BSTRACT arXiv:0711.0635v1 [math.DG] 5 Nov 2007 · 2020. 1. 23. · arXiv:0711.0635v1 [math.DG] 5 Nov 2007 SPECTRAL FLOW AND ITERATION OF CLOSED SEMI-RIEMANNIAN GEODESICS MIGUEL ANGEL

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SPECTRAL FLOW AND ITERATIONOF CLOSED SEMI-RIEMANNIAN GEODESICS

MIGUEL ANGEL JAVALOYES AND PAOLO PICCIONE

ABSTRACT. We introduce the notion of spectral flow along a periodic semi-Riemanniangeodesic, as a suitable substitute of the Morse index in the Riemannian case. We studythe growth of the spectral flow along a closed geodesic under iteration, determining itsasymptotic behavior.

1. INTRODUCTION

Closed geodesics are critical points of the geodesic actionfunctional in the free loopspace of a semi-Riemannian manifold(M, g); a very classical problem in Geometry is toestablish multiplicity of closed geodesics (see [17]). By “multiplicity of closed geodesics”,it is always meant multiplicity of “prime closed geodesics”, i.e., those geodesics that arenot obtained by iteration of another closed geodesic. One ofthe difficult aspects in thevariational theory of geodesics is precisely the question of distinguishing iterates. In acelebrated paper by R. Bott [9] it is studied the Morse index of a closed Riemannian ge-odesic; the main result is a formula establishing the growthof the index under iteration.This formula shows that, given a closed geodesicγ, its iteratesγ(N) either have Morseindex that grows linearly withN , or they have all null index. This has been used by Gro-moll and Meyer in another celebrated paper [15], where the authors develop an equivariantMorse theory to prove the existence of infinitely many prime closed geodesics in com-pact Riemannian manifolds whose free loop space has unbounded Betti numbers. Roughlyspeaking, the (uniform) linear growth of the Morse index of an iterate implies that if therewere only a finite number of prime closed geodesics, then the homology generated by theiriterates would not suffice to produce the homology of the entire free loop space. Refine-ments of this kind of results have appeared in subsequent literature (see [6, 17]).

More recently, an increasing interest has arised around thequestion of existence ofperiodic solutions of more general variational problems, and especially in the context ofsemi-Riemannian geometry. Recall that a semi-Riemannian manifold is a manifoldM en-dowed with a nondegenerate, but possibly non positive definite, metric tensorg. In thiscontext, the geodesic variational theory is extremely moreinvolved, even in the fixed end-point case (see [1, 3]), due to the strongly indefinite character of the action functional.When the metric tensor is Lorentzian, i.e., it has index equal to 1, and the metric is station-ary, i.e., time invariant, then it is possible to perform a certain reduction of the geodesicvariational problem that yields existence results similarto the positive definite case (see[8, 10, 13, 14, 19]). For instance, it is proven in [8] that anystationary Lorentzian manifoldhaving a compact Cauchy surface and whose free loop space hasunbounded Betti numbershas infinitely many distinct prime closed geodesics. A Bott type result on the Morse indexof an iterate has been proven in [16] for stationary Lorentzian metrics.

Dropping the stationarity assumption is at this stage a quite challenging task. The firstproblem that one encounters is the fact that the critical points of the geodesic action func-tional havetruly infinite Morse index, so that standard Morse theory fails. Inorder to

Date: October 16th, 2007.M. A. J. is sponsored by Fapesp; P. P. is partially sponsored by CNPq.

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http://arxiv.org/abs/0711.0635v1

2 M. A. JAVALOYES AND P. PICCIONE

develop Morse theory for strongly indefinite functionals (see [2]), one computes the di-mension of the intersection between the stable and the unstable manifolds as the differenceof a sort of generalized index function defined at each critical point. In the fixed endpointgeodesic case, such generalized index function can be described explicitly as a kind ofalgebraic count of the degeneracies of the index form along the geodesic. More precisely,this is the so-calledspectral flowof the path of index forms along the geodesic. Severalextensions of the Morse index theorem (see [13, 22]) show that this number is related to asymplectic invariant associated to a fixed endpoint geodesic, called the Maslov index. TheMaslov index is the natural substitute for the number of conjugate points along a geodesic,which may be infinite when the metric is non positive definite.

As to the periodic case, the notion of spectral flowsf(γ) of a closed geodesicγ hasbeen introduced only recently (see [7]); this is a generalization of the Morse index ofthe geodesic action functional in the Riemannian case. For the reader’s convenience, inSection 3 we will review briefly this definition, that is givenin terms of the choice of aperiodic frame along the geodesic. An explicit computationshows that the periodic spectralflow equals the fixed endpoint spectral flow plus aconcavity index, as in the original paperby M. Morse [20], plus a certain degeneracy term (see Theorem3.2).

The main purpose of the present paper is to establish the growth of the spectral flow un-der iteration of the closed geodesic, along the lines of [9].Given a closed semi-Riemanniangeodesicγ, we will show the existence of a functionλγ defined on the unit circleS1 andtaking values inZ (Definition 4.1), with the property that the spectral flow of theN -thiterateγ(N) of γ equals the sum of the values thatλγ takes at theN -th roots of unity (The-orem 5.3). This function is continuous, i.e., locally constant, at the points ofS1 \ 1 thatare not eigenvalues of the linearized Poincare mapPγ of γ (Proposition 4.4); the jump ofλγ at an eigenvalue ofPγ is bounded by the dimension of the corresponding eigenspace(Corollary 4.11). As in the Riemannian case, knowing the exact value of the jumps ofλγ

at each discontinuity point would determine entirely the functionλγ . It should be observedthat these discontinuities correspond to isolated degeneracy instants of ananalyticpath ofself-adjoint Fredholm operators, and the value of the jump equals the contribution of thedegeneracy instant to the spectral flow of the path. In principle, these jumps can be com-puted using higher order methods (see [11]), involving a finite number of derivatives of thepath. As to the pointz = 1, there is always a discontinuity ofλγ wheng is not positivedefinite (see Corollary 4.8); the value of the jump atz = 1 equals the index of the metrictensorg. Concerning thenullity of the iteratesγ(N), the semi-Riemannian case is totallyanalogous to the Riemannian case, where the question is reduced to studying the spectrumof the linearized Poincare map.

Using these properties of the spectral flow functionλγ , we then study the asymptotic be-havior of the sequenceN 7→ sf(γ(N)) ∈ Z, by first showing that the limitlim

N→∞

1N sf(γ(N))

exists and is finite (Proposition 6.6). More precisely, using a certain finite dimensional re-duction, we show thatsf(γ(N)) is the sum of a linear term inN , a uniformly boundedterm, and the term of a sequence which is either bounded or it satisfies a sort of uniformlinear growth inN (Proposition 6.1, Lemma 6.2 and Proposition 6.8). Whenγ is a hyper-bolic geodesic, i.e., whenPγ has no eigenvalue on the unit circle, then|sf(γ(N))| eithergrows linearly withN , or it is constant equal to the index of the metric tensorg (Proposi-tion 6.12). In view to the development of a full-fledged Morsetheory for semi-Riemannianclosed geodesics, the most important result is that the spectral flow of an iterateγ(N) iseither bounded, or it has a uniform linear growth (Proposition 6.7, Corollary 6.10). Thisimplies, in particular, that if a semi-Riemannian manifoldhas only a finite number of dis-tinct prime closed geodesics, then fork ∈ Z with |k| sufficiently large, the total numberof geometrically distinctclosed geodesics whose spectral flow is equal tok has to be uni-formly bounded (Proposition 6.11). This is the key point of Gromoll and Meyer celebratedRiemannian multiplicity result.

SPECTRAL FLOW AND ITERATION OF CLOSED GEODESICS 3

The results are obtained mostly by functional analytical techniques. Using periodicframes along the geodesic (Section 3), the problem is cast into the language of differentialsystems inRn and studied in the appropriate Sobolev space setting. Following Bott’s ideas,the spectral flow functionλγ is then obtained by considering a suitable complexificationofthe index form and of the space of infinitesimal variations ofthe geodesic (Subsections 4.1and 4.2). The central property ofλγ , that givessf(γ(N)) as a sum of the values ofλγ attheN -th roots of unity, is proved in Section 5; using Bott’s suggestive terminology, thisis called theFourier theorem. Its proof in the non positive definite case relies heavily ona very special property of the index form, which is that of being represented by a compactperturbation of a fixedsymmetryof the Hilbert space of variations ofγ. By a symmetryof a Hilbert space it is meant a self-adjoint operatorI whose squareI2 is the identity. Forpaths of the form symmetry plus compact, the spectral flow only depends on the endpointsof the path, which is used in the proof of the Fourier theorem.The question of continuityof λγ , which is quite straightforward in the positive definite case, is more involved in thegeneral semi-Riemannian case. At pointsz ∈ S1 \ 1, it is obtained by showing a per-turbation result for the spectral flow of paths of self-adjoint Fredholm operators restrictedto continuous families of closed subspaces of a fixed Hilbertspace (Corollary 2.3). Thedefinition and a few basic properties of spectral flow on varying domains are discussedpreliminarly in Section 2. As to the pointz = 1, where Corollary 2.3 doesnot apply, weuse a certain finite dimensional reduction formula for the spectral flow (Proposition 4.5),which was proved recently in [7] to show that the spectral flowfunction has inz = 1 a sortof artificial discontinuity whenγ is nondegenerate. The reduction formula is used also inlast section, where we obtain the iteration formula for the spectral flow (Proposition 6.1)and we prove estimates on its growth. For simplicity, in thispaper we will only considerorientation preserving closed geodesics; however, in Subsection 4.5 we discuss briefly howto deal with the non-orientation preserving case.

Future developments of the theory of periodic semi-Riemannian geodesics should in-clude an equivariant version of the strongly indefinite Morse theory, along the lines of [23].A preliminary important step would deal with the case of nondegenerate critical orbits; inthe context of periodic geodesics, this would apply to the so-calledbumpy metrics. Recallthat a metric is bumpy if all its closed geodesics are nondegenerate. Bumpy metrics aregeneric in the Riemannian setting (see [4, 5, 18, 24]); nothing is known with this respectin the nonpositive definite case.

2. SPECTRAL FLOW ON VARYING DOMAINS

Let H be a real or complex Hilbert space; we will denote byB(H) the Banach algebraof all bounded operators onH , by GL(H) the open subset ofB(H) consisting of all iso-morphisms, byO(H) the subgroup ofGL(H) consisting of all isometries, and byFsa(H)the set of all self-adjoint Fredholm operators onH . The adjoint of an operatorT onH willbe denoted byT ∗. Let us recall that the spectral flow is a integer invariant associated to acontinuous pathT : [a, b] → Fsa(H), which is:

(i) fixed endpoint homotopy invariant;(ii) additive by concatenation;(iii) invariant bycogredience, i.e., given two Hilbert spacesH1,H2, a continuous curve

T : [a, b] → Fsa(H1) and a continuous curveM : [a, b] → Iso(H1, H2) ofisomorphisms fromH1 to H2, then the spectral flow ofT on H1 coincides withthe spectral flow of[a, b] ∋ t 7→ MtTtM

∗t onH2.

We will denote bysf(T, [a, b]) the spectral flow of the curveT ; recall thatsf(T, [a, b]) is asort of algebraic count of the degeneracy instants of the pathTt ast runs froma to b. Detailson the definition and the basic properties of the spectral flowcan be found, for instance, inrefs. [7, 11, 21]. There are several conventions on how to compute the contribution of the


endpoints of the path, in case of degenerate endpoints; although making a specific choiceis irrelevant in the context of the present paper, we will follow the convention in [21].

Property (i) above holds in fact in a slightly more general form, as follows. Ifh : [a, b]×[c, d] → Fsa(H) is a continuous map such thatdim

(Ker(ha,s)

)anddim

(Ker(hb,s)

)are

constant for alls ∈ [c, d], then the spectral flow of the curve[a, b] ∋ t 7→ ht,c equals thespectral flow of the curve[a, b] ∋ t 7→ ht,d (see Corollary 2.3).

We will need to consider paths of Fredholm operators defined on varying domains.Let us consider the following setup: let[a, b] ∋ t 7→ Ht be a continuous path of closedsubspaces ofH . Recall that this means that, denoting byPt : H → H the orthogonalprojection ontoHt, then the curvet 7→ Pt is continuous relatively to the operator normtopology ofB(H). For instance, the kernels of a continuous familyt 7→ Ft of surjectivebounded linear maps fromH to some other Hilbert spaceH ′ form a continuous familyof closed subspaces ofH ([13, Lemma 2.9]). A simple lifting argument in fiber bundlesshows that there exists a continuous curvet 7→ Φt ∈ GL(H) and1 a closed subspaceH⋆

of H such thatΦt(H⋆) = Ht for all t. We will call the pair(Φ, H⋆) a trivialization of thepatht 7→ Ht. Assume now that[a, b] ∋ t 7→ Tt ∈ B(H) is a continuous curve with theproperty thatPtTt|Ht : Ht → Ht belongs toFsa(Ht) for all t. Then, given a trivialization(Φ, H⋆) for (Ht)t∈[a,b], for all t ∈ [a, b] the operatorP⋆Φ

∗tPtTtΦt|H⋆ : H⋆ → H⋆ belongs

to Fsa(H⋆), whereP⋆ is the orthogonal projection ontoH⋆. We can therefore give thefollowing definition:

Definition 2.1. The spectral flow of the pathT over the varying domains(Ht)t∈[a,b],denoted bysf

(T ; (Ht)t∈[a,b]

), is defined as the spectral flow of the continuous path[a, b] ∋

t 7→ P⋆Φ∗tPtTtΦt|H⋆ of self-adjoint Fredholm operators onH⋆.

Invariance by cogredience shows easily that the above definition does not depend onthe choice of the trivialization(Φ, H⋆) of (Ht)t∈[a,b]. Namely, assume that(Φ, H⋆) is

another trivialization of(Ht)t∈[a,b]. Denoting byP⋆ (resp.,P⋆) the orthogonal projection

ontoH⋆ (resp., ontoH⋆), and settingBt = P⋆Φ∗tPtTtΦt|H⋆ , Bt = P⋆Φ

∗tPtTtΦt| eH⋆

, and

Ψt = Φ∗t Φt, one has:

Bt =(Ψt| eH⋆

)∗Bt

(Ψt| eH⋆

)

for all t, hencesf(B, [a, b]) = sf(B, [a, b]).

Let us study how the spectral flow varies with respect to the domain.

Lemma 2.2. Let [a, b] ∋ t 7→ Tt ∈ B(H) be a continuous map and[c, d] ∋ s 7→ Hs bea continuous path of closed subspaces ofH , with the property thatPsTt|Hs ∈ Fsa(Hs)for all s and t. For all s ∈ [c, d], denote byhs the spectral flow of the path[a, b] ∋ t 7→PsTt|Hs of Fredholm operators onHs. Similarly, for allt ∈ [a, b] denote byvt the spectralflow of the (constant) path of Fredholm operatorsTt on the varying domains(Hs)s∈[c,d].Then:

(2.1) hc − hd = va − vb.

Proof. Choose a trivialization(Φ, H⋆) for (Hs)s∈[c,d], and define a continuous mapB :[a, b]× [c, d] → Fsa(H⋆) by:

Bs,t = P⋆Φ∗sPsTtΦs|H⋆ .

By definition, vt = sf(s 7→ Bs,t, [c, d]) andhs = sf(t 7→ Bs,t, [a, b]); formula (2.1)follows immediately from the homotopy invariance and the concatenation additivity of thespectral flow.

1In fact, one can find the curveΦt taking values inO(H), see [7].


Corollary 2.3. Under the assumptions of Lemma 2.2, ifKer(PsTa|Hs

)andKer

(PsTb|Hs

)

have constant dimension for alls ∈ [a, b], then the spectral flow oft 7→ PsTt|Hs onHs

does not depend ons.

Proof. This follows easily from the fact that curves of self-adjoint Fredholm operatorswith kernel of constant dimension have null spectral flow. Thus, under our assumptionsboth termsva andvb in (2.1) vanish.

3. SPECTRAL FLOW ALONG A CLOSED GEODESIC

We will recall from [7] the definition of spectral flow along a closed geodesic.

3.1. Periodic geodesics.We will consider throughout ann-dimensional semi-Riemannianmanifold(M, g), denoting by∇ the covariant derivative of its Levi–Civita connection, andbyR its curvature tensor, chosen with the sign conventionR(X,Y ) = [∇X ,∇Y ]−∇[X,Y ].Let γ : [0, 1] → M be a periodic geodesic inM , i.e.,γ(0) = γ(1) andγ(0) = γ(1). Wewill assume thatγ is orientation preserving, which means that the parallel transport alongγ is orientation preserving; the non orientation preservingcase can be studied similarly, asexplained in Subsection 4.5. IfM is orientable, then every closed geodesic is orientationpreserving. Moreover, given any closed geodesicγ, its two-fold iterationγ(2), defined byγ(2)(t) = γ(2t), is always orientation preserving. We will denote byD

dt the covariant dif-ferentiation of vector fields alongγ; recall that theindex formIγ is the bounded symmetricbilinear form defined on the Hilbert space of all periodic vector fields of Sobolev classH1

alongγ, given by:

(3.1) Iγ(V,W ) =

∫ 1

0

g(DdtV,

DdtW ) + g(RV,W ) dt,

where we setR = R(γ, ·)γ. Closed geodesics inM are the critical points of the geodesicaction functionalf(γ) = 1

2

∫ 1

0 g(γ, γ) dt defined in thefree loop spaceΩM of M ; ΩMis the Hilbert manifold of all closed curves inM of Sobolev classH1. The index formIγis the second variation off at the critical pointγ; unlessg is positive definite, the Morseindex of f at each non constant critical point is infinite. The notion ofMorse index isreplaced by the notion of spectral flow.

Let us denote byPγ : Tγ(0)M⊕Tγ(0)M → Tγ(0)M⊕Tγ(0)M thelinearized Poincaremapof γ, defined by:

Pγ(v, v′) =

(V (1), D

dtV (1)),

whereV is the unique Jacobi field alongγ such thatV (0) = v and DdtV (0) = v′. Fixed

points ofPγ correspond to periodic Jacobi fields alongγ. Moreover,Pγ preserves thesymplectic form of Tγ(0)M ⊕ Tγ(0)M defined by:

((v, v′), (w,w′)

)= g(v, w′)− g(v′, w).

3.2. Periodic frames and trivializations. Consider a smooth periodic orthonormal frameT alongγ, i.e., a smooth family[0, 1] ∋ t 7→ Tt of isomorphisms:

(3.2) Tt : Rn −→ Tγ(t)M

with T0 = T1, and

(3.3) g(Ttei, Ttej) = ǫiδij ,

whereeii=1,...,n is the canonical basis ofRn, ǫi ∈ −1, 1 andδij is the Kroneckersymbol. The existence of such frame is guaranteed by the orientability assumption on theclosed geodesic. The pull-back byTt of the metricg gives a symmetric nondegeneratebilinear formG onRn, whose index is the same as the index ofg; note that this pull-backdoes not depend ont, by the orthogonality assumption on the frameT. In the sequel, we


will also denote byG : Rn → Rn the symmetric linear operator defined by(Gv) · w; by(3.3),G satisfies:

(3.4) G2 = Id.

Moreover, the pull-back of the linearized Poincare mapPγ by the isomorphismT0 ⊕ T0 :Rn ⊕ Rn → Tγ(0)M ⊕ Tγ(0)M gives a linear endomorphism ofRn ⊕ Rn that will bedenoted byP.

For all t ∈ ]0, 1], define byHγt the Hilbert space of allH1-vector fieldsV alongγ|[0,t]

satisfying:T−10 V (0) = T−1

t V (t).

Observe that the definition ofHγt depends on the choice of the periodic frameT, however,

Hγ1 , which is the space of all periodic vector fields alongγ, does not depend onT. Al-

though in principle there is no necessity of fixing a specific Hilbert space inner product, itwill be useful to have one at disposal, and this will be chosenas follows. For allt ∈ ]0, 1],consider the Hilbert space:

H1per

([0, t],Rn

)=

V ∈ H1

([0, t],Rn) : V (0) = V (t)

;

a natural Hilbert space inner product inH1per

([0, t],Rn

)is given by:

(3.5) 〈V ,W 〉 = V (0) ·W (0) +

∫ t

0

V′(s) ·W

′(s) ds,

where· is the Euclidean inner product inRn. The mapΨt : Hγt → H1

per

([0, t],Rn

)

defined byΨt(V ) = V , whereV (s) = T−1s (V (s)) is a linear isomorphism; the space

Hγt will be endowed with the pull-back of the inner product (3.5)by the isomorphismΨt.

Denote byRt ∈ End(Rn) the pull-back byTt of the endomorphismRγ(t) = R(γ, ·)γ ofTγ(t)M :

Rt = T−1t Rγ(t) Tt;

observe thatt 7→ Rt is a smooth map ofG-symmetric endomorphisms ofRn. Finally,denote byΓt ∈ End(Rn) theChristoffel symbolof the frameT, defined by:

Γt(v) = T−1t

(DdtV

)−

d

dtV (t),

whereV is any vector field satisfyingV (t) = v, andV = Ψ−1t (V ). A straightforward

computation shows thatΓt isG-anti-symmetric for allt.The push-forward byΨt of the index formIγ onHγ

t is given by the bounded symmetricbilinear formIt onH1

per

([0, t],Rn

)defined by:

(3.6) It(V ,W ) =

∫ t

0

G(V

′(s),W

′(s)

)+G

(ΓsV (s),W

′(s)

)+G

(V

′(s),ΓsW (s)

)

+G(ΓsV (s),ΓsW (s)

)+G

(RsV (s),W (s)

)ds.

Finally, for t ∈ ]0, 1], we will consider the isomorphism

Φt : H1per

([0, t],Rn

)→ H1

per

([0, 1],Rn

),

defined byV 7→ V , whereV (s) = V (st), s ∈ [0, 1]. The push-forward byΦt of thebilinear formIt is given by the bounded symmetric bilinear formIt onH1

per

([0, 1],Rn

)

defined by:

(3.7)

It(V , W ) =1

t2

∫ 1

0

G(V ′(r), W ′(r)

)+ tG

(ΓtrV (r), W ′(r)

)+ tG

(V ′(r),ΓtrW (r)

)

+ t2G(ΓtrV (r),ΓtrW (r)

)+ t2G

(RtrV (r), W (r)

)dr.


3.3. Spectral flow of a periodic geodesic.For t ∈ ]0, 1], define the Fredholm bilinearformBt on the Hilbert spaceH1

per

([0, 1],Rn

)by setting:

(3.8) Bt = t2 · It.

From (3.7), one sees immediately that the map]0, 1] ∋ t 7→ Bt can be extended continu-ously tot = 0 by setting:

B0(V , W ) =

∫ 1

0

G(V ′(r), W ′(r)

)dr.

Observe thatKer(B0) isn-dimensional, and it consists of all constant vector fields.For allt ∈ [0, 1], the bilinear formIt onH1

per

([0, 1],Rn

)is represented with respect to the inner

product (3.5) by a compact perturbation of the symmetryJ of H1per

([0, 1],Rn

)given by

V 7→ GV .

Definition 3.1. Thespectral flowsf(γ) of the closed geodesicγ is defined as the spectralflow of the continuous path of Fredholm bilinear forms[0, 1] ∋ t 7→ Bt on the HilbertspaceH1

per

([0, 1],Rn

).

It is a non trivial fact that the definition of spectral flow along a closed geodesic doesnot depend on the choice of a periodic orthonormal frame along the geodesic. This resultis obtained in [7] by determining an explicit formula givingthe spectral flow in terms ofsome other integers associated to the geodesic, such as theMaslov indexand the concavityindex. The Maslov index is a symplectic invariant, which is computed as an intersectionnumber in the Lagrangian Grasmannian of a symplectic vectorspace; it will be denoted byiMaslov(γ).

Recall that aJacobi fieldalongγ is a smooth vector fieldJ alongγ that satisfies thesecond order linear equation:

D2

dt2J(t) = R(γ(t), J(t)

)γ(t), t ∈ [0, 1];

let us denote byJγ the2n-dimensional real vector space of all Jacobi fields alongγ. Letus introduce the following spaces:

J perγ =

J ∈ Jγ : J(0) = J(1), D

dtJ(0) =DdtJ(1)

,

J 0γ =

J ∈ Jγ : J(0) = J(1) = 0

, and

J ⋆γ =

J ∈ Jγ : J(0) = J(1)

.

It is well known thatJ perγ is the kernel of the index formIγ defined in (3.1), whileJ 0

γ is thekernel of the restriction of the index form to the space of vector fields alongγ vanishing atthe endpoints. We denote bynper(γ) andn0(γ) the dimensions ofJ per

γ andJ 0γ respectively.

The nonnegative integernper(γ) is the nullity ofγ as a periodic geodesic, i.e., the nullity ofthe Hessian of the geodesic action functional atγ in the space of closed curves. Observethatnper(γ) ≥ 1, asJ per

γ contains the one-dimensional space spanned by the tangent fieldJ = γ. Similarly, n0(γ) is the nullity of γ as a fixed endpoint geodesic, i.e., it is thenullity of the Hessian of the geodesic action functional atγ in the space of fixed endpointscurves inM . In this case,n0(γ) > 0 if and only if γ(1) is conjugate toγ(0) alongγ. Theindex of concavityof γ, that will be denoted byiconc(γ) is a nonnegative integer invariantassociated to periodic solutions of Hamiltonian systems. In our notations,iconc(γ) is equalto the index of the symmetric bilinear form:

(J1, J2) 7−→ g(DdtJ1(1)−

DdtJ1(0), J2(0)

)

defined on the vector spaceJ ⋆γ . It is not hard to show that this bilinear form is symmetric,

in fact, it is given by the restriction of the index formIγ toJ ⋆γ .


Theorem 3.2. Let (M, g) be a semi-Riemannian manifold and letγ : [0, 1] → M be aclosed oriented geodesic inM . Then, the spectral flowsf(γ) is given by the followingformula:

(3.9) sf(γ) = dim(J perγ ∩ J 0

γ

)− iMaslov(γ)− iconc(γ)− n−(g),

wheren−(g) is the index of the metric tensorg.

Proof. See [7, Theorem 5.6].

Formula (3.9) proves in particular that the definition of spectral flow for a periodicgeodesicγ does not depend on the choice of an orthonormal frame alongγ.

4. THE SPECTRAL FLOW FUNCTION

4.1. The basic data. As described in Section 3, the choice of a smooth periodic orthonor-mal frame along a closed orientation preserving semi-Riemannian geodesic produces thefollowing objects:

• a non degenerate symmetric bilinear formG onRn and a symplectic form onRn ⊕Rn defined by

((v, v′), (w,w′)

)= G(v, w′)−G(v′, w);

• a smooth1-periodic curveΓ : R → End(Rn) of G-anti-symmetric linear endo-morphisms ofRn;

• a smooth1-periodic curveR : R → EndG(Rn) of G-symmetric linear endomor-

phisms ofRn;• a linear endomorphismP : Rn ⊕Rn → Rn ⊕Rn that preserves the symplectic

form.

We will use a complexification of these data. More precisely,G will be extended bysesquilinearity onCn, Γ andR will be extended toC-linear endomorphisms ofCn, andPwill be extended toC-linear endomorphisms ofC2n. LetH be the complex Hilbert spaceH1

([0, 1],Cn) endowed with inner product:

〈V , W 〉 = V (0) · W (0) +

∫ 1

0

V ′(r) · W ′(r) dr,

where now· denotes the canonical Hermitian product inCn: v · w =∑n

j=1 vjwj . Givena unit complex numberz, letHz denote the Hilbert subspace ofH defined by:

(4.1) Hz =V ∈ H : V (1) = zV (0)

;

moreover, we will denote byHo the subspace:

(4.2) Ho =V ∈ H : V (0) = V (1) = 0

.

For t ∈ [0, 1], letBt : H×H → C denote the bounded Hermitian form defined by:

(4.3)

Bt(V , W ) =

∫ 1

0

G(V ′(r), W ′(r)

)+ tG

(ΓtrV (r), W ′(r)

)+ tG

(V ′(r),ΓtrW (r)

)

+ t2G(ΓtrV (r),ΓtrW (r)

)+ t2G

(RtrV (r), W (r)

)dr.

With the above data, the Jacobi equation alongγ gives the following second order linearhomogeneous equation for vector fields inCn:

(4.4) V ′′(r) + 2ΓrV′(r) + (Γ′

r + Γ2r −Rr)V (r) = 0.

The linear mapP : Cn ⊕ Cn → Cn ⊕ Cn is given by:

P(v, v′) =(V (1), V ′(1) + Γ0V (1)

),

whereV ∈ C2([0, 1],Cn

)is the unique solution of (4.4) satisfyingV (0) = v andV ′(0) =

v′ − Γ0v. We observe thatt → Bt can extended in the obviuos way fort ∈ [0,+∞), andBN corresponds with the index form associated to theN -th iterateγ(N).


4.2. The spectral flow function on the circle. As in Section 3, it is easy to see that for allt ∈ [0, 1] and allz ∈ S1, the restriction of the Hermitian formBt toHz×Hz is representedby a compact perturbation of the symmetryJ of H defined byV 7→ GV ; observe that eachHz is invariant byJ.

Definition 4.1. Thespectral flow functionλγ : S1 → N is defined by:

λγ(z) = spectral flow of[0, 1] ∋ t 7→ Bt on the spaceHz ×Hz.

Let us recall that the spectral flow of a continuous path of symmetric bilinear formsB on a real Hilbert spaceH equals the spectral flow of the continuous path of FredholmHermitian forms obtained by taking the sesquilinear extension of B on the complexifiedHilbert spaceHC. In particular,λγ(1) = sf(γ).

More generally, given an integerN ≥ 1, we can defineλγ(z;N) as the spectral flow ofthe patht 7→ BNt onHz ×Hz . Then:

(4.5) λγ(1, N) = sf(γ(N)

),

whereγ(N) is theN -th iterate ofγ.It will also be useful to introduce the following notation:

(4.6) λoγ = spectral flow of[0, 1] ∋ t 7→ Bt on the spaceHo ×Ho;

it is shown in [12] that

(4.7) λoγ = n0(γ)− n−(g)− iMaslov(γ).

4.3. Continuity and jumps of the spectral flow function. The reader will easily checkby an immediate partial integration argument that equation(4.4) characterizes the elementsin the kernel of the Hermitian formB1 in (4.3); more precisely:

Lemma 4.2. V ∈ H is in the kernel ofB1 : Hz ×Hz → C if and only if it satisfies(4.4)and the boundary conditions:

V (1) = z · V (0), and V ′(1) = z · V ′(0).

Thus,B1 is degenerate onHz if and only ifz is in the spectrum of the linearized PoincaremapPγ , anddim

(Ker(B1|Hz×Hz)

)= dim

(Ker(Pγ − z · Id)

).

Lemma 4.3. The mapS1 ∋ z 7→ Hz ⊂ H is a continuous2 map of closed subspaces ofH.

Proof. Hz is the kernel of a continuous familyFz : H → Cn of surjectivebounded linearmaps, defined byFz(V ) = V (1)− zV (0) (see [13, Lemma 2.9]).

Proposition 4.4. The spectral flow functionλγ is constant on every connected subsetAof S1 \ 1 that does not contain elements in the spectrum of the linearized Poincare mapPγ .

Proof. This follows easily from Corollary 2.3. By Lemma 4.2, the assumption on thespectrum of the Poincare map says thatB1 does not degenerates onHz , for all z ∈ A.Moreover, it is easy to see thatB0 is nondegenerate onHz for all z 6= 1 (while the kernelof B0|H1×H1 consists of all constant maps, and it has dimensionn). This concludes theproof.

We conclude thatλγ has a finite number of jumps onS1, that can occur only at thosepoints in the spectrum ofPγ that lie inS1 or atz = 1.

Let us study now the behavior ofλγ aroundz = 1; the result of Proposition 4.4 cannotbe extended toz = 1, becauseB0 is always degenerate onH1. We will show that, unlessthe metric tensorg is positive definite, thenλγ is indeed discontinuous atz = 1. To this

2In fact, the very same argument shows thatz 7→ Hz is a real analytic map. The same conclusion holds inLemma 4.7.


aim, we will need to determine an alternative description ofthe functionλγ based on afinite dimensional reduction for the computation of the spectral flow.

4.4. A finite dimensional reduction. By a symmetryof a Hilbert spaceH we mean abounded self-adjoint operatorJ on H satisfyingJ2 = 1. For paths[a, b] ∋ t 7→ Tt ∈Fsa(H) of Fredholm self-adjoint operators of the formTt = J + Kt, whereJ is a fixedsymmetry ofH andKt is a compact self-adjoint operator onH for all t, the spectral flowsf(T, [a, b]) depends only on the endpointsTa andTb. More precisely,sf(T, [a, b]) is therelative dimension of the generalized negative spaces ofTt at t = a and att = b. We wantto compare the spectral flow of a pathT onH with the spectral flow of its restriction to afinite codimensional subspace ofH . Let us recall the following result from [7]:

Proposition 4.5. Let T : [a, b] → Fsa(H) be a continuous curve where eachTt is acompact perturbation of a fixed symmetryJ of H , and letV ⊂ H be a closed subspace offinite codimension inH . SetBt = 〈Tt·, ·〉 andVt = (TtV)

⊥ = V⊥Bt ; then:

(4.8)sf(T, [a, b])−sf

(PVT |V , [a, b]

)= n−

(Ba|Va×Va

)+dim(V∩Va)−dim

(V ∩Ker(Ba)

)

− n−(Bb|Vb×Vb

)− dim(V ∩ Vb) + dim

(V ∩Ker(Bb)

),

wheren− denotes the index of a Hermitian (or symmetric in the real case) bilinear form.

Proof. See [7, Theorem 4.3].

We apply this result to the path of bilinear forms[0, 1] ∋ t 7→ Bt given in (4.3), to theHilbert spaceH = Hz in (4.1) and to the closed finite codimensional subspaceV = Ho

defined in (4.2), obtaining:

Proposition 4.6. For all z ∈ S1, the following equality holds:3

(4.9) λγ(z)− λoγ = (1− δz,1) · n−(g)− n0(γ) + dim

(J(1)γ (z)

)− n−(bz),

whereJ(1)γ (z) is the finite dimensional vector space:

J(1)γ (z) = Ho ∩Ker(B1|Hz×Hz)

=V ∈ C2

([0, 1],Cn

): V solution of (4.4), V (0) = V (1) = 0, V ′(1) = zV ′(0)

andbz is the Hermitian form on the finite dimensional space:

J(2)γ (z) =

V ∈ C2

([0, 1],Cn

): V solution of (4.4), V (1) = zV (0)

,

given by the restriction ofB1, or, more explicitely:

bz(V , W ) = G(zV ′(1)− V ′(0), W (0)

).

Proof. Formula (4.9) follows directly by applying Proposition 4.5to the above setup. It iseasily obtained after checking the following identities:

• H⊥B0o ∩Hz =

V : V (t) = A(z−1)t+A, for someA ∈ Cn

, thus the index of

the restriction ofB0 to such space is equal to0 if z = 1 and is equal ton−(G) =n−(g) if z 6= 1;

• Ho ∩(H

⊥B0o ∩Hz

)= 0;

• Ker(B0|Hz×Hz

)=

V : V ′′ = 0,

(V (1), V ′(1)

)= z ·

(V (0), V ′(0)

), such

space is0 if z 6= 1, and it consists of constant functions ifz = 1;

• Ho ∩(Ker

(B0|Hz×Hz

))= 0;

• Ker(B1|Hz×Hz

)=

V solution of (4.4):

(V (1), V ′(1)

)= z ·

(V (0), V ′(0)

);

3Hereδz,1 is the Kronecker symbol, equal to1 if z = 1 and to0 otherwise.


• Ho ∩Ker(B1|Hz×Hz

)= J

(1)γ (z);

• H⊥B1o ∩Hz = J

(2)γ (z);

• Ho ∩ H⊥B1o = Ker

(B1|Ho×Ho) =

V solution of (4.4), V (0) = V (1) = 0

.

We recall that the termn0(γ) in (4.9) is the nullity ofγ as afixed endpoints geodesic,i.e., the multiplicity ofγ(1) as conjugate point toγ(0) alongγ. In particular, it coincideswith the dimension of the space:

(4.10)V ∈ C2

([0, 1],Cn

): V solution of (4.4), V (0) = V (1) = 0

.

Observe that for allz ∈ S1, the spaceJ(1)γ (z) is contained in (4.10); it follows that

(4.11) − n0(γ) + dim(J(1)γ (z)

)≤ 0, ∀ z ∈ S

1.

Lemma 4.7. The mapz 7→ J(2)γ (z) ⊂ H is continuous at those pointsz ∈ S1 that are not

in the spectrum ofP.

Proof. It suffices to show thatz 7→ J(2)γ (z) is a continuous family of closed subspaces of

the finite dimensional closed subspaceS =V ∈ C2

([0, 1],Cn

): V is solution of (4.4)

of H. If we identify S ∼= Cn ⊕ Cn via the mapV 7→(V (0), V ′(0)

), thenJ(2)γ (z) is the

kernel of the linear mapLz = π1(P−z ·I) : Cn⊕Cn → Cn, whereπ1 : Cn⊕Cn → Cn

is the projection on the first summand. Clearly,z 7→ Lz is continuous. Ifz ∈ S1 is notin the spectrum ofP, thenP − z · I is an isomorphism, and thusLz is surjective, whichconcludes the proof (see [13, Lemma 2.9]).

Corollary 4.8. LetA be a connected subset ofS1 that containsz = 1, and that does notcontain any eigenvalue of the linearized Poincare mapPγ . Then,λγ is constant equal toλγ(1) + n−(g) onA \ 1.

Proof. Using formula (4.9), the reader will easily convince himself that the statement isequivalent to proving that the quantitydim

(J(1)γ (z)

)− n−(bz) is constant on every con-

nected subsetA of S1 that does not contain elements in the spectrum ofPγ . This follows

immediately from the continuity of the subspacesA ∋ z 7→ J(2)γ (z) proved in Lemma 4.7,

and from the following observations:

(a) Ker(bz) =W solution of (4.4): W (0) = W (1) = 0

, thusKer(bz) does not

depend onz ∈ A, andn−(bz) is constant onA;

(b) J(1)γ (z) ⊂ Ker(B1|Hz×Hz), hencedim

(J(1)γ (z)

)= 0 for all z ∈ A.

In order to prove (a), note that ifz is not in the spectrum ofPγ , then the mapJ(2)γ (z) ∋

V 7→ zV ′(1)− V ′(0) ∈ Cn is an isomorphism. It follows immediately from the definition

of bz thatKer(bz) =W ∈ J

(2)γ (z) : W (0) = 0

, which gives the desired conclusion.

Remark4.9. Note that, when1 is not in the spectrum ofPγ , the discontinuity atz = 1 ofλγ occurs only in the non Riemannian case.

4.5. Non orientation preserving closed geodesics.When the geodesicγ is non orienta-tion preserving, the definition of the spectral flow given in Definition 3.1 does not applybecause there is no periodic orthonormal frame along the geodesic. Let us now indicatebriefly how to modify the construction in order to get a well defined notion of spectral flowalso in this case. One can consider an arbitrary smooth orthonormal frameT = (Tt)t∈[0,1]

along the geodesic as in (3.2), which willnot satisfyT0 = T1; denote byS ∈ GL(Cn) thecomplexification of the isomorphismT−1

1 T0. Then, it would be natural to define the spec-tral flow sf(γ) as the spectral flow of the path of Fredholm bilinear forms[0, 1] ∋ t 7→ Bt


given in (3.8) on the space:

H1S

([0, 1],Cn

)=

V ∈ H1

([0, 1],Cn

): V (1) = SV (0)

.

(compare with Definition 3.1). However, with this definition, using Proposition 4.5 onechecks easily that formula (3.9) will not hold in general. More precisely, the right handside of (3.9) will contain an extra term, given by the index ofthe restriction of the metrictensorg on the image of the operatorS − Id. This is proved easily with the help ofProposition 4.5, as in the proof of Proposition 4.6. Namely,in this case, theB0-orthogonalspace toH1

0

([0, 1],Cn

)in H1

S

([0, 1],Cn

)is given by then-dimensional space of all affine

mapsV (t) = (S − Id)Bt + B, with B ∈ Cn. The restriction ofB0 to such space hasindex equal to the index of the restriction ofg to the image ofS − Id. Note thatS = Idin the orientation preserving case. Thus, one way to make thedefinition independent onthe orthogonal frame would be to restrict to framesT for which the operatorS = T−1

1 T0

is such thatg is positive semi-definite on the image ofS − Id. This is always possiblewheng is not negative definite, by a simple linear algebra argument. Or, more simply,one could definesf(γ) as the difference between the spectral flow of the patht 7→ Bt onH1

S

([0, 1],Cn

)minus the index of the restriction ofg to Im(S − Id), which by what has

been observed, is a quantity independent of the frame. With such definition, formula (3.9)holds also in the non orientation preserving case, and the entire theory developed in thispaper carries over to the non orientable case.

4.6. On the jumps of the spectral flow function. By the results of Proposition 4.4 andCorollary 4.8, we know thatλγ is a piecewise constant function onS1, whose jumps occurat the eigenvalues of the linearized Poincare map and at1. Thus,λγ is uniquely determinedonce we know its value at some given point, say atz = 1, and the value of the jump ofλγ

at each discontinuity point. When1 is not an eigenvalue ofPγ , then the jump ofλγ at1 isn−(g), as proved in Corollary 4.8, butlim

θ→0−λγ(e

iθ) = limθ→0+

λγ(eiθ). In order to give an

upper estimate for the jumps at the eigenvalues ofPγ , we need the following:

Lemma 4.10. Let I ∋ t 7→ Bt be a continuous curve of Hermitian forms on a HilbertspaceH , and letI ∋ t 7→ Ht be a continuous family of closed subspaces ofH , such thatBt|Ht×Ht is Fredholm for allt. Assume that a pointt0 in the interior ofI is an isolated de-generacy instant forBt|Ht×Ht . Then, forε > 0 small enough,

∣∣sf(B, (Ht)t∈[t0−ε,t0+ε]

)∣∣ ≤dim

[Ker

(Bt0 |Ht0×Ht0

)].

Proof. Using a trivialization for the pathI ∋ t 7→ Ht, it suffices to prove the result for thespectral flow of a continuous path of Fredholm Hermitian forms t 7→ Bt on a fixed HilbertspaceH . WhenH is finite dimensional, in which case the spectral flow is the variation ofthe index function, the result is elementary and well known.The infinite dimensional casecan be reduced to the finite dimensional one by an argument of functional calculus. Moreprecisely, for allt, let Tt be the self-adjoint Fredholm operator such thatBt = 〈Tt·, ·〉; letε > 0 be small enough so that the spectrum ofTt0 has empty intersection with]−2ε, 2ε[ \0. Then, fort sufficiently close tot0, the spectrum ofTt has empty intersection with[−ε, ε] \ 0. Thus, denoting byχ[−ε,ε] the characteristic function of[−ε, ε], the mapt 7→ Pt = χ[−ε,ε](Tt) is a map of finite rank projections which is continuous neart0; theimageHt of Pt is Tt-invariant, so thatKer(Bt) = Ker(Bt|Ht×Ht). By definition (see[21]), the spectral flow throught0 of the pathT is the variation of index of the restrictionof Bt toHt ×Ht, which reduces the general case to a finite dimensional one.

With this, we are now able to prove that the jump ofλγ atz ∈ S1 can be estimated withthe dimension of thez-eigenspace ofPγ .

Corollary 4.11. Leteiθ0 be an eigenvalue ofPγ . Then:∣∣∣ limθ→0+

λγ

(ei(θ0+θ)

)− lim

θ→0−λγ

(ei(θ0+θ)

)∣∣∣ ≤ dim(Ker(Pγ − eiθ0 · I)

).


Proof. By the concatenation additivity of the spectral flow and asB0|Hz×Hz is non-degenerate forz 6= 1, for θ > 0 small enough the differenceλγ

(ei(θ0+θ)

)− λγ

(ei(θ0+θ)

)

is equal to the spectral flow of the constant Fredholm Hermitian formB1 on the continuouscurve of closed subspaces[−θ, θ] ∋ ρ 7→ Hei(θ0+ρ) . By assumption, there is an isolateddegeneracy instant ofB1 at ρ = 0. By Lemma 4.10, the jump ofλγ at eiθ0 is less than orequal to the dimension ofKer

(B1|H

eiθ0×H

eiθ0

); this is equal todim

(Ker(Pγ − eiθ0 · I)

)

by Lemma 4.2.

5. THE FOURIER THEOREM

We will now fix an integerN ≥ 1 and we set:

ω = e2πiN .

For all k ≥ 0, givenVk ∈ Hωk we will assume thatVk is extended to a continuous mapVk : R → Cn by setting:

Vk(t+m) = ωkmVk(t), m ∈ Z, t ∈ [0, 1[ .

Lemma 5.1. The mapΦN : H1 →N−1⊕k=0

Hωk defined by:

ΦN (V ) = (V0, . . . , VN−1),

where:

(5.1) Vk(t) =1

N

N−1∑

j=0

ω−jkV

(t+ j

N

), t ∈ [0, 1],

is a linear isomorphism, whose inverseΨN :N−1⊕k=0

Hωk → H1 is given by:

ΨN(V0, . . . , VN−1) = V ,

where

(5.2) V (t) =

N−1∑

k=0

Vk(tN).

Proof. A matter of straightforward calculations, based on the identity:

N−1∑

j=0

ωjk =

N, if k ≡ 0 mod N ;

0, otherwise.

First, one needs to prove thatΦN is well defined, i.e., that the mapVk in (5.1) belongs toHωk :

Vk(1) =1

N

N−1∑

j=0

ω−kj V(1+jN

)=

1

N

N∑

j=1

ω−k(j−1)V(

jN

)

= ωk

1

N

N∑

j=1

ω−kj V(

jN

) =

ωk

N

V (1) +

N−1∑

j=1

ω−kj V(

jN

)

=ωk

N

N−1∑

j=0

ω−kj V(

jN

) = ωkVk(0),

i.e., Vk ∈ Hωk . Similarly, ΨN is well defined. Clearly,ΦN andΨN are linear andbounded.


In order to check thatΨN is a right inverse forΦN , set(W0, . . . , WN−1) = ΦN

ΨN(V0, . . . , VN−1). Then, for allk = 0, . . . , N − 1 and allt ∈ [0, 1]:

Wk(t) =1

N

N−1∑

j=0

N−1∑

l=0

ω−kj Vl(t+ j) =1

N

N−1∑

j=0

N−1∑

l=0

ω−kjωlj Vl(t)

=1

N

N−1∑

l=0

N−1∑

j=0

ωj(l−k)

Vl(t) =

N−1∑

l=0

δl,kVl(t) = Vk(t),

i.e.,ΦN ΨN is the identity ofN−1⊕k=0

Hωk .

Finally, to check thatΨN is a left inverse forΦN , setV = ΨN ΦN (W ) and compute:

V (t) =1

N

N−1∑

k=0

N−1∑

j=0

ω−kjW (t+ j)

=1

N

N−1∑

k=0

N−1∑

j=0

ω−kjW (t) =N−1∑

k=0

δk,0W (t) = W (t),

for all t ∈ [0, 1]. Thus,ΨN ΦN is the identity ofH1, which concludes the proof.

Lemma 5.2. GivenV , W ∈ H1, set:

ΦN (V ) = (V0, . . . , VN−1), and ΦN (W ) = (W0, . . . , WN−1),

with Vk, Wk ∈ Hωk for all k = 0, . . . , N − 1. Then, the following identities hold:

B0(V , W ) = N2N−1∑

k=0

B0(Vk, Wk), and BN (V , W ) = N2N−1∑

k=0

B1(Vk, Wk).

Proof. Again, a matter of direct calculations, as follows:

B0(V , W ) = N2∑

k,j

∫ 1

0

G(V ′k(rN), W ′

j(rN))dr

= N∑

k,j

∫ N

0

G(V ′k(s), W

′j(s)

)ds = N

∑

k,j,l

∫ l+1

l

G(V ′k(s), W

′j(s)

)ds

= N∑

k,j,l

∫ 1

0

G(V ′k(s+ l), W ′

j(s+ l))ds = N

∑

k,j,l

ω(k−j)l

∫ 1

0

G(V ′k(s), W

′j(s)

)ds

= N2∑

k

∫ 1

0

G(V ′k(s), W

′k(s)

)ds = N2

∑

k

B0(Vk, Wk).


Similarly,

BN (V , W ) = N2

∫ 1

0

∑

k,j

[G(V ′k(rN), W ′

j(rN))+G

(ΓNrVk(rN), W ′

k(rN))

+G(V ′k(rN),ΓNrWj(rN)

)+G

(ΓNrVk(rN),ΓNrWj(rN)

)

+G(RNrVk(rN), Wj(rN)

)]dr

= N2∑

k,j,l

∫ l+1N

lN

[G(V ′k(rN), W ′

j(rN))+G

(ΓNrVk(rN), W ′

j(rN))

+G(V ′k(rN),ΓNrWj(rN)

)+G

(ΓNrVk(rN),ΓNrWj(rN)

)

+G(RNrVk(rN), Wj(rN)

)]dr

= N∑

k,j,l

∫ l+1

l

[G(V ′k(s), W

′j(s)

)+G

(ΓsVk(s), W

′j(s)

)

+G(V ′k(s),ΓsWj(s)

)+G

(ΓsVk(s),ΓsWj(s)

)

+G(RsVk(s), Wj(s)

)]ds

= N∑

k,j,l

ω(k−j)l

∫ 1

0

[G(V ′k(s), W

′j(s)

)+G

(ΓsVk(s), W

′j(s)

)


)+G

(ΓsVk(s),ΓsWj(s)

)

+G(RsVk(s), Wj(s)

)]ds

= N2∑

k

∫ 1

0

[G(V ′k(s), W

′k(s)

)+G

(ΓsVk(s), W

′k(s)

)


)+G

(ΓsVk(s),ΓsWk(s)

)

+G(RsVk(s), Wk(s)

)]ds

= N2∑

k

B1(Vk, Wk).

Theorem 5.3(Fourier theorem).

sf(γ(N)

)=

N−1∑

k=0

λγ(ωk).

Proof. This follows immediately from (4.5), Lemmas 5.1, 5.2, and the following two ob-servations:

(1) the spectral flow of a path of compact perturbations of a fixed symmetry onlydepends on the endpoints of the path;

(2) the spectral flow is additive by direct sums.

6. A SUPERLINEAR ESTIMATE FOR THE SPECTRAL FLOW OF AN ITERATE

We will now use the Fourier theorem and formula (4.9) in orderto establish estimateson the growth of the spectral flow for theN -th iterate of a closed geodesicγ. We will usethe notations in Subsection 4.4; an immediate application of Proposition 4.6, Theorem 5.3and Eq. (4.7) gives the following:


Proposition 6.1. Given a closed geodesic and an integerN ≥ 1:

(6.1) sf(γ(N)

)= −N

(iMaslov(γ)

)− n−(g) +

N−1∑

k=0

dim(J(1)γ (ωk)

)−

N−1∑

k=0

n−(bωk),

whereω = e2πi/N .

Let us estimate the last two terms in formula (6.1).

Lemma 6.2. The quantity∑N−1

k=0 dim(J(1)γ (ωk)

)is uniformly bounded:

(6.2) 0 ≤N−1∑

k=0

dim(J(1)γ (ωk)

)≤ 2 dim(M).

Proof. If we identify the spaceS =V ∈ C2

([0, 1],Cn

): V is solution of (4.4)

with Cn ⊕ Cn via the mapV 7→(V (0), V ′(0)

), then for allz ∈ S1, J(1)γ (z) is iden-

tified with a subspace ofKer(Pγ − z · I). The conclusion follows from the fact that∑z∈C

dim(Ker(Pγ − z · I)

)≤ 2 dim(M).

A rough estimate for the term containing the index of the Hermitian formsbz is givenin the following:

Lemma 6.3. For all z ∈ S1, n−(bz) ≤ 2 dim(M) − n0(γ); if z is not in the spectrum ofPγ , thenn−(bz) ≤ dim(M)− n0(γ). It follows that

(6.3) 0 ≤N−1∑

k=0

n−(bωk) ≤ N ·[dim(M)− n0(γ)

]+ 4dim(M)2 − 2 n0(γ) dim(M).

Proof. Arguing as in the proof of Corollary 4.8, one proves easily that dim(J(2)γ (z)

)≤

2 dim(M), and thatdim(Ker(bz)

)≥ n0(γ). This last inequality follows from the fact

that the spaceW solution of (4.4) : W (0) = W (1) = 0

is always contained in the

kernel ofbz. This proves thatn−(bz) ≤ 2 dim(M) − n0(γ). Whenz ∈ S1 is not inthe spectrum ofPγ , then it is shown in the proof of Corollary 4.8 thatdim

(J(2)γ (z)

)=

dim(M), which gives the improved inequalityn−(bz) ≤ dim(M) − n0(γ). Inequality(6.3) follows now easily, observing that there are at most2 dim(M) eigenvalues ofPγ (onthe unit circle).

Corollary 6.4. SetCγ = 4dim(M)2 − 2 n0(γ) dim(M) + n−(g); then:

sf(γ(N)

)≥

[− iMaslov(γ) + n0(γ)− dim(M)

]·N − Cγ ,(6.4)

sf(γ(N)

)≤ −iMaslov(γ) ·N − n−(g) + 2 dim(M),(6.5)

for all N ≥ 1.

Proof. Follows easily from (6.1), (6.2) and (6.3).

Inequality (6.4) becomes interesting wheniMaslov(γ) < n0(γ) − dim(M), while (6.5)when iMaslov(γ) > 0. Thus, the question is understanding the asymptotic behavior ofsf(γN ) when

(6.6) n0(γ)− dim(M) ≤ iMaslov(γ) ≤ 0;

note thatn0(γ)− dim(M) ≤ −1, and thatCγ is bounded uniformly onγ:

2 dim(M)2 + 2dim(M) + n−(g) ≤ Cγ ≤ 4 dim(M)2 + n−(g).

By (3.9), (6.6) is equivalent to:

sf(γ) ≥ dim(J perγ ∩ J 0

γ

)− iconc(γ)− n−(g)(6.7)

sf(γ) ≤ dim(J perγ ∩ J 0

γ

)− iconc(γ)− n−(g) + dim(M)− n0(γ).


Lemma 6.5. If for somek ≥ 1, |sf(γ(k))| > 2 dim(M) + n−(g), then the sequenceN 7→ |sf(γ(kN))| has superlinear growth.

Proof. The inequality|sf(γ(k))| > 2 dim(M)+n−(g) implies that (6.7) is not satisfied bythe iterateγ(k); this follows easily considering the trivial inequalities:

dim(J perγ(k) ∩ J 0

γ(k)

)≤ dim

(J 0γ(k)

)≤ dim(M)− 1

andiconc

(γ(k)

)≤ dim

(J ⋆γ(k)

)≤ dim

(Jγ(k)

)= 2dim(M).

By Corollary 6.4,N 7→ |sf(γ(kN))| has superlinear growth.

The result of Lemma 6.5 is not yet satisfactory; we want to prove that ifsf(γ(N)) is notbounded, then the entire sequenceN 7→ |sf(γ(N))| has superlinear growth. Let us studymore precisely the behavior ofsf

(γ(N)

)asN → ∞:

Proposition 6.6. The limit:

(6.8) Lγ = limN→∞

1N sf

(γ(N)

)

exists, and it is finite.

Proof. Using (6.1), it suffices to show that the limit:

limN→∞

1

N

N−1∑

k=0

n−(be2πik/N

)

exists and is finite. By Lemma 4.7, the mapS1 ∋ z 7→ n−(bz) ∈ N is constant on everyconnected component ofS1 that does not contain elements in the spectrum ofPγ ; thus,this function is Riemann integrable onS1, and:

(6.9) 0 ≤ limN→∞

1

N

N−1∑

k=0

n−(be2πik/N

)=

1

2π

∫

S1

n−(beiθ ) dθ < +∞.

Clearly, the following inequality holds:

−iMaslov(γ) + n0(γ)− dim(M) ≤ Lγ ≤ −iMaslov(γ);

moreover, in the second inequality, the equality holds if and only if bz is positive semi-definite at each point ofS1 that does not belong to the spectrum ofPγ . With this, we canfinally prove the following:

Proposition 6.7. The sequence|sf(γ(N)

)| is either bounded or it has superlinear growth.

Proof. The thesis is equivalent to proving thatLγ = 0 if and only if the sequencesf(γ(N)

)

is bounded. The “if” part is trivial. Now, assume by contradiction thatLγ = 0 and thatthe sequencesf

(γ(N)

)is unbounded. By Lemma 6.5, there existsk ≥ 1 such that the

subsequenceN 7→ |sf(γ(kN)

)| has superlinear growth, i.e.,lim

N→∞

1N sf

(γ(kN)

)= kLγ 6=

0, which is a contradiction.

Our goal will now be to prove the occurrence of auniformsuperlinear growth for thespectral flow of an iterate. To this aim, we need to study the sequence:

BN =N−1∑

k=0

n−(be2πik/N

),

which is in a sense, the non trivial part in formula (6.1). A substantial improvement to theresult of Lemma 6.3 can be obtained as follows.


Proposition 6.8. The limit

(6.10) Kγ = limN→∞

1

NBN

exists, and it is a nonnegative real number. This number is zero if and only if BN isbounded, which occurs if and only ifn−(bz) vanishes almost everywhere onS1. If BN isnot bounded, then its superlinear growth is uniform in the following sense: there exist aconstantα ∈ R, such that for allN,P ∈ N:

(6.11) KγP − α ≤ BN+P − BN ≤ KγP + α,

Proof. The existence of the limit has already been established in the proof of Proposi-tion 6.6. From the equality in (6.9) one obtains easily that the limit is zero if and onlyif n−(bz) vanishes almost everywhere onS1, and as it has always a finite number of dis-continuities (the eigenvalues ofPγ), one deduces that this occurs precisely whenBN isbounded. As to the last statement, assume thateiθ1 , . . . , eiθk are all the eigenvalues ofPγ

in S1 \ 1, with 0 < θ1 < . . . < θk < 2π; setθ0 = 0 andθk+1 = 2π. Forj = 0, . . . , k,definedj as

dj = limθ→0+

n−(bei(θj+θ)

)≥ 0;

recalling (a) in the proof of Corollary 4.8,dj is the constant value of the mapn−(bz) in thearcAj =

eiθ : θ ∈ ]θj , θj+1[

. With these notations, we have:

(6.12) Kγ =1

2π

k∑

j=0

dj(θj+1 − θj);

by Lemma 6.3:

dj ≤ 2 dim(M)− n0(γ).

By the first part of the proof, the assumption thatBN is unbounded is equivalent to the factthat at least one of the term in the sum (6.12) is positive (i.e., Kγ > 0). Finally, defineconstantsaN andCN,j, for N ≥ 1 andj ∈ 0, 1, . . . , k by:

aN = cardinality ofj : Nθj ≡ 0 mod 2π

,

CN,j =

⌊N(θj+1 − θj)

2π

⌋,

where⌊·⌋ denotes the integer part function:⌊x⌋ = maxm ∈ Z : m ≤ x. Clearly,0 ≤ aN ≤ k + 1; moreover, for allN andj, the arcAj contains a number ofN -th rootsof unity which is at mostCN,j +1 and at leastCN,j − 1. With this in mind, we proceed to


the final calculation giving the desired uniform superlinear growth, as follows:

BN+P − BN =

N+P−1∑

l=0

n−(be2πil/(N+P)

)−

N−1∑

l=0

n−(be2πil/N

)

=

N+P−1∑

l=1

n−(be2πil/(N+P)

)−

N−1∑

l=1

n−(be2πil/N

)

≥

k∑

j=0

dj(CN+P,j − 1)−

k∑

j=0

dj(CN,j + 1)− aN maxz∈S1

[n−(bz)

]

by Lemma 6.3≥

k∑

j=0

dj(CN+P,j − CN,j − 2)− (k + 1)[2 dim(M)− n0(γ)

]

≥

k∑

j=0

dj(CN+P,j − CN,j)− 2( k∑

j=0

dj

)− (k + 1)

[2 dim(M)− n0(γ)

]

≥k∑

j=0

dj(θj+1 − θj)

2πP − 4

( k∑

j=0

dj

)− (k + 1)

[2 dim(M)− n0(γ)

]

by (6.12)≥ Kγ P − 5(k + 1)

[2 dim(M)− n0(γ)

].

This concludes the proof of the first inequality in (6.11). The second inequality in (6.11) isobtained similarly:

BN+P − BN ≤k∑

j=0

dj(CN+P,j + 1)−k∑

j=0

dj(CN,j − 1) + aN maxz∈S1

[n−(bz)

]

by Lemma 6.3≤

k∑

j=0

dj(CN+P,j − CN,j + 2) + (k + 1)[2 dim(M)− n0(γ)

]

≤

k∑

j=0

dj(CN+P,j − CN,j) + 2( k∑

j=0

dj

)+ (k + 1)

[2 dim(M)− n0(γ)

]

≤

k∑

j=0

dj(θj+1 − θj)

2πP + 4

( k∑

j=0

dj

)+ (k + 1)

[2 dim(M)− n0(γ)

]

by (6.12)≤ Kγ P + 5(k + 1)

[2 dim(M)− n0(γ)

].

From (6.1), (6.8), and (6.10) one obtains immediately:

Lγ = −Kγ − iMaslov(γ);

moreover, we can finally prove the uniform superlinear growth of sf(γ(N)

):

Proposition 6.9. With the notations of Corollary 6.4 and Proposition 6.8, thefollowinginequalities hold:

(6.13) Lγ · P − 2 dim(M)− α ≤ sf(γ(N+P )

)− sf

(γ(N)

)≤ Lγ · P + 2dim(M) + α.

Proof. Immediate from Propositions 6.1, 6.8 and Lemma 6.2.

Corollary 6.10. The sequencesf(γ(N)

)is either bounded or it has uniform linear growth.

Proof. We have seen in the proof of Proposition 6.7 that the sequencesf(γ(N)

)is bounded

if and only if Lγ = 0, so that the thesis follows from Proposition 6.9.


Denote byΛM the free loop space4 of M , and byf : ΛM → R the geodesic actionfunctional of(M, g), whose critical points are well known to be closed geodesics. Thereis an equivariant action of the orthogonal groupO(2) onΛM , obtained from the naturalaction ofO(2) on the parameter spaceS1. A critical O(2)-orbit of f consists of all closedgeodesics that are obtained by rotation and inversion of a given closed geodesic in(M, g);it is immediate that all the closed geodesics in the same critical orbit have equal spectralflow. Using equivariant Morse theory applied to the geodesicaction functional, Gromolland Meyer have proved that, in the Riemannian case, the contribution to the homology ofthe free loop spaceΛM in a fixed dimensionk is given only by those closed orbits whoseMorse index is an integer betweenk − dim(M) andk. A key point of their multiplic-ity result is that, assuming the existence of only a finite number of distinct closed primegeodesics, one has a uniformly bounded number of distinct orbits with a fixed Morse index([15, Corollary 2]). Aiming at the development of an equivariant Morse theory for stronglyindefinite functionals, we prove an extension of their result, replacing the Morse index withthe spectral flow.

Proposition 6.11.Let(M, g) be a semi-Riemannian manifold that has only a finite numberof distinct prime closed geodesics. Then, fork ∈ Z with |k| sufficiently large, the totalnumber of critical orbits of the geodesic action functionalf in the free loop spaceΛMhaving spectral flow equal tok is bounded uniformly ink.

Proof. Let γ1,. . . ,γr be the family of all distinct prime closed geodesics inM and fixsome integerk with |k| > 2 dim(M) + n−(g) (recall Lemma 6.5). We can remove fromthe family those geodesics whose spectral flow is not unbounded by iteration, and assumethat all these geodesics have iterates with unbounded spectral flow; in particular,Lγi 6= 0

for all i. For i = 1, . . . , r, let Ni ≥ 1 be the first integer such thatsf(γ(Ni)i

)= k; if

no such integerNi exists, we can remove alsoγi from the family. From (6.13) we obtaineasily that|sf

(γ(Ni+P )

)− k| > 1 when

P >1 + α+ 2dim(M)

|Lγi|.

Thus, there are at most:

r +

r∑

i=1

⌊1 + α+ 2dim(M)

|Lγi |

⌋

critical orbits off with spectral flow equal tok.

Finally, let us recall thatγ is said to behyperbolicif the linearized Poincare mapPγ

does not have eigenvalues on the unit circle. For hyperbolicgeodesics, the iteration formulafor the spectral flow has a simple expression:

Proposition 6.12. If γ is hyperbolic, then:

sf(γ(N)

)= Nsf(γ) + (N − 1) n−(g).

Proof. This follows easily from Corollary 4.8 and Theorem 5.3.

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DEPARTAMENTO DEMATEMATICA ,UNIVERSIDADE DE SAO PAULO ,RUA DO MATAO 1010,CEP 05508-900, SAO PAULO , SP, BRAZIL

E-mail address: [email protected]

DEPARTAMENTO DEMATEMATICA ,UNIVERSIDADE DE SAO PAULO ,RUA DO MATAO 1010,CEP 05508-900, SAO PAULO , SP, BRAZIL

E-mail address: [email protected]: http://www.ime.usp.br/˜piccione



http://arxiv.org/math.DG/0705.0589

BSTRACT arXiv:0711.0635v1 [math.DG] 5 Nov 2007 · 2020. 1. 23. · arXiv:0711.0635v1 [math.DG] 5 Nov 2007 SPECTRAL FLOW AND ITERATION OF CLOSED SEMI-RIEMANNIAN GEODESICS MIGUEL ANGEL

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