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 (M,F), a closed geodesic c:S1=ℝ/ℤ→M is prime if it is not a multiple covering (i.e., iteration) of any other closed geodesics. Here, the m-th iteration cm of c is defined by cm(t)=c(mt). The inverse curve c-1 of c is defined by c-1(t)=c(1-t) for t∈ℝ. Note that unlike on Riemannian manifolds, the inverse curve c-1 of a closed geodesic c on an irreversible Finsler manifold need not be a geodesic. We call two prime closed geodesics c and ddistinct if there is no θ∈(0,1) such that c(t)=d(t+θ) for all t∈ℝ. On a reversible Finsler (or Riemannian) manifold, two closed geodesics c and d are called geometrically distinct if c(S1)≠d(S1), 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 (M,F), denote by Pc the linearized Poincaré map of c. Then, Pc∈Sp(2n-2) is symplectic. For any M∈Sp(2k), we define the elliptic heighte(M) of M to be the total algebraic multiplicity of all eigenvalues of M on the unit circle 𝕌={z∈ℂ∣|z|=1} in the complex plane ℂ. Since M is symplectic, e(M) is even and 0≤e(M)≤2k. A closed geodesic c is called elliptic if e(Pc)=2(n-1), i.e., if all the eigenvalues of Pc are located on 𝕌, irrationally elliptic if it is elliptic and Pc is suitably homotopic to the ⋄-product of n-1 rotation (2×2) matrices with rotation angles being irrational multiples of π, hyperbolic if e(Pc)=0, i.e., all the eigenvalues of Pc are located away from 𝕌, infinitely degenerate if 1 is an eigenvalue of Pcm for infinitely many m∈ℕ, and, finally, non-degenerate if 1 is not an eigenvalue of Pc. A Finsler manifold (M,F) 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 [15] and Bangert [4] 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 [19] found some irreversible Finsler metrics on spheres with only finitely many closed geodesics and all closed geodesics being non-degenerate and elliptic (cf. [37]).
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 [5] that there exist at least two closed geodesics on every (S2,F). 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 [30], Rademacher has introduced the reversibility λ=λ(M,F) of a compact Finsler manifold as
λ
=
max
{
F
(
-
X
)
|
X
∈
T
M
,
F
(
X
)
=
1
}
≥
1
.
Then, in [31], 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 Sn with reversibility λ and flag curvature K satisfying (λ/(1+λ))2<K≤1, there exist at least n/2-1 closed geodesics with length <2nπ. If (9/4)(λ/(1+λ))2<K≤1 with λ<2, 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 ±1. Some similar results in the Riemannian case are obtained in [2, 3].
Recently, Wang proved in [33] that for every Finsler n-dimensional sphere Sn with reversibility λ and flag curvature K satisfying (λ/(1+λ))2<K≤1, 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 exp(-1πμ) with an irrational μ. The same author proved in [36] that for every Finsler n-dimensional sphere Sn for n≥6 with reversibility λ and flag curvature K satisfying (λ/(1+λ))2<K≤1, either there exist infinitely many prime closed geodesics or there exist [n/2]-2 closed geodesics possessing irrational mean indices. Furthermore, assuming that the metric F is bumpy, he showed in [35] that there exist 2[(n+1)/2] closed geodesics on (Sn,F). Also, in [35], he showed that for every bumpy Finsler metric F on Sn satisfying (9/4)(λ/(1+λ))2<K≤1, there exist two prime elliptic closed geodesics provided the number of closed geodesics on (Sn,F) is finite.
Very recently, Duan proved in [9] that for every Finsler n-dimensional sphere (Sn,F), n≥2, with reversibility λ and flag curvature K satisfying (λ/(1+λ))2<K≤1, 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 exp(θ-1) with θ being an irrational multiple of π. Furthermore, in [8], he proved that for every Finsler metric F on the n-dimensional sphere Sn, n≥3, with reversibility λ and flag curvature K satisfying (λ/(1+λ))2<K≤1, either there exist infinitely many closed geodesics or there exist always three prime closed geodesics and at least two of them are elliptic; when n≥6, these three distinct closed geodesics are non-hyperbolic. If the metric is bumpy, Duan and Long proved in [11] that on every bumpy Finsler three-dimensional sphere (S3,F), 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 (S3,F) with flag curvature K satisfying (λ/(1+λ))2<K≤1. Motivated by the results mentioned above, in this paper, we prove the following theorem.
Theorem 1.1
For every Finsler metric F on the three-dimensional sphere S3 with reversibility λ and flag curvature K satisfying (9/4)(λ/(1+λ))2<K≤1 with λ<2, if there exist exactly three prime closed geodesics, then two of them are irrationally elliptic and the third one is infinitely degenerate.
Remark 1.2
Note that Anosov conjectured in [1] that the lower bound of the number of distinct closed geodesics on a Finsler three-dimensional sphere (S3,F) is four, where Katok’s examples in [19] 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 (S3,F) is three. To our knowledge, it is not clear whether there exist some irreversible Finsler metrics on S3 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 [27], 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 ℕ, ℕ0, ℤ, ℚ, ℝ, and ℂ 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 ℚ-coefficients. For an S1-space X, we denote by X¯ the quotient space X/S1. We define the functions
(1.1)
[
a
]
=
max
{
k
∈
ℤ
|
k
≤
a
}
,
E
(
a
)
=
min
{
k
∈
ℤ
|
k
≥
a
}
,
φ
(
a
)
=
E
(
a
)
-
[
a
]
,
{
a
}
=
a
-
[
a
]
.
In particular, we have φ(a)=0 if a∈ℤ and φ(a)=1 if a∉ℤ.
2 Morse Theory and Morse Index of Closed Geodesics
2.1 Morse Theory for Closed Geodesics
Let M=(M,F) be a compact Finsler manifold. Then, the space Λ=ΛM of H1-maps γ:S1→M has a natural structure of Riemannian Hilbert manifolds on which the group S1=ℝ/ℤ acts continuously by isometries (cf. [20]). This action is defined by (s⋅γ)(t)=γ(t+s) for all γ∈Λ and s,t∈S1. For any γ∈Λ, the energy functional is defined by
(2.1)
E
(
γ
)
=
1
2
∫
S
1
F
(
γ
(
t
)
,
γ
˙
(
t
)
)
2
𝑑
t
and is C1,1 and invariant under the S1-action. The critical points of E of positive energies are precisely the closed geodesics γ:S1→M. The index form of the functional E is well-defined along any closed geodesic c on M, which we denote by E′′(c). As usual, we denote by i(c) and ν(c)-1 the Morse index and the nullity of E at c, respectively. In the following, we denote
Λ
κ
=
{
d
∈
Λ
|
E
(
d
)
≤
κ
}
,
Λ
κ
-
=
{
d
∈
Λ
|
E
(
d
)
<
κ
}
for all κ≥0. For a closed geodesic c, we set Λ(c)={γ∈Λ∣E(γ)<E(c)}.
Recall that the mean index ı^(c) and the S1-critical modules of cm are defined by
ı
^
(
c
)
=
lim
m
→
∞
i
(
c
m
)
m
,
C
¯
*
(
E
,
c
m
)
=
H
*
(
(
Λ
(
c
m
)
∪
S
1
⋅
c
m
)
/
S
1
,
Λ
(
c
m
)
/
S
1
;
ℚ
)
,
respectively.
We say that a closed geodesic satisfies the isolation condition if
(Iso)
the orbit
S
1
⋅
c
m
is an isolated critical orbit of
E
for all
m
∈
ℕ
.
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 ℤm={n/m∣0≤n<m} of S1 acts on C¯*(E,c). As studied in [29, p. 59], for all m∈ℕ, let H∗(X,A)±ℤm={[ξ]∈H∗(X,A)∣T∗[ξ]=±[ξ]}, where T is a generator of the ℤm-action. On S1-critical modules of cm, the following lemma holds (cf. [29, Satz 6.11], [5]).
Lemma 2.1
Suppose c is a prime closed geodesic on a Finsler manifold M satisfying (Iso). Then, there exist Ucm and Ncm, the so-called local negative disk and the local characteristic manifold at cm, respectively, such that ν(cm)=dimNcm and
C
¯
q
(
E
,
c
m
)
≡
H
q
(
(
Λ
(
c
m
)
∪
S
1
⋅
c
m
)
/
S
1
,
Λ
(
c
m
)
/
S
1
)
=
(
H
i
(
c
m
)
(
U
c
m
-
∪
{
c
m
}
,
U
c
m
-
)
⊗
H
q
-
i
(
c
m
)
(
N
c
m
-
∪
{
c
m
}
,
N
c
m
-
)
)
+
ℤ
m
,
where Ucm-=Ucm∩Λ(cm), Ncm-=Ncm∩Λ(cm).
When
ν
(
c
m
)
=
0
, there holds
C
¯
q
(
E
,
c
m
)
=
{
ℚ
if
i
(
c
m
)
-
i
(
c
)
∈
2
ℤ
and
q
=
i
(
c
m
)
,
0
otherwise.
When
ν
(
c
m
)
>
0
, there holds
C
¯
q
(
E
,
c
m
)
=
H
q
-
i
(
c
m
)
(
N
c
m
-
∪
{
c
m
}
,
N
c
m
-
)
ϵ
(
c
m
)
ℤ
m
,
where ϵ(cm)=(-1)i(cm)-i(c).
Define
k
j
(
c
m
)
≡
dim
H
j
(
N
c
m
-
∪
{
c
m
}
,
N
c
m
-
)
,
k
j
±
1
(
c
m
)
≡
dim
H
j
(
N
c
m
-
∪
{
c
m
}
,
N
c
m
-
)
±
ℤ
m
.
Then, we have the following lemma (cf. [29, 26, 33]).
Lemma 2.2
Let c be a prime closed geodesic on a Finsler manifold (M,F). Then, we have the following.
For any m∈ℕ, there holds kj(cm)=0 for j≠[0,ν(cm)].
For any m∈ℕ, there holds k0(cm)+kν(cm)(cm)≤1 and if k0(cm)+kν(cm)(cm)=1, then there holds kj(cm)=0 for j∈(0,ν(cm)).
For any m∈ℕ, there holds k0+1(cm)=k0(cm) and k0-1(cm)=0. In particular, if cm is non-degenerate, then there holds k0+1(cm)=k0(cm)=1 and k0-1(cm)=kj±1(cm)=0 for all j≠0.
Suppose that the nullities satisfy ν(cm)=ν(cn) for some integer m=np≥2 with n,p∈ℕ. Then, there holds kj(cm)=kj(cn) and kj±1(cm)=kj±1(cn) for any integer j.
Let (M,F) be a compact simply connected Finsler manifold with finitely many closed geodesics. Denote those prime closed geodesics on (M,F) with positive mean indices by {cj}1≤j≤k. Rademacher established in [28, 29] a celebrated mean index identity relating all cj 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]).
Theorem 2.3
Assume that there exist finitely many closed geodesics on (S3,F) and denote the prime closed geodesics with positive mean indices by {cj}1≤j≤k for some k∈ℕ. Then, we have the identity
(2.2)
∑
j
=
1
k
χ
^
(
c
j
)
ı
^
(
c
j
)
=
1
,
where
(2.3)
χ
^
(
c
j
)
=
1
n
(
c
j
)
∑
1
≤
m
≤
n
(
c
j
)
0
≤
l
≤
2
(
n
-
1
)
χ
(
c
j
m
)
=
1
n
(
c
j
)
∑
1
≤
m
≤
n
(
c
j
)
0
≤
l
≤
4
(
-
1
)
i
(
c
j
m
)
+
l
k
l
ϵ
(
c
j
m
)
(
c
j
m
)
∈
ℚ
and the analytical period n(cj) of cj is defined by (cf. [26] )
(2.4)
n
(
c
j
)
=
min
{
l
∈
ℕ
|
ν
(
c
j
l
)
=
max
m
≥
1
ν
(
c
j
m
)
with
i
(
c
j
m
+
l
)
-
i
(
c
j
m
)
∈
2
ℤ
for all
m
∈
ℕ
}
.
Set
Λ
¯
0
=
Λ
¯
0
S
3
=
{
constant point curves in
S
3
}
≅
S
3
.
Let (X,Y) be a space pair such that the Betti numbers bi=bi(X,Y)=dimHi(X,Y;ℚ) are finite for all i∈ℤ. As usual, the Poincaré series of (X,Y) is defined by the formal power series P(X,Y)=∑i=0∞biti. 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], [16], and also [12, Lemma 2.5]), and for Theorem 2.5, see [7, Theorem I.4.3].
Lemma 2.4
Let (S3,F) be a three-dimensional Finsler sphere. Then, the Betti numbers are given by
(2.5)
b
j
=
rank
H
j
(
Λ
S
3
/
S
1
,
Λ
0
S
3
/
S
1
;
ℚ
)
=
{
2
if
j
=
2
k
≥
4
,
1
if
j
=
2
,
0
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
.
Theorem 2.5
Let (M,F) be a Finsler manifold with finitely many closed geodesics, denoted by {cj}1≤j≤k. Set
M
q
=
∑
1
≤
j
≤
k
m
≥
1
dim
C
¯
q
(
E
,
c
j
m
)
,
q
∈
ℤ
.
Then, for every integer q≥0, there holds
(2.6)
M
q
-
M
q
-
1
+
⋯
+
(
-
1
)
q
M
0
≥
b
q
-
b
q
-
1
+
⋯
+
(
-
1
)
q
b
0
(2.7)
M
q
≥
b
q
.
2.2 Index Iteration Theory of Closed Geodesics
In [22], 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 [23]. Note that this index iteration formulae works for Morse indices of iterated closed geodesics (cf. [21] 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 Sn coincides with the index of a corresponding symplectic path.
As in [23], we denote
N
1
(
λ
,
b
)
=
(
λ
b
0
λ
)
,
λ
=
±
1
,
b
∈
ℝ
,
H
(
λ
)
=
(
λ
0
0
λ
-
1
)
,
λ
∈
ℝ
∖
{
0
,
±
1
}
,
R
(
θ
)
=
(
cos
θ
-
sin
θ
sin
θ
cos
θ
)
,
θ
∈
(
0
,
π
)
∪
(
π
,
2
π
)
,
and
N
2
(
exp
(
θ
-
1
)
,
B
)
=
(
R
(
θ
)
B
0
R
(
θ
)
)
,
θ
∈
(
0
,
π
)
∪
(
π
,
2
π
)
,
where
B
=
(
b
1
b
2
b
3
b
4
)
,
b
j
∈
ℝ
,
b
2
≠
b
3
.
Here, N2(exp(θ-1),B) is non-trivial if (b2-b3)sinθ<0 and trivial if (b2-b3)sinθ>0.
As in [23], the ⋄-sum (direct sum) of any two real matrices is defined by
(
A
1
B
1
C
1
D
1
)
2
i
×
2
i
⋄
(
A
2
B
2
C
2
D
2
)
2
j
×
2
j
=
(
A
1
0
B
1
0
0
A
2
0
B
2
C
1
0
D
1
0
0
C
2
0
D
2
)
.
For every M∈Sp(2n), the homotopy set Ω(M) of M in Sp(2n) is defined by
Ω
(
M
)
=
{
N
∈
Sp
(
2
n
)
|
σ
(
N
)
∩
𝕌
=
σ
(
M
)
∩
𝕌
≡
Γ
and
ν
ω
(
N
)
=
ν
ω
(
M
)
for all
ω
∈
Γ
}
,
where σ(M) denotes the spectrum of M, νω(M)≡dimℂkerℂ(M-ωI) for ω∈𝕌. The component Ω0(M) of P in Sp(2n) is defined by the path-connected component of Ω(M) containing M.
For Theorem 2.6 below, cf. [22, Theorem 7.8], [23, Theorems 1.2 and 1.3] and also [24, Theorem 1.8.10, Lemma 2.3.5, and Theorem 8.3.1].
Theorem 2.6
For every P∈Sp(2n-2), there exists a continuous path f∈Ω0(P) such that f(0)=P and
f
(
1
)
=
N
1
(
1
,
1
)
⋄
p
-
⋄
I
2
p
0
⋄
N
1
(
1
,
-
1
)
⋄
p
+
⋄
N
1
(
-
1
,
1
)
⋄
q
-
⋄
(
-
I
2
q
0
)
⋄
N
1
(
-
1
,
-
1
)
⋄
q
+
⋄
N
2
(
exp
(
α
1
-
1
)
,
A
1
)
⋄
⋯
⋄
N
2
(
exp
(
α
r
∗
-
1
)
,
A
r
∗
)
⋄
N
2
(
exp
(
β
1
-
1
)
,
B
1
)
⋄
⋯
⋄
N
2
(
exp
(
β
r
0
-
1
)
,
B
r
0
)
(2.8)
⋄
R
(
θ
1
)
⋄
⋯
⋄
R
(
θ
r
′
)
⋄
R
(
θ
r
′
+
1
)
⋄
⋯
⋄
R
(
θ
r
)
⋄
H
(
±
2
)
⋄
h
,
where θj/2π∈ℚ∩(0,1) for 1≤j≤r′ and θj/2π∉ℚ∩(0,1) for r′+1≤j≤r. The terms N2(exp(αj-1),Aj) are non-trivial and N2(exp(βj-1),Bj) are trivial, and the non-negative integers p-,p0,p+,q-,q0,q+,r,r∗,r0,h satisfy the equality
(2.9)
p
-
+
p
0
+
p
+
+
q
-
+
q
0
+
q
+
+
r
+
2
r
∗
+
2
r
0
+
h
=
n
-
1
.
Let
γ
∈
𝒫
τ
(
2
n
-
2
)
=
{
γ
∈
C
(
[
0
,
τ
]
,
Sp
(
2
n
-
2
)
)
|
γ
(
0
)
=
I
}
and denote the basic normal form decomposition of P≡γ(τ) by (2.8). Then, we have
i
(
γ
m
)
=
m
(
i
(
γ
)
+
p
-
+
p
0
-
r
)
+
2
∑
j
=
1
r
E
(
m
θ
j
2
π
)
-
r
(2.10)
-
p
-
-
p
0
-
1
+
(
-
1
)
m
2
(
q
0
+
q
+
)
+
2
∑
j
=
1
r
∗
φ
(
m
α
j
2
π
)
-
2
r
∗
,
(2.11)
ν
(
γ
m
)
=
ν
(
γ
)
+
1
+
(
-
1
)
m
2
(
q
-
+
2
q
0
+
q
+
)
+
2
ς
(
m
,
γ
(
τ
)
)
,
where
ς
(
m
,
γ
(
τ
)
)
=
r
-
∑
j
=
1
r
φ
(
m
θ
j
2
π
)
+
r
∗
-
∑
j
=
1
r
∗
φ
(
m
α
j
2
π
)
+
r
0
-
∑
j
=
1
r
0
φ
(
m
β
j
2
π
)
.
We have that i(γ,1) is odd if f(1)=N1(1,1), I2, N1(-1,1), -I2, N1(-1,-1) and R(θ); i(γ,1) is even if f(1)=N1(1,-1) and N2(ω,b); i(γ,1) can be any integer if σ(f(1))∩𝕌=∅.
The following is the common index jump theorem of Long and Zhu [27] (cf. [27, Theorems 4.1–4.3] and [24]).
Theorem 2.7
Let γk, k=1,…,q, be a finite collection of symplectic paths and Mk=γk(τk)∈Sp(2n-2). Suppose ı^(γk,1)>0 for all k=1,…,q. Then, for every k=1,…,q, there exist infinitely many (N,m1,…,mq)∈ℕq+1 such that
ν
(
γ
k
,
2
m
k
-
1
)
=
ν
(
γ
k
,
1
)
,
ν
(
γ
k
,
2
m
k
+
1
)
=
ν
(
γ
k
,
1
)
,
i
(
γ
k
,
2
m
k
-
1
)
+
ν
(
γ
k
,
2
m
k
-
1
)
=
2
N
-
(
i
(
γ
k
,
1
)
+
2
S
M
k
+
(
1
)
-
ν
(
γ
k
,
1
)
)
,
i
(
γ
k
,
2
m
k
+
1
)
=
2
N
+
i
(
γ
k
,
1
)
,
i
(
γ
k
,
2
m
k
)
≥
2
N
-
e
(
M
k
)
2
,
i
(
γ
k
,
2
m
k
)
+
ν
(
γ
k
,
2
m
k
)
≤
2
N
+
e
(
M
k
)
2
,
where SMk+(1) is the splitting number of Mk.
More precisely, by [27, (4.10) and (4.40)] , we have
(2.12)
m
k
=
(
[
N
M
ı
^
(
γ
k
,
1
)
]
+
χ
k
)
M
,
1
≤
k
≤
q
,
where χk=0 or 1 for 1≤k≤q and mkθ/π∈ℤ whenever exp(θ-1)∈σ(Mk) and θ/π∈ℚ for some 1≤k≤q. Furthermore, given M0∈ℕ, by the proof of [27, Theorem 4.1], we may further require M0|N (since the closure of the set {{Nv}∣N∈ℕ,M0|N} is still a closed additive subgroup of 𝕋h for some h∈ℕ, 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).
We also have the following properties in the index iteration theory (cf. [27, Theorem 2.2] or [24, Theorem 10.2.3]).
Theorem 2.8
Let γ∈𝒫τ(2n). Then, for any m∈ℕ, there holds
ν
(
γ
,
m
)
-
e
(
M
)
2
≤
i
(
γ
,
m
+
1
)
-
i
(
γ
,
m
)
-
i
(
γ
,
1
)
≤
ν
(
γ
,
1
)
-
ν
(
γ
,
m
+
1
)
+
e
(
M
)
2
,
where e(M) 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
(FCG)
there exist only finitely many closed geodesics
c
k
,
k
=
1
,
…
,
q
, on
(
S
3
,
F
)
with reversibility
λ
and flag curvature
K
satisfying
(
9
/
4
)
(
λ
/
(
1
+
λ
)
)
2
<
K
≤
1
with
λ
<
2
.
Then, we have an estimate on the index and on the mean index of ck.
Lemma 3.1
We have i(ck)≥2 and ı^(ck)>3 for k=1,…,q.
Proof.
By assumption, since the flag curvature K satisfies (9/4)(λ/(1+λ))2<K≤1, we can choose a δ in [31, Lemma 2] to satisfy
δ
>
9
4
(
λ
1
+
λ
)
2
and
ı
^
(
c
k
)
≥
2
δ
λ
+
1
λ
>
3
.
The claim i(ck)≥2 follows from [30, Theorem 3 and Lemma 3]. ∎
Combining Lemma 3.1 with Theorem 2.8, it follows that
(3.1)
i
(
c
k
m
+
1
)
-
i
(
c
k
m
)
-
ν
(
c
k
m
)
≥
i
(
c
k
)
-
e
(
P
c
k
)
2
≥
0
for all m∈ℕ. Here, the last inequality holds by the fact that e(Pck)≤4.
It follows from Lemma 3.1 and Theorem 2.7 that there exist infinitely many (q+1)-tuples of the form (N,m1,…,mq)∈ℕq+1 such that, for any 1≤k≤q, there holds
(3.2)
i
(
c
k
2
m
k
-
1
)
+
ν
(
c
k
2
m
k
-
1
)
=
2
N
-
(
i
(
c
k
)
+
2
S
M
k
+
(
1
)
-
ν
(
c
k
)
)
,
(3.3)
i
(
c
k
2
m
k
)
≥
2
N
-
e
(
P
c
k
)
2
,
(3.4)
i
(
c
k
2
m
k
)
+
ν
(
c
k
2
m
k
)
≤
2
N
+
e
(
P
c
k
)
2
,
(3.5)
i
(
c
k
2
m
k
+
1
)
=
2
N
+
i
(
c
k
)
.
Note that by [24, List 9.1.12] and the fact that ν(ck)=pk-+2pk0+pk+ we obtain
(3.6)
2
S
M
k
+
(
1
)
-
ν
(
c
k
)
=
2
(
p
k
-
+
p
k
0
)
-
(
p
k
-
+
2
p
k
0
+
p
k
+
)
=
p
k
-
-
p
k
+
.
So, by (3.1)–(3.6) and the fact that e(Pck)≤4, we have
(3.7)
i
(
c
k
m
)
+
ν
(
c
k
m
)
≤
2
N
-
i
(
c
k
)
-
p
k
-
+
p
k
+
,
for all
1
≤
m
<
2
m
k
,
(3.8)
i
(
c
k
2
m
k
)
+
ν
(
c
k
2
m
k
)
≤
2
N
+
e
(
P
c
k
)
2
≤
2
N
+
2
,
(3.9)
2
N
+
2
≤
i
(
c
k
m
)
,
for all
m
>
2
m
k
.
In addition, the precise formulae of i(ck2mk) and i(ck2mk)+ν(ck2mk) for k=1,…,q can be computed as follows (cf. [9, (3.16) and (3.21)] for the details):
i
(
c
k
2
m
k
)
+
ν
(
c
k
2
m
k
)
=
2
N
+
p
k
0
+
p
k
+
+
q
k
-
+
q
k
0
(3.10)
i
(
c
k
2
m
k
)
=
2
N
-
S
M
k
+
(
1
)
-
C
(
M
k
)
+
2
Δ
k
,
(3.11)
+
2
r
k
0
′
-
2
(
r
k
∗
-
r
k
∗
′
)
+
2
r
k
′
-
r
k
+
2
Δ
k
where rk, rk∗, and rk0 denote the number of normal forms R(θ), N2(exp(α-1),A), and N2(exp(β-1),B) in (2.8) of Theorem 2.6 with P=Pck, k=1,2, respectively, and rk′, rk∗′, and rk0′ denote the number of normal forms R(θ), N2(exp(α-1),A), and N2(exp(β-1),B) with θ,α,β being the rational multiples of π in (2.8) of Theorem 2.6 with P=Pck, k=1,2, respectively, and
(3.12)
Δ
k
≡
∑
0
<
{
m
k
θ
/
π
}
<
δ
S
M
k
-
(
exp
(
θ
-
1
)
)
≤
r
k
-
r
k
′
+
r
k
∗
-
r
k
∗
′
,
C
(
M
k
)
≡
∑
θ
∈
(
0
,
2
π
)
S
M
k
-
(
exp
(
θ
-
1
)
)
,
where δ>0 is a small enough number (cf. [27, (4.43)]) and the estimate of Δk follows from the inequality [9, (3.18)].
Under the assumption (FCG), using [9, Theorem 1.1], we have that there exist at least two elliptic closed geodesics c1 and c2 on (S3,F) whose flag curvature satisfies (λ/(1+λ))2<K≤1. 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.
Lemma 3.2
Under the assumption (FCG), there exist at least two elliptic closed geodesics c1 and c2 on (S3,F) whose flag curvature satisfies (λ/(1+λ))2<K≤1. Moreover, there exist infinitely many pairs of (q+1)-tuples of the form (N,m1,m2,…,mq)∈ℕq+1 and (N′,m1′,m2′,…,mq′)∈ℕq+1 such that
(3.13)
i
(
c
1
2
m
1
)
+
ν
(
c
1
2
m
1
)
=
2
N
+
2
,
C
¯
2
N
+
2
(
E
,
c
1
2
m
1
)
=
ℚ
,
(3.14)
i
(
c
2
2
m
2
′
)
+
ν
(
c
2
2
m
2
′
)
=
2
N
′
+
2
,
C
¯
2
N
′
+
2
(
E
,
c
2
2
m
2
′
)
=
ℚ
,
and
(3.15)
p
k
-
=
q
k
+
=
r
k
∗
=
r
k
0
-
r
k
0
′
=
h
k
=
0
,
k
=
1
,
2
,
(3.16)
r
1
-
r
1
′
=
Δ
1
≥
1
,
r
2
-
r
2
′
=
Δ
2
′
≥
1
,
(3.17)
Δ
k
+
Δ
k
′
=
r
k
-
r
k
′
,
k
=
1
,
2
,
where we can require 2|N or 2|N′ as remarked in Theorem 2.7 and
(3.18)
Δ
k
′
≡
∑
0
<
{
m
k
′
θ
/
π
}
<
δ
S
M
k
-
(
exp
(
θ
-
1
)
)
,
k
=
1
,
2
.
Proof.
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 c1 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 c1 and c2 are symmetric. ∎
Lemma 3.3
Under the assumption (FCG), for the two elliptic closed geodesics c1, c2 found in Lemma 3.2, there holds
(3.19)
k
ν
(
c
k
n
(
c
k
)
)
ϵ
(
c
k
n
(
c
k
)
)
(
c
k
n
(
c
k
)
)
=
1
,
k
j
ϵ
(
c
k
n
(
c
k
)
)
(
c
k
n
(
c
k
)
)
=
0
for all 0≤j<ν(ckn(ck)), k=1,2, and then χ^(ck)≤1 for k=1,2.
Proof.
We only give the proof for c1. The proof for c2 is identical.
First, by (3.13) and Lemma 2.1, we have
1
=
dim
C
¯
2
N
+
2
(
E
,
c
1
2
m
1
)
=
dim
H
2
N
+
2
-
i
(
c
1
2
m
1
)
(
N
c
1
2
m
1
∪
{
c
1
2
m
1
}
,
N
c
1
2
m
1
)
ϵ
(
c
1
2
m
1
)
ℤ
2
m
1
=
dim
H
ν
(
c
1
2
m
1
)
(
N
c
1
2
m
1
∪
{
c
1
2
m
1
}
,
N
c
1
2
m
1
)
ϵ
(
c
1
2
m
1
)
ℤ
2
m
1
=
k
ν
(
c
1
2
m
1
)
ϵ
(
c
1
2
m
1
)
(
c
1
2
m
1
)
,
which implies that
k
j
ϵ
(
c
1
2
m
1
)
(
c
1
2
m
1
)
=
0
for any 0≤j<ν(c12m1) by Lemma 2.2 (ii). In addition, note that since n(c1)|2m1 and ν(c12m1)=ν(c1n(c1)) by (2.4) and (2.12), there holds
k
j
ϵ
(
c
1
2
m
1
)
(
c
1
2
m
1
)
=
k
j
ϵ
(
c
1
n
(
c
1
)
)
(
c
1
n
(
c
1
)
)
for any 0≤j≤ν(c12m1) by Lemma 2.2 (iv). Thus, (3.19) holds.
Note that by (3.16), the linearized Poincaré map Pck of the elliptic closed geodesic ck is conjugate to R(θ1)⋄R(θ2) or R(θ1)⋄N1(λ,b) for some θ1/2π∈(0,1)\ℚ, λ=±1, and b=0,1. Then,
(3.20)
ν
(
c
k
m
)
=
0
for all m<n(ck). In fact, when Pck is conjugate to R(θ1)⋄N1(1,b), we have n(ck)=1. By (2.4), (3.20) holds. When Pck is conjugate to R(θ1)⋄N1(-1,b), we have n(ck)=2. By (2.4), (3.20) also holds. When Pck is conjugate to R(θ1)⋄R(θ2), (3.20) holds by (2.4).
Then, (3.20) yields
k
0
ϵ
(
c
k
m
)
(
c
k
m
)
=
1
,
k
j
ϵ
(
c
k
m
)
(
c
k
m
)
=
0
for all 0<j≤4 and for 1≤m<n(ck), which together with (2.3) and (3.19) gives
χ
^
(
c
k
)
=
1
n
(
c
k
)
(
(
-
1
)
i
(
c
k
n
(
c
k
)
)
+
ν
(
c
k
n
(
c
k
)
)
+
∑
1
≤
m
<
n
(
c
k
)
(
-
1
)
i
(
c
k
m
)
)
≤
1
.
∎
Proof of Theorem 1.1.
In order to prove Theorem 1.1, based on [33, Theorem 1.5] (cf. also [8, Theorem 1.1]), we make the assumption that
(TCG)
there exist exactly two elliptic distinct closed geodesics
c
1
,
c
2
possessing all properties listed
in Lemmas 3.2 and 3.3 and a third closed geodesic
c
3
on
(
S
3
,
F
)
with reversibility
λ
and
flag curvature
K
satisfying
(
9
/
4
)
(
λ
/
(
1
+
λ
)
)
2
<
K
≤
1
with
λ
<
2
.
Claim 1
c
1
m
has no contribution to the Morse-type numbers M2N+1, M2N, and M2N-1 for any m∈ℕ, c2m has possible contribution to the Morse-type numbers M2N+1, M2N, or M2N-1 only when m=2m2, and this time c22m2 has no contribution to M2N+1 and M2N-1, but contributes at most one to M2N.
In fact, by (3.16), for k=1,2, the linearized Poincaré map Pck of the elliptic closed geodesic ck is conjugate to R(θ1)⋄R(θ2) or R(θ1)⋄N1(λ,b) for some θ1/2π∈(0,1)\ℚ, λ=±1, and b=0,1. Combining this fact with Lemma 3.1 and (3.7), we have
(3.21)
i
(
c
k
m
)
+
ν
(
c
k
m
)
≤
2
N
-
i
(
c
k
)
-
p
k
-
+
p
k
+
≤
2
N
-
1
for m<2mk, k=1,2, where the equality in (3.21) holds if and only if Pck is conjugate to R(θ1)⋄N1(1,-1) and i(ck)=2, but i(ck)∈2ℕ-1 when Pck is conjugate to R(θ1)⋄N1(1,-1), thus the equality in (3.21) does not hold. Then,
(3.22)
i
(
c
k
m
)
+
ν
(
c
k
m
)
≤
2
N
-
2
for m<2mk, k=1,2. Combining Lemma 2.2 (i) with (3.9) and (3.22), we know that ckm has no contribution to the Morse-type numbers M2N+1, M2N, and M2N-1 for m≠2mk, where k=1,2. Note that by (3.13) and (3.19), c12m1 has also no contribution to M2N+1, M2N, and M2N-1.
On one hand, there holds
ν
(
c
2
2
m
2
)
=
ν
(
c
2
2
m
2
′
)
by the choices of m2 and m2′ in (2.12) of Theorem 2.7. On the other hand, it yields
i
(
c
2
2
m
2
′
)
=
i
(
c
2
2
m
2
)
(
mod
2
)
by (2.10) of Theorem 2.6. So, i(c22m2)+ν(c22m2) is even since i(c22m2′)+ν(c22m2′) is even by (3.14) of Lemma 3.2, and then c22m2 has no contribution to M2N+1 and M2N-1 by (3.19). If c22m2 has contribution to M2N, then c22m2 contributes exactly one to M2N by (3.19). Hence, Claim 1 holds.
Claim 2
c
3
m
has no contribution to the Morse-type numbers M2N+1, M2N, and M2N-1 for any m≠2m3.
First, by (3.9) and Lemma 2.2 (i), we know that c3m has no contribution to the Morse-type numbers M2N+1, M2N, and M2N-1 for m>2m3.
On the other hand, from Lemma 3.1 and (3.1)–(3.2) along with the fact that ν(c32m3-1)=ν(c3), we have
i
(
c
3
m
)
+
ν
(
c
3
m
)
≤
i
(
c
3
2
m
3
-
1
)
=
2
N
-
(
i
(
c
3
)
+
2
S
M
3
+
(
1
)
)
≤
2
N
-
2
for all 1≤m<2m3-1, which, together with Lemma 2.2 (i), implies that c3m has no contribution to the Morse-type numbers M2N+1, M2N, and M2N-1 for any m<2m3-1.
Now, we prove Claim 2 by contradiction. We can assume that c32m3-1 has contribution to the Morse-type numbers M2N+1, M2N, or M2N-1, i.e.,
(3.23)
∑
q
=
2
N
-
1
2
N
+
1
dim
C
¯
q
(
E
,
c
2
m
3
-
1
)
≥
1
.
Note that, by (3.2) and (3.6), we have
(3.24)
i
(
c
3
2
m
3
-
1
)
+
ν
(
c
3
2
m
3
-
1
)
=
2
N
-
i
(
c
3
)
-
p
3
-
+
p
3
+
≤
2
N
-
2
+
2
=
2
N
,
which, together with Lemma 2.2 (i) and the assumption (3.23), gives i(c32m3-1)+ν(c32m3-1)=2N or 2N-1.
We continue the proof by distinguishing two cases.
Case 1. i(c32m3-1)+ν(c32m3-1)=2N. In this case, by (3.24), Pc3 is conjugate to N1(1,-1)⋄2 and i(c3)=2, ν(c3m)=2 for all m≥1. Then, using Theorem 2.6, we obtain
(3.25)
i
(
c
3
m
)
+
ν
(
c
3
m
)
-
2
=
i
(
c
3
m
)
=
m
i
(
c
3
)
=
m
(
i
(
c
3
)
+
ν
(
c
3
)
-
2
)
=
2
m
for all m≥1.
Now, by (2.5) and (2.7), we obtain M2N≥b2N=2, which, together with Claim 1, implies that c3m must have contribution to M2N for some m∈ℕ, i.e.,
(3.26)
∑
m
≥
1
dim
C
¯
2
N
(
E
,
c
3
m
)
≥
1
.
Thus, c32m3-1 has contribution to M2N and kν(c3)(c3)=1, since otherwise c32m3-1 contributes to M2N-1 and k1(c3)≠0, and then c3m has no contribution to M2N for any m∈ℕ by (3.25), which contradicts (3.26). Now, kν(c3)(c3)=1 and (3.25) imply that c3 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. i(c32m3-1)+ν(c32m3-1)=2N-1. In this case, by (3.24), one of the following cases may happen.
For (i), we have that Pc3 is conjugate to N1(1,-1)⋄2, which implies that i(c3) is even, thus case (i) cannot happen.
Noticing that i(c3)=2 is even in case (ii), we have that Pc3 is conjugate to N1(1,-1)⋄H(2). So, by Theorem 2.6, we have
(3.27)
i
(
c
3
m
)
+
ν
(
c
3
m
)
=
m
i
(
c
3
)
+
ν
(
c
3
m
)
=
2
m
+
1
for m≥1. Now, in this case it follows from (3.23) that c32m3-1 has contribution to M2N-1 and then kν(c3)(c3)=1, which together with (3.27) implies that c3m has no contribution to M2N for any m∈ℕ, which in turn contradicts (3.26). This completes the proof of Claim 2.
Claim 3
c
2
2
m
2
has no contribution to M2N.
In fact, c22m2 contributes otherwise exactly one to M2N by Claim 1. By (2.5) and (2.7), M2N≥b2N=2, and then c32m3 must have contribution to M2N by Claims 1 and 2. Thus, c32m3 has no contribution to M2N+2 and M2N-2 by (3.3)–(3.4) and Lemma 2.2 (ii). So, we obtain that
(3.28)
-
M
2
N
+
1
+
M
2
N
-
M
2
N
-
1
=
∑
0
≤
l
≤
4
(
-
1
)
i
(
c
3
2
m
3
)
+
l
k
l
ϵ
(
c
3
2
m
3
)
(
c
3
2
m
3
)
+
1
.
On the other hand, by (2.6) and Lemma 2.4, we have
(3.29)
M
2
N
+
1
-
M
2
N
+
M
2
N
-
1
≥
b
2
N
+
1
-
b
2
N
+
b
2
N
-
1
=
-
2
.
Combining (3.28) and (3.29), we get
(3.30)
χ
(
c
3
2
m
3
)
=
∑
0
≤
l
≤
4
(
-
1
)
i
(
c
3
2
m
3
)
+
l
k
l
ϵ
(
c
3
2
m
3
)
(
c
3
2
m
3
)
≤
1
.
Note that since n(c3)|2m3 and ν(c32m3)=ν(c3n(c3)) by (2.4) and (2.12), there holds
k
j
ϵ
(
c
3
2
m
3
)
(
c
3
2
m
3
)
=
k
j
ϵ
(
c
3
n
(
c
3
)
)
(
c
3
n
(
c
3
)
)
for any 0≤j≤ν(c32m3) by Lemma 2.2 (iv). Then, it follows from (2.4) and (3.30) that
(3.31)
χ
(
c
3
n
(
c
3
)
)
=
χ
(
c
3
2
m
3
)
≤
1
.
Now, we can obtain that
(3.32)
χ
(
c
3
m
)
≤
1
for all 1≤m<n(c3).
In fact, if c3 is totally degenerate, i.e., if 1 is the unique eigenvalue of Pc3, then n(c3)=1 and (3.32) holds by (3.31).
If c3 is not totally degenerate, by (2.4), either ν(c3m)<2 for 1≤m<n(c3) or ν(c3m0)=2 for 1≤m0<n(c3) with Pc3m0 conjugating to I⋄R(θ) for some θ/2π∈ℚ and i(c3m0)∈2ℕ. In any case, (3.32) follows from Lemma 2.2 (ii).
Now, we combine (3.31) and (3.32) to get χ^(c3)≤1, which, together with Lemma 3.1 and Lemma 3.3, implies that
∑
j
=
1
3
χ
^
(
c
j
)
ı
^
(
c
j
)
<
1
3
+
1
3
+
1
3
=
1
,
which contradicts the identity (2.2) in Theorem 2.3. Hence, Claim 3 holds.
Claim 4
c
1
and c2 are irrationally elliptic.
By (3.16) and (3.17), there holds Δ2=0. Then, together with the fact that r2∗=0 from (3.15), it follows from (3.16) and (3.11) that
=
2
N
+
(
p
2
0
+
p
2
+
+
q
2
-
+
q
2
0
+
2
r
2
0
′
+
r
2
′
)
-
(
r
2
-
r
2
′
)
(3.33)
2
N
≥
i
(
c
2
2
m
2
)
+
ν
(
c
2
2
m
2
)
(3.34)
≥
2
N
-
2
,
where (3.33) holds by the fact that p20+p2++q2-+q20+2r20′+r2′≤1 from (2.9) and (3.16) and r2-r2′≥1 from (3.16), and the equality in (3.34) holds if and only if r2-r2′=2. On the other hand, by Claim 3, we have i(c22m2)+ν(c22m2)≠2N and by (3.14), we have that
i
(
c
2
2
m
2
)
+
ν
(
c
2
2
m
2
)
is even since it has the same parity with i(c22m2′)+ν(c22m2′). Thus, by (3.33), we obtain i(c22m2)+ν(c22m2)≤2N-2, which together with (3.34) implies r2-r2′=2, i.e., c2 is irrationally elliptic. By the symmetry of c1 and c2, we also obtain that c1 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 Pc3m0 for some m0∈ℕ, then 1 must be an eigenvalue of Pc32lm0 for any l∈ℕ by (2.11) of Theorem 2.6. So, if c3 is not infinitely degenerate, then all iterates c3m of c3 with m∈ℕ are non-degenerate and then all closed geodesics ck, k=1,2,3, 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, c3 is infinitely degenerate. ∎
