## 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.

*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

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 *i* coming from time reversal. We call *orbifold of oriented geodesics* and *orbifold of non-oriented geodesics*, and prove the following rigidity result.

*For a Besse 2-orbifold *

- (i)
,$\mathcal{O}\cong {S}^{2}/G$ ${\mathcal{O}}_{g}\cong {S}^{2}/{G}^{\times}$ *and*${\mathcal{O}}_{g}/i\cong {S}^{2}/{G}^{*}$ *as orbifolds, where*$G<O\left(3\right)$ *is a finite subgroup,*${G}^{\times}=\{\mathrm{det}\left(g\right)g:g\in G\}<\mathrm{SO}\left(3\right)$ *and* .${G}^{*}=\langle G,-1\rangle <O\left(3\right)$ - (ii)
*pq**odd,* ,$\mathcal{O}\cong {S}^{2}(p,q)$ ${\mathcal{O}}_{g}\cong {S}^{2}(\frac{p+q}{2},\frac{p+q}{2})$ *and* .${\mathcal{O}}_{g}/i\cong \mathbb{R}{\mathbb{P}}^{2}\left(\frac{p+q}{2}\right)$ - (iii)
*pq**even,* ,$\mathcal{O}\cong {S}^{2}(p,q)$ ${\mathcal{O}}_{g}\cong {S}^{2}(\frac{p+q}{\kappa},\frac{p+q}{\kappa})$ *and*${\mathcal{O}}_{g}/i\cong {D}^{2}(\frac{p+q}{\kappa};)$ *with*κ*being*1*or*2*depending on whether*$p+q$ *is odd or even.* - (iv)
*pq**odd,* ,$\mathcal{O}\cong {D}^{2}(;p,q)$ ${\mathcal{O}}_{g}\cong {S}^{2}(2,2,\frac{p+q}{2})$ *and* .${\mathcal{O}}_{g}/i\cong {D}^{2}(2;\frac{p+q}{2})$ - (v)
*pq**even,* ,$\mathcal{O}\cong {D}^{2}(;p,q)$ ${\mathcal{O}}_{g}\cong {S}^{2}(2,2,\frac{p+q}{\kappa})$ *and*${\mathcal{O}}_{g}/i\cong {D}^{2}(;2,2,\frac{p+q}{\kappa})$ *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 *M*, one from the geodesic flow and another one from the projection

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

## 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.

An *n*-dimensional Riemannian orbifold*U* of *x* in *n*-dimensional Riemannian manifold *M* and a finite group Γ that acts by isometries on *M* such that *U* with the restricted metric and

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 *local group* of *x*. Riemannian orbifolds are stratified by manifolds. The *k*-dimensional stratum consists of those points

The underlying topological space *orientable* if and only if *l* isolated singularities in the interior of *k* isolated corner-reflector singularities on the boundary of

The metric quotient of a Riemannian 2-orbifold by a finite group of isometries is again a Riemannian 2-orbifold [21]. If *metric double*

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

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*.

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

We need the following concept.

A *covering orbifold* of a Riemannian orbifold *U* isometric to some

For instance, if a finite group *G* acts isometrically on a Riemannian 2-orbifold

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

### 2.2 Besse metrics on 2-orbifolds

In [16] a Besse *p* and *q* with

The quotient *horizontal geodesics* on

Another construction of Besse metrics on *p* and *q* is similar to the construction of non-standard Zoll metrics on

and let a Riemannian metric on

Then the metric completion of *p* and *q* are defined differently therein. Since the metric is invariant under a reflection in

*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 *t*. Note that if *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:

Let

### 2.3 2-orbifolds that admit Besse metrics

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

*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.

*A Besse 2-orbifold is compact.*

First suppose that *M* are geodesics that project to the geodesics on *M* have a common period, say *l*, and thus so have the geodesics on

Since

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

*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.
*

By Proposition 2.5 it remains to prove the only if direction. So let

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) *M*. Then the orbits are circles and define a decomposition of *M* into so-called *fibers*. If some element of *exceptional (or singular) of order * if its isotropy subgroup of

*k*. Since

*M*. The metric quotient

*M*together with a chosen orientation and its decomposition into fibers defines a

*Seifert fiber space*of type

The fibers of a Seifert fiber space of type

A Seifert fiber space (of type *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 *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

*The classification of smooth, almost free *

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

Let *M* be *k*-foldly covered by *k*-fold Lie group covering. Then the action of *M* via *M* of type

Finally, note that the classifications of Seifert fibered spaces (of type

### 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 *lens space* is defined as a quotient of

An alternative description of lens spaces and Seifert fiberings on them works as follows. Suppose we have two solid tori *meridians**longitudes**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

### 2.6 Finite group actions on ${S}^{1}$ and ${S}^{2}$

We will encounter isometric actions of finite groups on Riemannian 2-orbifolds

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 *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 *x* in the positive direction, one *encounters**same or in different directions*, cf. Remark 2.8. We will need the following statement.

*Let *

*h*be a homeomorphism of

*M*of finite order

*h*to each fiber has order

*n*. Then

*h*rotates all fibers in the same direction.

By the connectedness assumption it suffices to prove the conclusion for a standard fibered solid torus with only regular fibers, i.e. for *h*. Indeed, if two fibers were rotated in different directions, then the images *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*.
∎

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 *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 *k* corresponds to (an equivalence class of reparametrizations of) a geodesic on *k*-times shorter than the period of a generic geodesic on *order k* and we call it

*exceptional*if

*regular*otherwise. We will see that in this situation

A non-orientable Besse 2-orbifold *s*. Since the *s*, the auxiliary metric on *s* invariant (cf. Section 2.4) so that *s* induces an orientation preserving isometry *s* either acts trivially on

*Suppose that *

*s*as an element of

The projection of a geodesic on *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) *s* (i.e. fixed as an element of *s* has order 2). It is pointwise fixed by *s* if and only if for some (and then each)

Examples for the two possible cases in the lemma are given by a reflection and an inversion 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 *i* is orientation-preserving and interchanges fibers of *i* (cf. preceding paragraph and Section 2.4). Then *i* induces an involutive orientation-reversing isometry of

For a Besse 2-orbifold *orbifold of oriented geodesics* to be *orbifold of non-oriented geodesics* to be

Recall that we sometimes view geodesics on

We call a geodesic on *self-inverse* if it is a branch point of the covering *i* as a point on

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

*A geodesic on a Besse 2-orbifold *

By definition a geodesic γ on *i* as an element of *i* reverses the orientation of *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 *i*-invariant, we can talk about periods of geodesic trajectories. By the *(non)-self-inverse geodesic periods* of

By the *(labeled) geodesic periods* of a Besse 2-orbifold

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

To summarize, in case of a Besse 2-orbifold with isolated singularities the geodesic periods and the data encoded in the covering *i* acts as a reflection or an inversion on the topological sphere *i* correspond to the *i* correspond to the

## 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

### Step (a)

Recall from Section 2.7 that the unit tangent bundle

*The unit tangent bundle *

To prove the lemma we choose an equator of *M* decomposes accordingly into the preimages *M* can be recovered from these full tori by a specification of the gluing homeomorphism *M* if the attaching map ψ satisfies

### Step (b)

Recall from Section 2.7 that *M* induced by the projection *M* and *M*. We denote these lifts by *M* as well as their lifts

### Step (c)

We have the commutative diagram

where the outer vertical projections are coverings of Riemannian orbifolds. The upper horizontal projections induce surjections

In the case of

### Step (d)

Let

*The lift *

The claim follows from the respective property of the action of *i* on the fibers of

If *pq* is odd, then *pq* the same conclusion will follow from the subsequent lemma.

*The actions of *

Let *i* leaves the fibers of *i* the map *M* implies that γ and

Indeed, now we can show the following:

*The involution i does not fix singular points on *

By Lemma 2.16 we only need to consider the case that *pq* is even. Suppose that a singular point on *i*. We have seen in (c) that this singular point has a single preimage in

Consequently, in any case we have

### Step (e)

An example of a Seifert fibering on

Since the actions of

and Seifert fiberings *r*-th roots of unity in

*Let *

*k*. If there are no exceptional fibers, i.e. if

*r*is odd or even. In particular,

*r*is divisible by 4 if

*k*is even.

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.

Let

Since *k* and where *t* and some

- (i)
,$\epsilon =1$ ,$k=1$ .${b}_{1}={b}_{2}=0$ - (ii)
,$\epsilon =-1$ even, and$2k=r$ .${b}_{1}={b}_{2}=1$ - (iii)
,$\epsilon =-1$ odd, and$k=r$ .${b}_{1}={b}_{2}=2$

Since these data completely determine the fiber-homomorphism type of

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:

- (i)
with$(L(r,1),\mathcal{F})=M(0;(1,r))=(L(r,1),{\mathcal{F}}^{+})$ .$k=1$ - (ii)
with$(L(r,1),\mathcal{F})=M(0;(k,1),(k,1))=(L(r,1),{\mathcal{F}}^{-})$ even.$r=2k$ - (iii)
with$(L(r,1),\mathcal{F})=M(0;(k,\frac{1+k}{2}),(k,\frac{1-k}{2}))=(L(r,1),{\mathcal{F}}^{-})$ odd.$r=k$

### Step (f)

In case of a sphere, i.e. when *pq* is odd or even. In particular, it follows that the geodesic periods are given by *pq* is odd or even as discussed in Section 2.7.

Observe that in the case

where *p*.

### 3.2 The real projective plane

In this subsection we apply the above analysis to the real projective plane *M* are defined as in Section 3.1, Step (b). The fibering

We choose a lift

*The group Γ is normalized by the map *

(a) Illustration of an argument in Lemma 3.9. (b) Illustration of an analogous argument. Note that

As a lift of an involution of the unit tangent bundle *M* the map *I* of order 4 generated by

We prove that *M* that is induced by the geodesic flow and that defines *i* acts freely on

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 *s* be the deck transformation of the covering *s* on *s*. If *pq* is odd, then there are no self-inverse geodesics *s*. For even *pq* the geodesics in *c* that lies in *p* in a direction *U* of *v* in *x* in a direction *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* is not invariant under *s*. Hence, in any case *s* fixes precisely two points on *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*:

*The geodesics contained in *

Suppose a geodesic *c* in *x* in *c*. By Lemma 3.5 we can assume that *pq* is odd. By our assumptions the fiber *c* and the fiber *s* while the fiber *s* to *s* is the identity on

We apply Lemma 2.13 in order to determine the geodesic periods. Recall that *s* preserves the orbifold structure of

If *s*. Hence, *pq* is even or odd since by Lemma 2.16 these are the conditions for the geodesics in *i* or not.

If *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 *pq* is even, it follows as above that

Observe that in the case

where

### 3.4 Orientable spherical orbifolds

The orientable spherical orbifolds are the quotients of *n*, dihedral groups *G*, endowed with a Besse metric. In other words, we have a *G*-invariant Besse metric on *G*. After collapsing the fibers this covering induces a covering of orbifolds *G*, i.e. we have *G* on

Theorem B (ii) and (iii) in the case *G* acts non-trivially as a rotation on *G* is a rotation, it follows that *G* acts effectively on *G* and *i* on *G*, as an abstract group, in *i* acts freely on

To compute the geodesic periods, recall from Section 2.7 how they are determined by the covering *i* preserves the orders of the local groups of

### Case 2, 3

If *n* there exist two geodesics on *n* and passing a branch point of the covering *i*. Hence, we have

For odd *n* geodesics on

*An exceptional geodesic on *

*n*.

From our discussion above and Lemma 2.16 we know that a geodesic of order *n* on

### Case 4

If *c* of order 3 on *c* hit a singularity of order 2 on *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

### 3.5 Non-orientable spherical orbifolds

Suppose that *G* of *G*-invariant Besse metric on *G* and set *G* on

In the case *G* acts effectively on *G*. The fact that a finite subgroup of *i* acts on

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 *s* either acts trivially on *s*-invariant geodesics on *G* that defines

### Case 8

In the case of

where

Since

### Case 9

In the case of

where

In particular, the exceptional geodesics on *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

### Case 10

In the case of

where

In particular, invariant under *s* are an exceptional geodesic of order *s*. Therefore, the invariant exceptional geodesic is not pointwise fixed, only the regular geodesic in

### Case 11

In the case of

where

In particular, invariant under *s* are the exceptional geodesic of order *s* only leaves two geodesics invariant. We conclude that the geodesic periods are

### Case 12

In the case of

where

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

In Section 3.4, Case 2, we have seen that the three exceptional geodesics of order 2 on

### Case 14

In the case of

In particular, all geodesics on *s*, but only the two exceptional geodesics in *s* (recall that

## 4 Conjugacy of induced contact structures

Let *M*. The form α is the restriction of the pullback of the canonical Liouville form on

*Suppose that ${g}_{0}$ and g are Besse metrics with the same minimal period on a 2-orbifold $O$ with only isolated singularities. Let ${\alpha}_{0}$ and α be the corresponding contact forms on ${T}^{1}(O,{g}_{0})$ and ${T}^{1}(O,g)$, respectively. Then there exists a diffeomorphism*

*such that *

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

that conjugates the *B* satisfying

where *B* and hence

Now, as in [1], one can show that

and

## A.1 Rigidity on the real projective plane

For a Riemannian metric *g* on

We recall its proof and show how it implies rigidity for Besse metrics on

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 *G* in terms of the Schönflies notation, see for example [19]). Recall that

geod. periods | |||||

( | |||||

01 | |||||

01’ | |||||

02 | |||||

03 | |||||

04 | |||||

05 | |||||

06 | |||||

( | |||||

07 | |||||

07’ | |||||

08 | |||||

09 | |||||

10 | |||||

11 | |||||

12 | |||||

( | |||||

13 | |||||

14 | |||||

15 | |||||

16 | |||||

17 | |||||

18 | |||||

19 |

Suppose that *g* is some Riemannian metric on *G* with its Haar measure μ and define

and *G*-invariant Riemannian metric on

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

In particular, this proves Pu’s inequality, since we have *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 *l* is the minimal geodesic period [33], cf. [3, Proposition 2.24].) It follows that *g* is proportional to

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

- [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.

- [6]↑
M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature, Grundlehren Math. Wiss. 319, Springer, Berlin 1999.

- [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.

- [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.

- [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.

- [29]↑
H. Seifert, Topologie dreidimensionaler gefaserter Räume, Acta Math. 60 (1933), no. 1, 147–238.

- [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.