), regardless of the parity of n.

Proof.

From our discussion above and Lemma 2.16 we know that a geodesic of order n on 𝒪 has to hit a singular point of even order. By Lemma 3.5 it has to be a branch point of the covering 𝒪^𝒪. ∎

Case 4

If G=T, then 𝒪,𝒪gS2(2,3,3). A geodesic c of order 3 on 𝒪 (corresponding to a singular point of order 3 on 𝒪g) is three-foldly covered by a geodesic on 𝒪~S2 that is invariant under a subgroup of T of order 3. If c hit a singularity of order 2 on 𝒪, the trajectory of its lift would be invariant under a subgroup of T of order 6 and thus invariant under T since there are no subgroups of order 6 in T. However, then the order of c would have to be at least 6 which is a contradiction. Hence, c does not hit a singularity of even order and thus, using Lemma 2.16, we have 𝒪g/iD2(3;2) and the geodesic periods are (1,2,3¯).

3.5 Non-orientable spherical orbifolds

Suppose that 𝒪 is a quotient of S2 by a finite subgroup G of O(3) that does not preserve the orientation. Suppose further that 𝒪 is endowed with a Besse metric. In other words, we have a G-invariant Besse metric on S2 and we denote this Besse manifold by 𝒪~. Let G+ be the orientation preserving subgroup of G and set 𝒪^=𝒪~/G+. By Section 3.4 we know that 𝒪~g:=(𝒪~)gS2 is the simply connected covering orbifold of 𝒪^g and that 𝒪^g𝒪~g/G+. Recall from Section 2.7 that 𝒪g was defined as 𝒪g=𝒪^g/(G/G+). Hence, 𝒪~g is also the simply connected covering orbifold of 𝒪g and we have 𝒪g𝒪~g/G with respect to the induced action of G on 𝒪~g.

In the case -1G it follows from Section 3.2 that 𝒪 has constant curvature and that 𝒪g𝒪^g. In the case -1G we claim that G acts effectively on 𝒪~g. Indeed, in this case the ord(g)/2-th power of an element gG with det(g)=-1 is a reflection and this reflection does not leave invariant the geodesics that hit its fixed point set perpendicularly. Hence, in this case we have 𝒪gS2/G× for a finite subgroup G× of SO(3) that is abstractly isomorphic to G. The fact that a finite subgroup of SO(3) is determined by its abstract isomorphism type implies that G×={det(g)g:gG}. Hence, in any case we have 𝒪gS2/G×, and 𝒪g/iS2/G* with G*=G,-1 since i acts on 𝒪~ as an inversion by Lemma 2.16. In particular, we have 𝒪g/i𝒪 as orbifolds if -1G.

It remains to prove rigidity of the geodesic periods in the respective cases and to compute them. Let s be the deck transformation of the covering 𝒪^𝒪. As seen in Section 2.6 the involution s either acts trivially on 𝒪^g or fixes precisely two points. As seen above, it acts trivially if and only if -1G. In order to determine the geodesic periods of 𝒪, we have to identify the s-invariant geodesics on 𝒪 and decide whether they are pointwise fixed or not. Then Lemma 2.13 tells us how the geodesic periods of the quotient look like. We go through the cases with -1G listed under numbers 8–12 in Table 1 and use the Schönflies notation to specify the group G that defines 𝒪S2/G (cf. Table 1 in the appendix). Note that in all cases 𝒪g/i is uniquely determined as an orbifold by its double covers 𝒪g and 𝒪. Case 7 with G+=Cn has already been treated in Section 3.3. The cases with -1G, listed under numbers 13–19 in Table 1, in which a Besse metric has constant curvature, can be treated analogously. We only give details on Case 14 as an example. We frequently use Lemma 2.16 and the identification of geodesics as elements of the orbifold of geodesics without explicitly mentioning it.

Case 8

In the case of 𝒪2(2n)S2/S4n we have

𝒪^S2(2n,2n)𝒪^g,

where S4n4nC4n. Hence,

𝒪gS2/C4nS2(4n,4n)and𝒪g/iD2(4n;).

Since 𝒪 has only isolated singularities, the geodesic periods are (1,4n¯) (cf. Section 2.7).

Case 9

In the case of 𝒪D2(2n+1;)S2/C2n+1h we have

𝒪^S2(2n+1,2n+1)𝒪^gand𝒪^g/i2(2n+1),

where C2n+1h2n+1×24n+2C4n+2. Hence,

𝒪gS2/C4n+2S2(4n+2,4n+2)and𝒪g/iD2(4n+2;).

In particular, the exceptional geodesics on 𝒪^ are invariant under s. Since the two geodesics in the fixed point set of s are also invariant, we see that these must be the exceptional geodesics. It follows that the geodesic periods are (1,2n+1¯).

Case 10

In the case of 𝒪D2(2;2n)S2/D2nd we have

𝒪^S2(2,2,2n)𝒪^gand𝒪^g/iD2(;2,2,2n),

where D2nd𝔇4nD4n. Hence,

𝒪gS2/D4nS2(2,2,4n)and𝒪g/iD2(;2,2,4n).

In particular, invariant under s are an exceptional geodesic of order 2n and a regular geodesic. By Lemma 3.12 this exceptional geodesic of order 2n hits a singularity of order 2 that is not fixed by s. Therefore, the invariant exceptional geodesic is not pointwise fixed, only the regular geodesic in Fix(s)𝒪^ is so. We conclude that the geodesic periods are (1,2,4n).

Case 11

In the case of 𝒪D2(;2,2,2n+1)S2/D2n+1h we have

𝒪^S2(2,2,2n+1)𝒪^gand𝒪^g/iD2(2;2n+1),

where D2n+1h𝔇2n+1×2D4n+2. Hence,

𝒪gS2/D4n+2S2(2,2,4n+2)and𝒪g/iD2(;2,2,4n+2).

In particular, invariant under s are the exceptional geodesic of order 2n+1 and a regular geodesic. These geodesics are contained in Fix(s)𝒪^, since there are two geodesics in Fix(s) and since s only leaves two geodesics invariant. We conclude that the geodesic periods are (1,2,2n+1).

Case 12

In the case of 𝒪D2(;2,3,3)S2/Td we have

𝒪^S2(2,3,3)𝒪^gand𝒪^g/iD2(3;2),

where Td𝔖4O. Hence,

𝒪gS2/OS2(2,3,4)and𝒪g/iD2(;2,3,4).

In particular, invariant under s is the exceptional geodesic of order 2 and a regular geodesic. We claim that the former is not pointwise fixed by s. To see this, we consider a three-fold orbifold covering

S2(2,2,2)𝒪𝒪^S2(2,3,3).

In Section 3.4, Case 2, we have seen that the three exceptional geodesics of order 2 on 𝒪 are defined by minimizing segments c1, c2, c3 connecting the three singularities of order 2 pairwise. Due to the minimizing property, their trajectories have a trivial intersection. Indeed, if c1 and c2 intersected away from the singularities, the intersection point would divide c1 and c2 in equal proportions by their minimizing property. But this would contradict the uniqueness of minimizing segments between singularities of order 2 (any such segment gives rise to an exceptional closed geodesic oscillating between the singularities of even order as discussed in Section 3.4, Case 2, but in total there are only three exceptional geodesics on 𝒪). Hence, the three exceptional geodesics on 𝒪 do not pass through the branch points of the covering 𝒪𝒪^. In other words, the exceptional geodesic of order 2 on 𝒪^ does not hit a singularity of order 3 and is thus not contained in Fix(s). The other, regular invariant geodesic is the unique geodesic in Fix(s)𝒪^ which is pointwise fixed. We conclude that the geodesic periods are (1,3,4).

Case 14

In the case of 𝒪D2(2n;)S2/C2nh we have

𝒪^S2(2n,2n)𝒪^g𝒪gand𝒪g/i𝒪.

In particular, all geodesics on 𝒪 are invariant under s, but only the two exceptional geodesics in Fix(s) are pointwise fixed by s (recall that 𝒪 has constant curvature in this case). Therefore, the exceptional geodesics on 𝒪^ project to geodesics of the same period, while all other geodesics project to geodesics of half the period. We conclude that the geodesic periods are (1,n¯).

4 Conjugacy of induced contact structures

Let 𝒪 be a Besse 2-orbifold with only isolated singularities. As in the manifold case, see [1, Appendix B] and [11, Theorem 1.5.2], the unit tangent bundle M=T1𝒪 carries a natural contact 1-form α whose Reeb flow is the geodesic flow on M. The form α is the restriction of the pullback of the canonical Liouville form on T*𝒪-𝒪 via the isomorphism T𝒪-𝒪T*𝒪-𝒪 induced by the metric (here we regard 𝒪T(*)𝒪 as the zero-section). The aim of this section is to prove the following result.

Theorem 4.1.

Suppose that g0 and g are Besse metrics with the same minimal period on a 2-orbifold O with only isolated singularities. Let α0 and α be the corresponding contact forms on T1(O,g0) and T1(O,g), respectively. Then there exists a diffeomorphism

φ:T1(𝒪,g0)T1(𝒪,g)

such that φ*α=α0.

Proof.

For the 2-sphere this result is proven in [1, Appendix B]. We explain how the same methods can be used to prove the more general case.

In case of the real projective plane and of spindle orbifolds we have seen in Lemma 3.6 that the S1-actions on T1𝒪 induced by the geodesic flows are conjugated by a diffeomorphism. Given Lemma 2.9, the same statement is true for the remaining orbifolds in question, i.e. the spherical orbifolds S2(2,2,n), S2(2,3,3), S2(2,3,4) and S2(2,3,5), since their unit tangent bundles admit up to orientation only one Seifert fibering [17, Theorem 10.2, p. 72]. Hence, in each case there exists a diffeomorphism

ψ:T1(𝒪,g0)T1(𝒪,g)

that conjugates the S1-actions induced by the geodesic flows. By assumption on the minimal periods the Reeb vector fields of α0 and α1:=ψ*α coincide. Moreover, we claim that α0 and α1 define the same orientation, i.e. that ψ preserves the orientations defined by α0dα0 and αdα. It is sufficient to show that the pulled back contact forms α~0 and α~1 on the universal cover S3 define the same orientation. The Reeb flows of α~0 and α~1 coincide and are defined by an S1-action (cf. Remark 2.10). From Sections 3.1 and 3.4 we know that S3/S1S2 as orbifolds. In other words, the S1-action defines a principal S1-bundle with base B=S3/S1S2. By (the easy part of) a theorem by Boothby and Wang, see [5] and [11, Theorem 7.2.5], α~0 and α~1 are connection 1-forms of this principal bundle, whose curvature forms ω0 and ω1 are area forms on B satisfying

p*ω0=dα~0andp*ω1=dα~1,

where p:S3B is the natural projection. Moreover, -[ω02π]=-[ω12π] is the Euler class of the principal S1-bundle. In particular, ω0 and ω1 induce the same orientation on B and hence α~0 and α~1 induce the same orientation on S3.

Now, as in [1], one can show that αt=tψ*α+(1-t)α0 is a contact form for every t[0,1] and apply Moser’s argument to find a one-parameter family of diffeomorphisms ϕt:T1(𝒪,g0)T1(𝒪,g0), t[0,1], such that ϕt*αt=α0 for every t[0,1]. In particular,

α0=ϕ1*α1=ϕ1*ψ*α

and φ=ψϕ1 is the desired diffeomorphism. ∎

Appendix

A.1 Rigidity on the real projective plane

For a Riemannian metric g on 𝒪=2 we denote its corresponding area measure by νg, its total area by Ag and the length of a shortest non-contractible loop by ag. The following inequality is due to Pu [25]:

Ag2πag2.

We recall its proof and show how it implies rigidity for Besse metrics on 2. This argument was explained to us by A. Abbondandolo.

Table 1

Orbifolds of geodesics and (labeled) geodesic periods of Besse 2-orbifolds. For definitions and notations see Sections 2.1 and 2.7. Expressions that only hold in the good orbifold case, i.e. when p=q, are stated in parentheses. For the good orbifolds 𝒪=S2/G, G<O(3), appearing in the table the second column specifies G in terms of the Schönflies notation, see for example [19]). Recall that G+=GSO(3), G×={gG:det(g)g} and G*=G,-1. A detailed discussion of the finite subgroups of O(3) and their relations can for instance be found in [19].

(G<O(3))𝒪(S2/G)𝒪g(S2/G×)𝒪g/i(S2/G*)geod. periods
(G<SO(3), G=G+)(S2/G)(S2/G*)
(Cp)S2(p,q)
012(p+q), 2pqS2/Cp+q2S2/Cp+q2h(1,p+q2¯)
2(p+q), 2pqS2/Cp+q2S2/Sp+q(p+q2¯)
01’2(p+q), 2pqS2/Cp+qS2/Cp+qh(1,p+q¯)
02D2nS2(2,2,2n)S2/D2nS2/D2nh(1,2,2,2n)
03D2n+1S2(2,2,2n+1)S2/D2n+1S2/D2n+1d(1,2¯,2n+1)
04TS2(2,3,3)S2/TS2/Th(1,2,3¯)
05OS2(2,3,4)S2/OS2/Oh(1,2,3,4)
06IS2(2,3,5)S2/IS2/Ih(1,2,3,5)
(-1GSO(3), GG×)(S2/G×)(S2/G*)
(Cpv)D2(;p,q)
072(p+q), 2pqS2/Dp+q2S2/Dp+q2h(1,p+q2)
2(p+q), 2pqS2/Dp+q2S2/Dp+q2d(1,p+q2)
07’2(p+q), 2pqS2/Dp+qS2/Dp+qh(1,2,p+q)
08S4n2(2n)S2/C4nS2/C4nh(1,4n¯)
09C2n+1hD2(2n+1;)S2/C4n+2S2/C4n+2h(1,2n+1¯)
10D2ndD2(2;2n)S2/D4nS2/D4nh(1,2,4n)
11D2n+1hD2(;2,2,2n+1)S2/D4n+2S2/D4n+2h(1,2,2n+1)
12TdD2(;2,3,3)S2/OS2/Oh(1,3,4)
(-1G=G*G+×2)(S2/G+)(S2/G)
13S4n+22(2n+1)S2/C2n+1S2/S4n+2(2n+1¯)
14C2nhD2(2n;)S2/C2nS2/C2nh(1,n¯)
15D2n+1dD2(2;2n+1)S2/D2n+1S2/D2n+1d(1,2n+1)
16D2nhD2(;2,2,2n)S2/D2nS2/D2nh(1,n)
17ThD2(3;2)S2/TS2/Th(1,3¯)
18OhD2(;2,3,4)S2/OS2/Oh(1,2,3)
19IhD2(;2,3,5)S2/IS2/Ih(1,3,5)

Suppose that g is some Riemannian metric on 2 and let g0 be the standard Riemannian metric on 2 of constant curvature 1. The group G=SO(3) acts on 2 in its standard way. By the uniformization theorem there is some positive smooth function φ on 2 such that g=φg0. We endow G with its Haar measure μ and define

φ¯=(G(g*φ)12𝑑μ)2

and g¯=φ¯g0. By construction g¯ is a G-invariant Riemannian metric on M=2 and hence has constant curvature. We claim that AgAg¯ and ag¯ag . Indeed, we have

Ag¯=Mφ¯𝑑νg0=M(G(h*φ)12𝑑μ)2𝑑νg0M(Gh*φ𝑑μ)𝑑νg0=G(Mh*φ𝑑νg0)𝑑μ=GAg𝑑μ=Ag,

where we have applied the Cauchy–Schwarz inequality. Moreover, with a shortest non-contractible loop (and hence geodesic) γ on 2 with respect to g¯ we have

ag¯=01φ¯(γ(s))12γ˙(s)g0𝑑s=01(G((h*φ)(γ(s)))12γ˙(s)g0𝑑μ)𝑑s=G(01((h*φ)(γ(s)))12γ˙(s)g0𝑑s)𝑑μGag𝑑μ=ag.

In particular, this proves Pu’s inequality, since we have Ag¯=2πag¯2 for the metric of constant curvature g¯ (see [25]). Now suppose that g is Besse. Theorem 4.1 implies that the same equality also holds for the Besse metric g. In fact, after normalizing g such that ag=π, Theorem 4.1 implies vol(T12,α0dα0)=vol(T12,αdα) which in turn implies Ag=2π by fiberwise integration with respect to T122. (Alternatively, this follows from a theorem of Weinstein: The two-fold covering (S2,g^) of (2,g) has area 2Ag and the minimal geodesic period is 2ag due to the fact that 𝒪gS2. Now the theorem by Weinstein says that for a Besse metric g^ on S2 we have area(S2,g^)=l2π, where l is the minimal geodesic period [33], cf. [3, Proposition 2.24].) It follows that Ag¯=Ag, i.e. we have equality in the Cauchy–Schwarz inequality implying that φ is constant. Hence, g is proportional to g0 and has constant curvature.

Acknowledgements

The author would like to thank Alexander Lytchak for drawing his attention to the subject. He is grateful to Alberto Abbondandolo for explaining to him how this paper’s result combined with work by Pu implies rigidity on the real projective plane (see Section A.1). He thanks the referee for critical comments and suggestions that helped to improve the exposition. The research in this paper is part of a project in the SFB/TRR 191 “Symplectic Structures in Geometry, Algebra and Dynamics”. A partial support is gratefully acknowledged.

References

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    V. Guillemin, The Radon transform on Zoll surfaces, Adv. Math. 22 (1976), no. 1, 85–119.

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    V. Guillemin, A. Uribe and Z. Wang, Geodesics on weighted projective spaces, Ann. Global Anal. Geom. 36 (2009), no. 2, 205–220.

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    T. Konno, Unit tangent bundle over two-dimensional real projective space, Nihonkai Math. J. 13 (2002), no. 1, 57–66.

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    H. Seifert, Topologie dreidimensionaler gefaserter Räume, Acta Math. 60 (1933), no. 1, 147–238.

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    O. Zoll, Über Flächen mit Scharen geschlossener geodätischer Linien, Math. Ann. 57 (1903), no. 1, 108–133.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    A. Abbondandolo, B. Bramham, U. L. Hryniewicz and P. A. S. Salomão, A systolic inequality for geodesic flows on the two-sphere, Math. Ann. 367 (2017), no. 1–2, 701–753.

  • [2]

    A. Adem, J. Leida and Y. Ruan, Orbifolds and stringy topology, Cambridge Tracts in Math. 171, Cambridge University Press, Cambridge 2007.

  • [3]

    A. L. Besse, Manifolds all of whose geodesics are closed, Ergeb. Math. Grenzgeb. (3) 93, Springer, Berlin 1978.

  • [4]

    M. Boileau, S. Maillot and J. Porti, Three-dimensional orbifolds and their geometric structures, Panor. Synthèses 15, Société Mathématique de France, Paris 2003.

  • [5]

    W. M. Boothby and H. C. Wang, On contact manifolds, Ann. of Math. (2) 68 (1958), 721–734.

  • [6]

    M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature, Grundlehren Math. Wiss. 319, Springer, Berlin 1999.

  • [7]

    M. G. Brin, Seifert fibered spaces, Notes for a course given in the Spring of 1993.

  • [8]

    D. Burago, Y. Burago and S. Ivanov, A course in metric geometry, Grad. Stud. Math. 33, American Mathematical Society, Providence 2001.

  • [9]

    M. W. Davis, Lectures on orbifolds and reflection groups, Transformation groups and moduli spaces of curves, Adv. Lect. Math. (ALM) 16, International Press, Somerville (2011), 63–93.

  • [10]

    U. Frauenfelder, C. Lange and S. Suhr, A Hamiltonian version of a result of Gromoll and Grove, preprint (2016), https://arxiv.org/abs/1603.05107v2.

  • [11]

    H. Geiges, An introduction to contact topology, Cambridge Stud. Adv. Math. 109, Cambridge University Press, Cambridge 2008.

  • [12]

    H. Geiges and C. Lange, Seifert fibrations of lens spaces, Abh. Math. Semin. Univ. Hambg. (2017), 10.1007/s12188-017-0188-z.

  • [13]

    L. W. Green, Auf Wiedersehensflächen, Ann. of Math. (2) 78 (1963), 289–299.

  • [14]

    D. Gromoll and K. Grove, On metrics on S 2 {S^{2}} all of whose geodesics are closed, Invent. Math. 65 (1981/82), no. 1, 175–177.

  • [15]

    V. Guillemin, The Radon transform on Zoll surfaces, Adv. Math. 22 (1976), no. 1, 85–119.

  • [16]

    V. Guillemin, A. Uribe and Z. Wang, Geodesics on weighted projective spaces, Ann. Global Anal. Geom. 36 (2009), no. 2, 205–220.

  • [17]

    M. Jankins and W. D. Neumann, Lectures on Seifert manifolds, Brandeis Lect. Notes 2, Brandeis University, Waltham 1983.

  • [18]

    T. Konno, Unit tangent bundle over two-dimensional real projective space, Nihonkai Math. J. 13 (2002), no. 1, 57–66.

  • [19]

    L. D. Landau and E. M. Lifshitz, Mechanics, Course Theor. Phys. 1, Pergamon Press, Oxford 1960.

  • [20]

    C. Lange, Characterization of finite groups generated by reflections and rotations, J. Topol. 9 (2016), no. 4, 1109–1129.

  • [21]

    C. Lange, Orbifolds from a metric viewpoint, preprint (2018), https://arxiv.org/abs/1801.03472.

  • [22]

    C. Lange, Equivariant smoothing of piecewise linear manifolds, Math. Proc. Camb. Philos. Soc. (2017), 10.1017/S0305004117000275.

  • [23]

    A. Lytchak and G. Thorbergsson, Curvature explosion in quotients and applications, J. Differential Geom. 85 (2010), no. 1, 117–139.

  • [24]

    C. Pries, Geodesics closed on the projective plane, Geom. Funct. Anal. 18 (2009), no. 5, 1774–1785.

  • [25]

    P. M. Pu, Some inequalities in certain nonorientable Riemannian manifolds, Pacific J. Math. 2 (1952), 55–71.

  • [26]

    M. Radeschi and B. Wilking, On the Berger conjecture for manifolds all of whose geodesics are closed, Invent. Math. 210 (2017), no. 3, 911–962.

  • [27]

    F. Raymond, Classification of the actions of the circle on 3-manifolds, Trans. Amer. Math. Soc. 131 (1968), 51–78.

  • [28]

    P. Scott, The geometries of 3-manifolds, Bull. Lond. Math. Soc. 15 (1983), no. 5, 401–487.

  • [29]

    H. Seifert, Topologie dreidimensionaler gefaserter Räume, Acta Math. 60 (1933), no. 1, 147–238.

  • [30]

    E. Swartz, Matroids and quotients of spheres, Math. Z. 241 (2002), no. 2, 247–269.

  • [31]

    W. P. Thurston, The geometry and topology of three-manifolds, Princeton University Press, Princeton 1979.

  • [32]

    A. W. Wadsley, Geodesic foliations by circles, J. Differential Geom. 10 (1975), no. 4, 541–549.

  • [33]

    A. Weinstein, On the volume of manifolds all of whose geodesics are closed, J. Differential Geom. 9 (1974), 513–517.

  • [34]

    B. P. Zimmermann, On finite groups acting on spheres and finite subgroups of orthogonal groups, Sib. Èlektron. Mat. Izv. 9 (2012), 1–12.

  • [35]

    O. Zoll, Über Flächen mit Scharen geschlossener geodätischer Linien, Math. Ann. 57 (1903), no. 1, 108–133.

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    Left: A (p,q)-spindle orbifold S2(p,q). Right: A (p,q)-half-spindle orbifold D2(;p,q) (no assumptions on p and q). (p,1)-spindle orbifolds are also known as teardrops. The orbifolds in the picture are bad if and only if pq.

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    Gluing construction in Lemma 3.1. Note that the curve c1 is homotopic to the curve c2-1, that is, c2 traversed in the opposite direction.

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    Illustration of an argument in Section 3.1, Step (f), in the case p+q=5. Note that γ and i~ commute by Lemma 3.4.

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