The existence of two non-contractible closed geodesics on every bumpy Finsler compact space form

Let $M=S^n/ \Gamma$ and $h$ be a nontrivial element of finite order $p$ in $\pi_1(M)$, where the integer $n\geq2$, $\Gamma$ is a finite group which acts freely and isometrically on the $n$-sphere and therefore $M$ is diffeomorphic to a compact space form. In this paper, we establish first the resonance identity for non-contractible homologically visible minimal closed geodesics of the class $[h]$ on every Finsler compact space form $(M, F)$ when there exist only finitely many distinct non-contractible closed geodesics of the class $[h]$ on $(M, F)$. Then as an application of this resonance identity, we prove the existence of at least two distinct non-contractible closed geodesics of the class $[h]$ on $(M, F)$ with a bumpy Finsler metric, which improves a result of Taimanov in [Taimanov 2016] by removing some additional conditions. Also our results extend the resonance identity and multiplicity results on $\mathcal{R}P^n$ in [arXiv:1607.02746] to general compact space forms.


Introduction
Let M = S n /Γ and h be a nontrivial element of finite order p in π 1 (M ), where the integer n ≥ 2, Γ is a finite group which acts freely and isometrically on the n-sphere and therefore M is diffeomorphic to a compact space form which is typically a non-simply connected manifold. In particular, if Γ = Z 2 , then S n /Γ is the n-dimensional real projective space RP n . Motivated by the works [44], [12] and [25] about closed geodesics on Finsler RP n , and based on Taimanov's work [39] on rational equivariant cohomology of non-contractible loops on S n /Γ, this paper is concerned with the multiplicity of closed geodesics on Finsler S n /Γ. It is well known (cf. Chapter 1 of Klingenberg [20]) that c is a closed geodesic or a constant Based on it, many important results on this subject have been obtained (cf. [1], [16]- [17], [33]- [34]).
In particular, in 1969 Gromoll and Meyer [15] used Morse theory and Bott's index iteration formulae [7] to establish the existence of infinitely many distinct closed geodesics on M , when the Betti number sequence {β k (ΛM ; Q)} k∈Z is unbounded. Then Vigué-Poirrier and Sullivan [45] further proved in 1976 that for a compact simply connected manifold M , the Gromoll-Meyer condition holds if and only if H * (M ; Q) is generated by more than one element.
However, when {β k (ΛM ; Q)} k∈Z is bounded, the problem is quite complicated. In 1973, Katok [19] endowed some irreversible Finsler metrics to the compact rank one symmetric spaces S n , RP n , CP n , HP n and CaP 2 , each of which possesses only finitely many distinct prime closed geodesics (cf. also Ziller [46], [47]).
On the other hand, Franks [13] and Bangert [4] together proved that there are always infinitely many distinct closed geodesics on every Riemannian sphere S 2 (cf. also Hingston [17], Klingenberg [21]). These results imply that the metrics play an important role on the multiplicity of closed geodesics on those manifolds.
In 2004, Bangert and Long [6] (published in 2010) proved the existence of at least two distinct closed geodesics on every Finsler S 2 . Subsequently, such a multiplicity result for S n with a bumpy Finsler metric was proved by Duan and Long [8] and Rademacher [36] independently. Furthermore in a recent paper [10], Duan, Long and Wang proved the same conclusion for any compact simplyconnected bumpy Finsler manifold. We refer the readers to [9]- [11], [18], [30], [35] [40]- [41] and the references therein for more interesting results and the survey papers of Long [29], Taimanov [38], Burns and Matveev [3] and Oancea [32] for more recent progresses on this subject.
Motivated by the studies on simply connected manifolds, in particular, the resonance identity proved by Rademacher [33], and based on Westerland's works [42], [43] on loop homology of RP n , Xiao and Long [44] in 2015 investigated the topological structure of the non-contractible loop space and established the resonance identity for the non-contractible closed geodesics on RP 2n+1 by use of Z 2 coefficient homology. As an application, Duan, Long and Xiao [12] proved the existence of at least two distinct non-contractible closed geodesics on RP 3 endowed with a bumpy and irreversible Finsler metric. Subsequently in [39], Taimanov used a quite different method from [44] to compute the rational equivariant cohomology of the non-contractible loop spaces in compact space forms S n /Γ and proved the existence of at least two distinct non-contractible closed geodesics on RP 2 endowed with a bumpy and irreversible Finsler metric. Then in [24], Liu combined Fadell-Rabinowitz index theory with Taimanov's topological results to get many multiplicity results of non-contractible closed geodesics on positively curved Finsler RP n . Very recently, Liu and Xiao [25] established the resonance identity for the non-contractible closed geodesics on RP n , and together with [12] and [39] proved the existence of at least two distinct non-contractible closed geodesics on every bumpy RP n with n ≥ 2.
Based on the works of [10] and [25], it is natural to ask whether every bumpy Finsler compact space form possesses two distinct closed geodesics on each of its nontrivial classes. This paper gives a positive answer to this question. To this end, we first establish the following resonance identity in section 2. Comparing with Theorem 1.1 of [25], the difficulties mainly lie in that the parity of the order p of the nontrivial element h in π 1 (M ) is unknown which yields that the computations of critical modules of non-contractible closed geodesics are very complicated (cf. (1.1) where the mean Euler numberχ(c j ) of c j is defined bŷ , i(c j ) andî(c j ) are the Morse index and mean index of c j respectively.
In particular, if the Finsler metric F on M = S n /Γ is bumpy, then (1.1) has the following simple form (ii) When Γ = Z 2 , then S n /Γ is the n-dimensional real projective space RP n and p = 2, one can easily check that for RP n , the results of the above Theorems 1.1-1.3 are just the results of Theorems 1.1-1.2 and Corollary 1.1 of [25]. So the main results of this paper are generalizations of those of [25]. Note that the only non-trivial group which acts freely on S 2n is Z 2 and S 2n /Z 2 = RP 2n (cf. P.5 of [39]), then we only need to prove Theorem 1.1 for the case when n is odd.
This paper is organized as follows. In section 2, we apply Morse theory to the non-contractible loops of the class [h] and establish the resonance identity of Theorem 1.1. Then in section 3, we firstly recall the precise iteration formulae of Morse indices for orientable closed geodesics, and combine it with Theorem 1.1 to investigate the Morse indices for closed geodesics on S n /Γ and build a bridge between the Morse indices and a division of an interval, then our problem are reduced to a problem in Number Theory and we review some theories about a special system of irrational numbers associated to our problem developed in [25]. In section 4, we draw support from the well known Kronecker's approximation theorem and other techniques in Number theory to give the proof of Theorem 1.2. Finally in section 5, for the reader's convenience, we give the proof of Theorem 3.2 about a special system of irrational numbers as an appendix.
In this paper, let N, N 0 , Z, Q and Q c denote the sets of natural integers, non-negative integers, integers, rational numbers and irrational numbers respectively. We also use notations E(a) = for any a ∈ R.
Throughout this paper, we use Q coefficients for all homological and cohomological modules.
2 Resonance identity of non-contractible closed geodesics on (S n /Γ, F ) Let M = (M, F ) be a compact Finsler manifold, the space Λ = ΛM of H 1 -maps γ : S 1 → M has a natural structure of Riemannian Hilbert manifolds on which the group S 1 = R/Z acts continuously by isometries. This action is defined by (s · γ)(t) = γ(t + s) for all γ ∈ Λ and s, t ∈ S 1 . For any γ ∈ Λ, the energy functional is defined by Morse index and nullity of E at c. In the following, we denote by For m ∈ N we denote the m-fold iteration map φ m : Λ → Λ by φ m (γ)(t) = γ(mt), for all γ ∈ Λ, t ∈ S 1 , as well as γ m = φ m (γ). If γ ∈ Λ is not constant then the multiplicity m(γ) of γ is the order of the isotropy group {s ∈ S 1 | s · γ = γ}. For a closed geodesic c, the mean indexî(c) is defined as usual byî(c) = lim m→∞ i(c m )/m. Using singular homology with rational coefficients we consider the following critical Q-module of a closed geodesic c ∈ Λ: In the following we let M = S n /Γ and h be a nontrivial element of finite order p in π 1 (M ), where the integer n ≥ 2, Γ acts freely and isometrically on the n-sphere and therefore M is diffeomorphic to a compact space form. Then the free loop space ΛM possesses a natural decomposition where Λ g M is the connected components of ΛM whose elements are homotopic to g. We set where c is a minimal closed geodesic of class [h] and c = γ t for a prime closed geodesic γ with t ∈ N .
Then d has multiplicity tp(m − 1) + t, the subgroup Z tp(m−1)+t = { l tp(m−1)+t | 0 ≤ l < tp(m − 1) + t} of S 1 acts on C * (E, d). As studied in p.59 of [34], for all m ∈ N, let H * (X, where T is a generator of the Z tp(m−1)+t -action. On S 1 -critical modules of c p(m−1)+1 , the following lemma holds: Lemma 2.1 (cf. Satz 6.11 of [34] and [6]) Suppose c is a non-contractible minimal closed geodesic of class [h] on a Finsler manifold M = S n /Γ satisfying (Iso). Then there exist U γ tp(m−1)+t and N γ tp(m−1)+t , the so-called local negative disk and the local characteristic manifold at c p(m−1)+1 respectively, such that ν(c p(m−1)+1 ) = dim N γ tp(m−1)+t and As usual, for m ∈ N and l ∈ Z we define the local homological type numbers of c p(m−1)+1 by Based on works of Rademacher in [33], Long and Duan in [30] and [9], we define the analytical period n c of the closed geodesic c by Note that here in order to simplify the study for non-contractible closed geodesics of class [h] on M = S n /Γ, we have slightly modified the definition in [30] and [9] by requiring the analytical period to be integral multiple of 2p. Then by the same proofs in [30] and [9], we have For more detailed properties of the analytical period n c of a closed geodesic c, we refer readers to the two Section 3s in [30] and [9].
As in [5], we have Then we haveî Proof: First, we claim that Theorem 3 in [5] for M = S n /Γ can be stated as: Assume by contradiction. Similarly as in [5], we can choose a different c ∈ Λ h M , if necessary, and Consider the following commutative diagram where m ≡ 1(mod p) and ψ m : Λ h M → Λ h M is the m-fold iteration map. By similar arguments as those in [5], there is A > 0 such that the map with k 0 = p and k 1 , k 2 , . . . , k s therein. Here note that the required m ≡ 1(mod p) and so c m ∈ On the other hand, we define Then by Corollary 1 of [5], there existsm ∈ N\{1} such that no k ∈ K dividesm and ψm * • i * vanishes. In particular, E(cm) > A and none of the On the other hand, by Lemma 7.1 of [34], there exists a k(c) ∈ pN such that ν(c m+k(c) ) = ν(c m ) for all m ∈ N. Specially we obtain ν(c mk(c)+1 ) = ν(c) for all m ∈ N and then elements of ker(E ′′ (c mk(c)+1 )) are precisely mk(c)+1st iterates of elements of ker(E ′′ (c)). Thus by the Gromoll-Meyer theorem in [14], the behavior of the restriction of E to ker(E ′′ (c mk(c)+1 )) is the same as that of the restriction of E to ker(E ′′ (c)). Then together with the fact i(c m ) = 0 for all m ∈ N, we obtain that c mk(c)+1 is a local minimum of E in Λ h M for every m ∈ N. Because M is compact and possessing finite fundamental group (π 1 (M ) is finite for the spherical space forms!), there must exist infinitely many distinct non-contractible closed geodesics of the class [h] on M by the above variant of Corollary 2 on p.386 of [5]. Then it yields a contradiction and proves (2.7).
In [39], Taimanov calculated the rational equivariant cohomology of the spaces of non-contractible loops in compact space forms which is crucial for us to prove Theorem 1.1 and can be stated as follows.
Then the S 1 -equivariant Poincaré series of Λ h M is given by which yields Betti numbers . (2.10) (ii) When n = 2k is even, the S 1 -cohomology ring of Λ h M has the form Then the S 1 -equivariant Poincaré series of Λ h M is given by which yields Betti numbers 0, otherwise. .
Now we give the proof of the resonance identity in Theorem 1.1. hv (M ) = {c 1 , . . . , c r } for some integer r > 0 when the number of distinct non-contractible minimal closed geodesics of the class [h] on M = S n /Γ is finite. Note also that by Lemma 2.2 we haveî(c j ) > 0 for all 1 ≤ j ≤ r. In the following proof of Theorem 1.1, we assume n = 2k + 1 for k ∈ N by Remark 1.1 (iii), then M is orientable.
The Morse series of Λ h M is defined by (2.13) In fact, by (2.6), we have Hence Claim 1 follows by (2.14) and (2.15).
We now use the method in the proof of Theorem 5.4 of [31] to estimate By (2.14), Lemma 3.1 below and the fact that n j ∈ 2N, we obtain On the one hand, we have On the other hand, we have Thus we obtain Since m q is bounded, we then obtain where P S 1 ,q (Λ h M ; Q)(t) is the truncated polynomial of P S 1 (Λ h M ; Q)(t) with terms of degree less than or equal to q. Thus by (2.10) we get , ∀ n ∈ 2N + 1.
3 Preliminary for the proof of Theorem 1.2

Index iteration formulae for closed geodesics
In [26] of 1999, Y. Long established the basic normal form decomposition of symplectic matrices.
Based on it, he further established the precise iteration formulae of Maslov ω-indices for symplectic paths in [27], which can be related to Morse indices of either orientable or non-orientable closed geodesics in a slightly different way (cf. [22], [23] and Chap. 12 of [28]). Roughly speaking, the orientable (resp. non-orientable) case corresponds to i 1 (resp. i −1 ) index, where i 1 and i −1 denote the cases of ω-index with ω = 1 and ω = −1 respectively (cf. Chap. 5 of [28]). Since we have assume the manifold M = S n /Γ is odd dimensional in Theorem 1.2, then M is orientable and we only state the precise index iteration formulae of orientable closed geodesics in the following.
Throughout this section we write i 1 (γ) as i(γ) for short.
For the reader's convenience, we briefly review some basic materials in Long's book [28].
Let γ 0 and γ 1 be two symplectic paths in Sp(2N − 2) connecting the identity matrix I to P and f (1) satisfying γ 0 ∼ ω γ 1 . Then it has been shown that i ω (γ m 0 ) = i ω (γ m 1 ) for any ω ∈ S 1 = {z ∈ C | |z| = 1}. Based on this fact, we always assume without loss of generality that each P c appearing in the sequel has the form (3.1).

3)
where we denote by

A variant of Precise index iteration formulae
In this subsection, we give a variant of the precise index iteration formulae in section 3.1 which makes them more intuitive and enables us to apply the Kronecker's approximation theorem to study the multiplicity of non-contractible closed geodesics of the class [h].
To prove Theorem 1.2, in the following of this paper, we always assume that there exists only one non-contractible minimal closed geodesic c of the class [h] on S 2n+1 /Γ with a bumpy irreversible metric F , which is then just the well known minimal point of the energy functional E on Λ h M satisfying i(c) = 0. We can suppose that c = γ t for some prime closed geodesic γ and t ∈ N, then we also have i(γ) = 0 since γ is also a local minimal point of the energy functional E.
The proof is complete.   on S 2n+1 /Γ with a bumpy irreversible metric F , where the order of h is p with p ≥ 2. Then there exist an integerp ≥ 2 andθ 1 ,θ 2 , . . . ,θ k in Q c with 2 ≤ k ≤ 2n such that kp ∈ 2N and k j=1θ j = 1 2 k + 2n p(n + 1) , In addition, c m has contribution to the Morse-type number {m q | q ∈ N 0 } if and only if m ≡

1(modp).
Proof: From (3.4), we haveî Now we prove in two cases:   Let m =p(n + 1)l +pL + 1 with l ∈ N and L ∈ Z. By (3.6) and (3.7) we obtain i(c m ) = 2nl + k + (pL + 1) 2n p(n + 1) where in the first identity we use the fact kp ∈ 2N, in the last identity for notational simplicity, we denote by (3.14) Since k j=2 {mθ j } ∈ Q c , we obtain by (3.13) that for 1 ≤ i ≤ k − 2, The following lemma will be also needed in the proof of Theorem 1.2 for S 2n+1 /Γ in Section 4.
where the fact k ≤ 2n is used.

The system of irrational numbers
In this subsection, we review some properties of a system of irrational numbers associated to our proof of Theorem 1.2, all the details can be found in section 4 of [25]. Let α = {α 1 , α 2 , . . . , α m } be a set of m irrational numbers. As usual, we have and is linearly dependent over Q otherwise. The rank of α is defined to be the number of elements in a maximal linearly independent subset of α, which we denote by rank(α).
Take arbitrarily η ∈ Q and make the following natural η-action to the system (3.20): which is obviously induced by the transformation η(θ) = θ + η. Then, we get a new system where the third equality we have used the condition (3.21). For simplicity of writing, we also denote the new system (3.24) by (3.20) η meaning that it comes from (3.20) by an η-action.
Definition 3.2 For every η ∈ Q, the absolute difference number of (3.20) η is defined to be the The effective difference number of (3.20) is defined by Two systems of irrational numbers with rank 1 are called to be equivalent, if their effective difference numbers are the same.
Remark 3.2 By the definition of an η-action in (3.23), it can be checked directly that η 1 • η 2 = η 1 + η 2 for every η 1 and η 2 in Q. So every system of irrational numbers with rank 1 is equivalent to the one which comes from itself by an η-action.
The following theorem is concerned with the lower estimate on the effective difference number of (3.20) and will play a crucial role in our proof of Theorem 1.2 in Section 4. For the reader's convenience, we give its proof as an appendix in Section 5.  In this section, we prove our main Theorem 1.2. By Theorem 3.1, we only need to prove Theorem 1.2 for M = S 2n+1 /Γ with a bumpy irreversible Finsler metric F which is involved in the irrational system {θ 1 ,θ 2 , . . . ,θ k } with 2 ≤ k ≤ 2n satisfying (3.6). For sake of readability, we divide it into two cases according to whether rank(θ 1 ,θ 2 , . . .θ k ) = 1 or not. We will give in details the proof for the first case. Based on the well known Kronecker's approximation theorem in Number theory, the second one can be then proved quite similarly and so we only sketch it.
Proof of Theorem 1.2: We carry out the proof in two cases.
As we have mentioned in Section 3.3, the irrational system (3.6) with r = 1 can be seen as a special case of (3.20) satisfying (3.21) and (3.22).
Since any η-action with η ∈ Q to (3.20), if necessary, does no substantive effect on our following arguments, by Theorem 3.2 and Remark 3.1 we can assume without loss of generality that with k 1 ≥ 1 due to (3.22), and denote by ξ j = r j q j for 1 ≤ j ≤ k 1 . Letq = q 1 q 2 · · · q k 1 and m l =p(n + 1)ql + 1 with l ∈ N, wherep is given by Lemma 3.  for some θ ∈ Q c . Then the set {{m l θ} | l ∈ N} is dense in [0, 1]. For every L ∈ Z, we introduce the auxiliary function and denote for simplicity by f = f 0 , which contains only finitely many discontinuous points.
Let a and b in (0, 1) be two real numbers sufficiently close to 0 and 1 respectively. Then, and by similar computation, where the second identity we have used k j=1 p j = 0. Proof: (i) By (4.5) and the assumption, Here and below, we write A ≈ B, if A and B can be chosen to be as close to each other as we want. Since the length of each interval in (3.16) with L = 0 is less than or equal to 1, so f (a) and f (b) must lie in different ones, For the case of k = 2, since the length of each interval of (3.16) with L = 0 is less than 1, (i) follows immediately. The rest case is k ≥ 3, which still contains three subcases. (3.6) and (3.14), we have TakeN > 2(n + 1) in (ii) of Lemma 4.1 and observe that In fact, by (2.9),β q = 0 whenever q is even, then by Lemma 3.2, for the even integer 2nql By definition, m l 1 =p(n + 1)ql 1 + 1 ≡ 1(modp), then ∃ L i ∈ Z such thatm = m l 1 +pL i . But by Case 2: r = rank(θ 1 ,θ 2 , . . .θ k ) ≥ 2.
Our basic idea for proving Case 2 is to construct an irrational system with rank 1 associated to (4.10), which plays the essential role in our sequel arguments due to the following result. Proof: we need only take s 2 = s 3 = · · · = s r = 0.
(ii) g L (a 1 , a 2 , . . . , a r ) and g L (b 1 , b 2 , . . . , b r ) lie in the same interval of (3.16) for any 1 ≤ |L| ≤N , including g L (0, 0, . . . , 0). a 1 , a 2 , . . . , a r (resp. b 1 , b 2 , . . . , b r ) are independent, we can select them by such a way that the decimal functions in g(a 1 , a 2 , . . . , a r ) and g(b 1 , b 2 , . . . , b r ) are mainly determined by a 1 and b 1 respectively. For instance, this can be realized by requiring a l (resp. b l ) with 2 ≤ l ≤ r to be much smaller than a 1 (resp. 1 − b 1 ). The rest proof is then similar as that in Lemma 4.1-(i), with g in stead of f therein. Due to Lemma 4.3, the rest proof is then almost word by word as that in Case 1 and so we omit the tedious details. We complete the proof of Theorem 1.2.

Appendix
For the reader's convenience, we give the proof of Theorem 3.2 as an appendix in this section.
Proof: Take η ∈ Q arbitrarily and recall the definition of η-action in (3.23). Then the equation θ k = p k θ contributes sgn(p k ) to the absolute difference number of (5.1) η if and only if η(0) = {0 − p k η} = {−p k η} = 0, that is η ∈ Z |p k | , which is also the sufficient and necessary condition such that the equationŝ contribute sgn(p k ) to the absolute difference number of (5.2) η . Since the other equations with 1 ≤ j ≤ k − 1 in (5.1) and (5.2) are the same, so do their contributions to the absolute difference numbers of (5.1) η and (5.2) η . As a result, the absolute difference numbers of (5.1) η and (5.2) η are equal for any η ∈ Q which yields that the effective difference numbers of (5.1) and (5.2) are the same and so they are equivalent.
It then follows immediately that (5.3) is equivalent to (5.4).
Proof of Theorem 3.2: We carry out the proof with two steps.