1 Introduction and Main Results
A closed curve on a Finsler manifold is a closed geodesic if it is locally the shortest path connecting any two nearby points on this curve. As usual, on any Finsler manifold , a closed geodesic is prime if it is not a multiple covering (i.e., iteration) of any other closed geodesics. Here, the m-th iteration of c is defined by . The inverse curve of c is defined by for . Note that unlike on Riemannian manifolds, the inverse curve of a closed geodesic c on an irreversible Finsler manifold need not be a geodesic. We call two prime closed geodesics c and d distinct if there is no such that for all . On a reversible Finsler (or Riemannian) manifold, two closed geodesics c and d are called geometrically distinct if , i.e., if their image sets in M are distinct. We shall omit the word distinct when we talk about more than one prime closed geodesic.
For a closed geodesic c on an n-dimensional manifold , denote by the linearized Poincaré map of c. Then, is symplectic. For any , we define the elliptic height of M to be the total algebraic multiplicity of all eigenvalues of M on the unit circle in the complex plane . Since M is symplectic, is even and . A closed geodesic c is called elliptic if , i.e., if all the eigenvalues of are located on , irrationally elliptic if it is elliptic and is suitably homotopic to the -product of rotation matrices with rotation angles being irrational multiples of π, hyperbolic if , i.e., all the eigenvalues of are located away from , infinitely degenerate if 1 is an eigenvalue of for infinitely many , and, finally, non-degenerate if 1 is not an eigenvalue of . A Finsler manifold is called bumpy if all the closed geodesics on it are non-degenerate.
There is a famous conjecture in Riemannian geometry which claims that there exist infinitely many closed geodesics on any compact Riemannian manifold. This conjecture has been proved except for compact rank-one symmetric spaces. The results of Franks  and Bangert  imply that this conjecture is true for any Riemannian 2-sphere (cf. [17, 18]). But once one moves to the Finsler case, the conjecture becomes false. It was quite surprising when Katok  found some irreversible Finsler metrics on spheres with only finitely many closed geodesics and all closed geodesics being non-degenerate and elliptic (cf. ).
Recently, index iteration theory of closed geodesics (cf. [6, 22, 23, 24]) has been applied to study the closed geodesic problem on Finsler manifolds. For example, Bangert and Long show in  that there exist at least two closed geodesics on every . After that, a great number of multiplicity and stability results has appeared (cf. [10, 11, 12, 13, 14, 25, 26, 31, 32, 33, 35, 34, 36] and the references therein).
In , Rademacher has introduced the reversibility of a compact Finsler manifold as
Then, in , he obtained some results on the multiplicity and the length of closed geodesics and their stability properties. For example, letting F be a Finsler metric on with reversibility λ and flag curvature K satisfying , there exist at least closed geodesics with length . If with , then there exists an elliptic-parabolic closed geodesic, i.e., its linearized Poincaré map is split into two-dimensional rotations and a part whose eigenvalues are . Some similar results in the Riemannian case are obtained in [2, 3].
Recently, Wang proved in  that for every Finsler n-dimensional sphere with reversibility λ and flag curvature K satisfying , either there exist infinitely many prime closed geodesics or there exists one elliptic closed geodesic whose linearized Poincaré map has at least one eigenvalue which is of the form with an irrational μ. The same author proved in  that for every Finsler n-dimensional sphere for with reversibility λ and flag curvature K satisfying , either there exist infinitely many prime closed geodesics or there exist closed geodesics possessing irrational mean indices. Furthermore, assuming that the metric F is bumpy, he showed in  that there exist closed geodesics on . Also, in , he showed that for every bumpy Finsler metric F on satisfying , there exist two prime elliptic closed geodesics provided the number of closed geodesics on is finite.
Very recently, Duan proved in  that for every Finsler n-dimensional sphere , , with reversibility λ and flag curvature K satisfying , either there exist infinitely many closed geodesics or there exist at least two elliptic closed geodesics and each linearized Poincaré map has at least one eigenvalue of the form with θ being an irrational multiple of π. Furthermore, in , he proved that for every Finsler metric F on the n-dimensional sphere , , with reversibility λ and flag curvature K satisfying , either there exist infinitely many closed geodesics or there exist always three prime closed geodesics and at least two of them are elliptic; when , these three distinct closed geodesics are non-hyperbolic. If the metric is bumpy, Duan and Long proved in  that on every bumpy Finsler three-dimensional sphere , either there exist two non-hyperbolic prime closed geodesics or there exist at least three prime closed geodesics.
Note that Wang proved in [33, Theorem 1.5] that there exist at least three distinct closed geodesics on with flag curvature K satisfying . Motivated by the results mentioned above, in this paper, we prove the following theorem.
For every Finsler metric F on the three-dimensional sphere with reversibility λ and flag curvature K satisfying with , if there exist exactly three prime closed geodesics, then two of them are irrationally elliptic and the third one is infinitely degenerate.
Note that Anosov conjectured in  that the lower bound of the number of distinct closed geodesics on a Finsler three-dimensional sphere is four, where Katok’s examples in  show that this lower bound can be attained. However, Ziller in [37, pp. 155–156] conjectured that the lower bound of the number of distinct closed geodesics on a Finsler three-dimensional sphere is three. To our knowledge, it is not clear whether there exist some irreversible Finsler metrics on with exactly three distinct closed geodesics. This is an interesting problem.
Our proof of Theorem 1.1 in Section 3 contains mainly three ingredients: the common index jump theorem of , Morse theory, and some new symmetric information about the index jump. In addition, we also follow some ideas from our recent preprints [9, 8, 13].
In this paper, let , , , , , and denote the sets of natural integers, non-negative integers, integers, rational numbers, real numbers, and complex numbers, respectively. We use only singular homology modules with -coefficients. For an -space X, we denote by the quotient space . We define the functions
In particular, we have if and if .
2 Morse Theory and Morse Index of Closed Geodesics
2.1 Morse Theory for Closed Geodesics
Let be a compact Finsler manifold. Then, the space of -maps has a natural structure of Riemannian Hilbert manifolds on which the group acts continuously by isometries (cf. ). This action is defined by for all and . For any , the energy functional is defined by
and is and invariant under the -action. The critical points of E of positive energies are precisely the closed geodesics . The index form of the functional E is well-defined along any closed geodesic c on M, which we denote by . As usual, we denote by and the Morse index and the nullity of E at c, respectively. In the following, we denote
for all . For a closed geodesic c, we set .
Recall that the mean index and the -critical modules of are defined by
We say that a closed geodesic satisfies the isolation condition if
Note that if the number of prime closed geodesics on a Finsler manifold is finite, then all closed geodesics satisfy (Iso).
If c has multiplicity m, then the subgroup of acts on . As studied in [29, p. 59], for all , let , where T is a generator of the -action. On -critical modules of , the following lemma holds (cf. [29, Satz 6.11], ).
Suppose c is a prime closed geodesic on a Finsler manifold M satisfying (Iso). Then, there exist and , the so-called local negative disk and the local characteristic manifold at , respectively, such that and
where , .
When , there holds
When , there holds
Let c be a prime closed geodesic on a Finsler manifold . Then, we have the following.
For any , there holds for .
For any , there holds and if , then there holds for .
For any , there holds and . In particular, if is non-degenerate, then there holds and for all .
Suppose that the nullities satisfy for some integer with . Then, there holds and for any integer j.
Let be a compact simply connected Finsler manifold with finitely many closed geodesics. Denote those prime closed geodesics on with positive mean indices by . Rademacher established in [28, 29] a celebrated mean index identity relating all with the global homology of M (cf. [29, Section 7], especially Satz 7.9 therein) for compact simply connected Finsler manifolds. Here, we give a brief review on this identity (cf. [29, Satz 7.9] and also [12, 26, 33]).
Assume that there exist finitely many closed geodesics on and denote the prime closed geodesics with positive mean indices by for some . Then, we have the identity
and the analytical period of is defined by (cf.  )
Let be a space pair such that the Betti numbers are finite for all . As usual, the Poincaré series of is defined by the formal power series . We need the following well-known version of results on Betti numbers and the Morse inequality. For Lemma 2.4 below, see [28, Theorem 2.4 and Remark 2.5], , and also [12, Lemma 2.5]), and for Theorem 2.5, see [7, Theorem I.4.3].
Let be a three-dimensional Finsler sphere. Then, the Betti numbers are given by
Let be a Finsler manifold with finitely many closed geodesics, denoted by . Set
Then, for every integer , there holds
2.2 Index Iteration Theory of Closed Geodesics
In , Long established the basic normal form decomposition of symplectic matrices. Based on this result, he further established the precise iteration formulae of indices of symplectic paths in . Note that this index iteration formulae works for Morse indices of iterated closed geodesics (cf.  and [24, Chapter 12]). Since every closed geodesic on a sphere must be orientable, then, by [21, Theorem 1.1], the initial Morse index of a closed geodesic on a Finsler coincides with the index of a corresponding symplectic path.
As in , we denote
Here, is non-trivial if and trivial if .
As in , the -sum (direct sum) of any two real matrices is defined by
For every , the homotopy set of M in is defined by
where denotes the spectrum of M, for . The component of P in is defined by the path-connected component of containing M.
For every , there exists a continuous path such that and
where for and for . The terms are non-trivial and are trivial, and the non-negative integers satisfy the equality
and denote the basic normal form decomposition of by (2.8). Then, we have
We have that is odd if , , , , and ; is even if and ; can be any integer if .
Let , , be a finite collection of symplectic paths and . Suppose for all . Then, for every , there exist infinitely many such that
where is the splitting number of .
More precisely, by [27, (4.10) and (4.40)] , we have
where or 1 for and whenever and for some . Furthermore, given , by the proof of [27, Theorem 4.1], we may further require (since the closure of the set is still a closed additive subgroup of for some , where we use notation as in [27, (4.21)]. Then, we can use the proof of [27, Theorem 4.1, Step 2] to get N).
Let . Then, for any , there holds
where is the elliptic height defined in Section 1.
3 Proof of Theorem 1.1
In this section, we prove our main theorem by using the mean index equality in Theorem 2.3, the Morse inequality in Theorem 2.5, and the index iteration theory developed by Long and his coworkers, especially a new observation on a symmetric property for closed geodesics in the common index jump intervals, i.e., Lemma 3.2.
First, we make the assumption that
Then, we have an estimate on the index and on the mean index of .
We have and for .
By assumption, since the flag curvature K satisfies , we can choose a δ in [31, Lemma 2] to satisfy
The claim follows from [30, Theorem 3 and Lemma 3]. ∎
for all . Here, the last inequality holds by the fact that .
Note that by [24, List 9.1.12] and the fact that we obtain
In addition, the precise formulae of and for can be computed as follows (cf. [9, (3.16) and (3.21)] for the details):
where , , and denote the number of normal forms , , and in (2.8) of Theorem 2.6 with , , respectively, and , , and denote the number of normal forms , , and with being the rational multiples of π in (2.8) of Theorem 2.6 with , , respectively, and
Under the assumption (FCG), using [9, Theorem 1.1], we have that there exist at least two elliptic closed geodesics and on whose flag curvature satisfies . The next lemma (cf. [9, Section 3]) lists some properties of these two closed geodesics which will be useful in the proof of Theorem 1.1.
Under the assumption (FCG), there exist at least two elliptic closed geodesics and on whose flag curvature satisfies . Moreover, there exist infinitely many pairs of -tuples of the form and such that
where we can require or as remarked in Theorem 2.7 and
In fact, all these properties have already been obtained in [9, Section 3] and here we only list references. More precisely, (3.13) follows from [9, Claim 1] and the arguments between [9, (3.25) and (3.26)], (3.14) follows from [9, Claim 3] and similar arguments as those for between [9, (3.25) and (3.26)], (3.15) and (3.16) follow from [9, (3.25), Claim 2, and Claim 3], and, finally, (3.17) follows from [9, (3.31)] and (3.15). In one word, the properties of and are symmetric. ∎
Under the assumption (FCG), for the two elliptic closed geodesics , found in Lemma 3.2, there holds
for all , , and then for .
We only give the proof for . The proof for is identical.
First, by (3.13) and Lemma 2.1, we have
which implies that
Note that by (3.16), the linearized Poincaré map of the elliptic closed geodesic is conjugate to or for some , , and . Then,
Then, (3.20) yields
Proof of Theorem 1.1.
has no contribution to the Morse-type numbers , , and for any , has possible contribution to the Morse-type numbers , , or only when , and this time has no contribution to and , but contributes at most one to .
for , . Combining Lemma 2.2 (i) with (3.9) and (3.22), we know that has no contribution to the Morse-type numbers , , and for , where . Note that by (3.13) and (3.19), has also no contribution to , , and .
On one hand, there holds
by (2.10) of Theorem 2.6. So, is even since is even by (3.14) of Lemma 3.2, and then has no contribution to and by (3.19). If has contribution to , then contributes exactly one to by (3.19). Hence, Claim 1 holds.
has no contribution to the Morse-type numbers , , and for any .
for all , which, together with Lemma 2.2 (i), implies that has no contribution to the Morse-type numbers , , and for any .
Now, we prove Claim 2 by contradiction. We can assume that has contribution to the Morse-type numbers , , or , i.e.,
We continue the proof by distinguishing two cases.
for all .
Thus, has contribution to and , since otherwise contributes to and , and then has no contribution to for any by (3.25), which contradicts (3.26). Now, and (3.25) imply that satisfies the condition of Hingston’s result (cf. [17, Proposition 1] and [33, Theorem 4.2]), which yields the existence of infinitely many closed geodesics which contradicts the assumption (TCG).
Case 2. . In this case, by (3.24), one of the following cases may happen.
For (i), we have that is conjugate to , which implies that is even, thus case (i) cannot happen.
Noticing that is even in case (ii), we have that is conjugate to . So, by Theorem 2.6, we have
for . Now, in this case it follows from (3.23) that has contribution to and then , which together with (3.27) implies that has no contribution to for any , which in turn contradicts (3.26). This completes the proof of Claim 2.
has no contribution to .
In fact, contributes otherwise exactly one to by Claim 1. By (2.5) and (2.7), , and then must have contribution to by Claims 1 and 2. Thus, has no contribution to and by (3.3)–(3.4) and Lemma 2.2 (ii). So, we obtain that
Now, we can obtain that
for all .
and are irrationally elliptic.
is even since it has the same parity with . Thus, by (3.33), we obtain , which together with (3.34) implies , i.e., is irrationally elliptic. By the symmetry of and , we also obtain that is irrationally elliptic. Thus, Claim 4 is true.
To conclude with the proof of Theorem 1.1, first note that if 1 is an eigenvalue of for some , then 1 must be an eigenvalue of for any by (2.11) of Theorem 2.6. So, if is not infinitely degenerate, then all iterates of with are non-degenerate and then all closed geodesics , , and their iterates are non-degenerate by Claim 4. Using [35, Theorem 1.2], we get four prime closed geodesics, which contradicts the assumption (TCG). Hence, is infinitely degenerate. ∎
The authors would like to sincerely thank Professor Yiming Long for his valuable help and his encouragement. The authors also thank him sincerely for his comments, suggestions, and helpful discussions about the closed geodesic problem. Finally, the authors sincerely thank the referee for her/his careful reading, valuable comments, and suggestions on this paper.
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About the article
Published Online: 2016-01-14
Published in Print: 2016-02-01
Funding Source: National Natural Science Foundation of China
Award identifier / Grant number: 11131004
Award identifier / Grant number: 11471169
Award identifier / Grant number: 11401555
Award identifier / Grant number: 11371339
Funding Source: China Postdoctoral Science Foundation
Award identifier / Grant number: 2014T70589
The first author is partially supported by the NSFC (grant nos. 11131004, 11471169), the LPMC of MOE of China, and Nankai University. The second author is partially supported by the NSFC (grant nos. 11401555, 11371339), the China Postdoctoral Science Foundation (grant no. 2014T70589), and the CUSF (grant no. WK3470000001).