arXiv:1903.10147v2 [math.DG] 6 Sep 2019 (1.1) energ The [10]. Hilbert a α of structure the admits It oissatn on ihtesm agn direction. tangent same the with geodes point closed starting A its a geodesics. to along to moves meaning manifol physical object Riemannian a re falling gives a which curve freely in a a relativity, points is general two Geodesic of between path spaces. shortest Euclidean the on line" "straight a hma rtclpit feeg ucinlo h reloop free the on functional energy of Λ( points C critical as geodesics. Hamiltonian them orient closed in closed orbits shortest a periodic or of studying longest volume in the the roles m of on important a lengths bounds of the place geometry of to and possible is on it constraints powerful put they γ ytkn trtsof iterates taking by ne h iceato ntefe opsaedfie ytrans by defined space loop free the on action circle the under orsod to corresponds ∈ h oino edsco imninmnflsi generali a is Riemannian on geodesic of notion The nodrt netgt h xsec fcoe edsc,g geodesics, closed of existence the investigate to order In geode closed non-constant perspective, topological a From h rtclpit fteeeg ucinlaecoe geod closed are functional energy the of points critical The ahcoe geodesic closed Each M Λ ) o simply (or r endrsetvl by respectively defined are fteeitneo lsdgeodesics. ne closed a of gives existence result poss this the fastest) with l of along on topology (resp. string slowest modeled how with discuss is geodesic homology closed level isolated on product) Goresky-Hingston Abstract. H LSDGOEI RBE N H STRING THE AND PROBLEM GEODESIC CLOSED THE E ( α = ) Λ S nti ae,w hwta h hsSlia rdc (respe product Chas-Sullivan the that show we paper, this In facmatReana manifold Riemannian compact a of ) 1 Z aiyo lsdgoeis aldthe called geodesics, closed of family 0 γ S 1 , 1 g = Λ γ γ ( × m α sas soitdwt nifiiefml fcoe geodesic closed of family infinite an with associated also is ′ Λ ( for t { ) → α α , m ∈ Λ ′ ( , 1. = W t )) ( e PRODUCTS 1 ± dt Introduction 2 RNMAITI ARUN , πit 2 1 ([0 , γ , and ± , ) 2 1] 1 , → M , · · · L ,β β, ( o h iceato n trtso a of iterates and action circle the So, . ) α | α = ) ( 0 = (0) s = ) Z 0 M 1 beidxgot ae We rate. growth index ible γ α p esetv nquestions on perspective w edsci uvdspacetime, curved in geodesic ( orbit (1) fdimension of s g dynamics. sc.Ec lsdgeodesics closed Each esics. ci edscta returns that geodesic a is ic oeesotnlk oview to like often eometers n egho nelement an of length and y pc.Tefe opspace loop free The space. isaeipratbecause important are sics + ( } α oe edsc loplay also geodesics losed rsnigi oesense some in presenting clgoer fan of geometry ocal lation . t ′ ( ) .I isenstheory Einstein’s In d. bemnfl nterms in manifold able of aino h oinof notion the of sation t . nfl.Frinstance, For anifold. ) α , γ n eoe by denoted and ′ ( t tvl the ctively )) n dt. sdfie by defined is γ ¯ s, , 2 ARUN MAITI geodesic produces geometrically the same object, meaning, their images are the same subset of M. The problem of showing the existence of infinitely many geometrically distinct closed geodesics on any Riemannian manifold, is also known as the closed geodesic problem. Here closed geodesics are always understood to be non-constant. In §2, we review some of the important results on this problem. The main approach to resolve this problem has been to use topological complexity of the free loop space to force the existence of critical points of the energy functional. Topological complexity of a space often reflects on its homology groups(for instance, on the growth of Betti numbers) and algebraic structures on them. String products are part of a bunch of algebraic operations on homology of the free loop space, introduced by M. Chas and D. Sullivan in [4, 15]. By string products we shall mean the Chas-Sullivan product and the Goresky-Hingston product on homology and cohomology of the free loop space respectively. ≤a

Λ≤a = α Λ F (α) a and Λ

2. Known results The existence of at least one closed geodesic on any compact manifold was shown by Lyusternik and Fet [11], extending the works of Birkhoff [1]. If the fundamental group is non-trivial then one merely needs to look at the closed curve representations with minimal length. For simply connected space one uses Birkhoff’s minimax argument. The existence of infinitely many geometrically distinct closed geodesics on surfaces is shown using a similar minimax argument as of Birkhoff’s. Rademacher has shown that a simply connected compact manifold with a generic Riemannian metric admits infinitely many geometrically distinct closed geodesics [14]. The main obstacle in finding geometrically distinct closed geodesics is that the iterates of closed geodesics may also contribute to the homology of the free loop space. The contributions are often controlled by their index and nullity. The following theorem due to R. Bott [2], gives us estimates of indices of iterates. Theorem 2.1. Let γ be a closed geodesic of a n-dimensional compact manifold M and m let λm and νm are the index and the nullity of the m-fold iterate γ respectively. Then ν 2(n 1) for all m and m ≤ − λm mλ1 (m 1)(n 1), (2.1) | − | ≤ − − λ + ν m(λ + ν ) (m 1)(n 1), | m m − 1 1 | ≤ − − Furthermore, if γ and γ2 are nondegenerate, then for any k 0, there exists a constant C such that for all m C , ≥ k ≥ k

λm mλ (m 1)(n 1) k, (2.2) | − | ≤ − − − λ + ν m(λ + ν) (m 1)(n 1) k. | m m − ≤ − − − The contributions of a geodesic to local level homologies are better understood than homology of the whole loop space. We recall below some of the well known results on this. Let γ be a closed geodesic and Γ− γ¯ be the negative bundle whose fibers are the negative eigenspaces of Hessians of F .→ We then have the following from [13]. Theorem 2.2. Let γ be nondegenerate closed geodesic on a compact manifold M of − index λ. Let G = Z when the negative bundle Γ γ¯ is oriented, and G = Z2 otherwise. Then the local level homology groups are given→ by G for i = λ,λ +1 H (Λ

A consequence of the above theorem, is the following celebrated theorem of Gromoll and Meyer [6]. Theorem 2.4. Let M be a compact simply connected Riemannian manifold such that the sequence of Betti numbers βi(Λ(M)) is unbounded (with some field of coefficients). Then M admits infinitely many non constant, geometrically distinct closed geodesics. In [16], Vigué-Poirier and Sullivan showed that the assumption of the above theorem is satisfied when the manifold is sufficiently complicated, more precisely: Theorem 2.5. For a simply connected compact manifold M, then the cohomology al- gebra of M with rational coefficients has at least two generators if and only if the Betti numbers of Λ(M) with rational coefficients are unbounded. This is in particular true for globally symmetric spaces of rank > 1 [17].

Among the spaces which do not satisfy the assumptions of the Gromoll-Meyer the- orem, are the symmetric spaces of rank 1, which includes our most familiar spaces n-spheres. Below we state two theorems of N. Hingston [7, 8], which can be applied to show that any Riemannian 2-sphere admits infinitely many closed geodesics. Theorem 2.6. Let γ be a closed geodesic on an n-dimensional compact manifold M with F (γ) = a. With the notations as above, assume that H (Λ

3. Applications of string products The string products, first defined in [4, 15], brought a new dimension to the study of homology of the free loop space. We briefly recall the definitions. ,Let A ֒ B be a co-dimension m embedding into a finite dimensional manifold B then the induced→ Gysin map is the composition

∩µ H (B) H (B, B A) H − (A), p → p − −→ p m where µ is the Thom class of a normal bundle V A, V a tubular neighbourhood of A in B. → For t (0, 1), let Θ = α Λ α(0) = α(t) , cut : Θ Λ Λ cuts a loop at time t, ∈ t { ∈ | } t t → × THE CLOSED GEODESIC PROBLEM AND THE STRING PRODUCTS 5 and Θ= (α, β) Λ Λ α(0) = β(0) , concat :Θ Λ denotes the concatenation. In [15], the authors{ ∈ showed× that| the Gysin} map can also→ be defined for the embeddings .Θ ֒ Λ codim n and Θ ֒ Λ Λ codim n t → → × Then the maps concat ,Λ Λ ֓ Θ Λ × ←− −−−→ induces the Chas-Sullivan product, denoted by , •

EZ Gysin map concat∗ (3.1) H (Λ) H (Λ) H (Λ Λ) H − (Θ) H − (Λ), i × j −−→ i+j × −−−−−−→ i+j n −−−−→ i+j n where EZ is the Einlenberg-Zilber map. We also have the maps

codim n cutt ,(Λ ֓ Θ Λ Λ for t (0, 1 ←−−−− t −−→ × ∈ or equivalently, the maps

cut 1 Γt codim n 2 ,[Λ Λ ֓ Θ 1 Λ Λ, for t [0, 1 (3.2) −→ ←−−−− 2 −−→ × ∈ where Γt :Λ Λ is a reparametrization such that cut 1 Γt = cutt on Θt ( see [5] for → 2 ◦ example of such a Γt), which induce

# (cut 1 ) # i 2 i Gysin map i+n Γt i+n (3.3) Pt : C (Λ Λ) C (Θ 1 ) C (Λ) C (Λ). × −−−−→ 2 −−−−−−→ −−→ Pt, t [0, 1] defines a homotopy between P0 and P1 i.e. P δ + δP = P0 P1. P0, P1 ∈ ∗ − vanishes on C (Λ, Λ0), cochains relative to constant loops. Thus, precomposing with cross product, P induces a product on cohomology, denoted by ⊛,

⊛ Hi(Λ, Λ ) Hj(Λ, Λ ) Hi+j+n−1(Λ, Λ ), 0 × 0 −→ 0 known as the Goresky-Hingston product. Let Fp be the finite field with p elements where p> 0 is a prime integer. In [9], using the Chas-Sullivan product, Jones and McCleary extended Theorem 2.5 for coefficients Fp, and consequently, proved the following. Theorem 3.1. Let M be a simply connected compact manifold. If the cohomology ∗ algebra H (M, Fp) cannot be generated by one element, then any Riemannian metric on M has infinite many geometrically distinct closed geodesics. In [5], the authors showed that the C-S product extends (compatible with the product on homology of Λ) naturally to a product on level homologies as follows:

≤a

H∗(Λ

•m <(m+1)a m <(m)a x H − − (Λ γ , Λ ) ∈ mj n(m 1) ∪ is a non-trivial class for all m 1. Denote by λm = index(γ), νm = nullity(γ). It follows then from the Gromoll and≥ Meyer theorem 2.3 that

(3.8) λ mj n(m 1) λ + ν +1 for all m 1, m ≤ − − ≤ m m ≥ and in particular, (3.9) λ j λ + ν +1. 1 ≤ ≤ 1 1 Since ν 2n 1 for all m, it follows that the average index of γ m ≤ − λ (3.10) λ¯ = lim m = j n. m→∞ m − On the other hand, we have the following from Bott’s index estimate 2.1 m(λ + ν ) (m 1)(n 1) λ + ν for all m 1, 1 1 − − − ≤ m m ≥ which implies (3.11) λ + ν n +1 λ.¯ 1 1 − ≤ Combining the above equations 3.9, 3.10 and 3.11, we get

j = λ1 + ν1 +1. THE CLOSED GEODESIC PROBLEM AND THE STRING PRODUCTS 7

So Equation 3.9 can be rewritten as (3.12) λ m(λ + ν ) (m 1)(n 1)+1 λ + ν +1. m ≤ 1 1 − − − ≤ m m Now we claim that λ + ν = m(λ + ν ) (m 1)(n 1) for all m 1. m m 1 1 − − − ≥ Suppose our claim is false, then it follows again from 2.1 that for some positive integers k and C we have λ + ν = k(λ + ν ) (k 1)(n 1) + C. k k 1 1 − − − Then for any positive integer r, we have

λ + ν r(k(λ + ν ) (k 1)(n 1) + C) (r 1)(n 1) rk rk ≥ 1 1 − − − − − − = λ rk(λ + ν ) (rk 1)(n 1) + rC ν . ⇒ rk ≥ 1 1 − − − − rk This contradicts the first inequality of Equation 3.12 for large r. This proves our claim, hence completes the proof of the theorem.  Proof of Theorem 1.2. Let x Hj(Λ

x⊛m Hmj+(m−1)(n−1)(Λ<(m+1)a γm, Λ<(m)a) ∈ ∪ is a non-trivial class for all m 1. It follows then from the theorem of Gromoll and Meyer 2.3 that ≥ (3.13) λ mj +(m 1)(n 1) λ + ν +1 for all m 1, m ≤ − − ≤ m m ≥ Using a very similar argument as in the last theorem, we get j = λ1 = index(γ), and we can also prove that γ has maximal index growth, i.e., λ = mλ +(m 1)(n 1) for all m 1. m 1 − − ≥  The following corollary is essentially a restatement of Theorems 12.6 and 12.7 in [5] put together. Corollary 3.3. Let M be a compact Riemannian manifold with F :Λ R defined as above. If there is a non-nilpotent level homology class with respect to the→ C-S product then M has infinitely many closed geodesics. An analogous statement holds true for cohomology with the G-H product. ≤a

The following corollary is essentially a restatement of Theorems 7.3 and 11.3 in [5] put together. Corollary 3.4. Let M be a compact manifold. Then for a generic Riemannian matric on M, with F :Λ R defined as above, for every a 0, every local level homology class

4. Acknowledgement The author would like to thank H. B. Rademacher and N. Hingston, for some valuable suggestions. The author was supported by the National Board of Higher Mathemat- ics(No. 2018/R&D-II/ 8872) during the academic year 2018–2019.

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