arXiv:1707.09618v2 [math.DG] 27 Oct 2017 oiiescinlcurvature. sectional positive hsi ieetal ucin h rtclpit r cl are points critical the function, differentiable X a is This (1) M ae at based osdrteeeg functional energy the consider { atclrfor particular ( relo space loop free sthe is fptsbetween paths of h23.Nww use we Now ch.2.3]. E RTCLVLE FHMLG LSE FLOOPS OF CLASSES HOMOLOGY OF VALUES CRITICAL H γ ( e od n phrases. and words Key 2010 Date o opc imninmanifold Riemannian compact a For o rtclpoint critical a For ; 1 Λ = ∈ γ pts on -paths) γ = ) X boueycontinuous absolutely length 2017-10-27. : ahmtc ujc Classification. Subject Mathematics M ifolds Abstract. olw rmrslsb uiooi vndmnin n Ball and dimensions even dimensions. odd in in Sugimoto Ziller & by Thorbergsson results from follows oooycaso oetpstv ieso ntesaeo l of space the point in a dimension positive at lowest of class homology length edsclo ihbs point base with loop geodesic a yLsenk&Fti h eea ae ti h anrsl of result main the is It case. a numbers general the of the that case existenc in in the Fet Birkhoff & of by Lusternik proof presented by the length of positive idea of the geodesic is This geodesic. closed nyfrterudmti nthe on metric round the for only ; a p E } ,goeisjoining geodesics ), ne h diinlassumption additional the Under ( (if . γ ( ) of ,g M, crl p X ≤ Λ = Λ M. p = c. ( Ω = a p ) ,g M, p q } esuycmatadsml-once imninman- Riemannian simply-connected and compact study We euetefloigntto o ulvlsets: sublevel for notation following the use We ihpstv etoa curvature sectional positive with N OIIECURVATURE POSITIVE AND ∈ ecnie ntesqe h olwn usae:the subspaces: following the sequel the in consider We n ban the obtains one and resp. p c M ) M X M crl ∈ resp. E = q ). edsclos opsae relo pc,Mretheory, Morse space, loop free space, loop loops, geodesic ri h relo pc n a en critical a define can one space loop free the in or ASBR RADEMACHER HANS-BERT p odnt n ftespaces the of one denote to X ( ( Ω : ,g M, { X γ ehave we crl γ = ) 1. ,

The critical value (or critical level) cr(h) of a relative homology class h ≤b ∈ Hk X, X , Zs is defined as follows: (2)  ≤a ≤b ≤b cr(h) = inf a b ; h Image Hk(X , X ; Zs) Hk(X, X ; Zs) , n ≥ ∈  −→ o cf. [15, ch.2.1] or [11, Sec.3]. Then h H X, X≤b; Z is nontrivial if and ∈ k s only if cr(h) > b. For a non-trivial class h Hk (X; Zs) there is a critical ∈ point c X with E(c)= cr(h) resp. L(c)= 2E(c). ∈ For a compact and simply-connected manifoldp M = M n of dimension n let k = k(M) 1, 2,...,n 1 be the unique number, such that M is k- connected, but∈ not { (k+1)-connected.− } Hence we have the following statement about the homotopy groups: πj(M) = 0 for 1 j k and πk+1(M) = 0 . Then for X =Ω M resp. X = ΛM there is a smallest≤ ≤ prime number s6 P pq ∈ such that H X, X0; Z = 0. This follows since π (Ω M) = π (M) resp. k s 6 j p ∼ j+1 πj (ΛM) = πj+1 (M) πj (M) , cf. [15, Cor. 2.1.5]. We define the critical ∼ ⊕ length crl (M, g) , resp. the critical length crlp (M, g) at p as follows: (3) crl (M, g) := max 2cr(h) ; h H ΛM, Λ0M; Z ∈ k(M) s np o and crl cr Z (4) p (M, g) := max 2 (h) ; h Hk(M) (ΩpM; s) . np ∈ o If the manifold M is simply-connected and homeomorphic to a sphere Sn 0 then there is a uniquely determined generator h H − ΛM, Λ M; Z = ∈ n 1 2 ∼ Z2, resp. hp Hn−1 (ΩpM; Z2) = Z2 such that crl (M, g) = cr(h), resp. ∈ ∼ crlp (M, g) = cr(hp). For a metric of positive sectional curvature we obtain the following result for the critical length resp. the critical length at a point. In particular we show that the maximal possible value is only attained for the round metric on a sphere of constant sectional curvature 1 : Theorem 1. Let (M n, g) be a compact and simply–connected Riemannian manifold of positive sectional curvature K 1. Then the critical lengths crl (M, g) and crl (M, g) for p M satisfy: ≥ p ∈ (a) For all p M : crl (M, g) 2π and crl (M, g) 2π. ∈ p ≤ ≤ (b) If for some p M : crlp (M, g) > π or if crl (M, g) > π then the manifold is homeomorphic∈ to the n-dimensional sphere Sn.

(b) If crlp (M, g) = 2π for some p M, or if crl (M, g) = 2π then (M, g) is isometric to the round sphere of constant∈ sectional curvature 1. Remark 1. (a) Note that the conclusions of the Theorem remain true if we replace crl (M, g) by 2 diam(M, g), where diam(M, g) is the diameter of the Riemannian manifold. Then Part (a) is Bonnet’s theorem, cf. [7, Thm.1.31], Part (b) is the generalized sphere theorem [9] by Grove & Shiohama and Part (c) is Toponogov’s maximal diameter theorem [7, Thm. 6.5], resp. [20]. But one cannot reduce the proof of the Theorem to the statements for the di- ameter as examples constructed by Balacheff, Croke & Katz [1] show: There is a one-parameter family g ,t 0 of smooth Zoll metrics on the 2-sphere t ≥ such that the following holds: The metric g0 is the round metric of constant 2 curvature 1 and for t > 0 the length (S , gt) of a shortest closed geodesic satisfies (S2, g ) > 2 diam(S2, g ). L L t t CRITICALVALUESANDPOSITIVECURVATURE 3

(b) Since crl (M, g) > 0 one obtains the existence of a non-trivial closed geodesic of length crl (M, g) , resp. a geodesic loop of length crlp (M, g) . This idea goes back to Birkhoff [5] in the case of a sphere and to Lyusternik-Fet [17] in the general case. Therefore we call crl (M, g) also the Birkhoff-length of the Riemannian manifold. 0 (c) Since for the number k = k(M) we have πk ΛM, Λ M ∼= πk (ΩpM) ∼= πk+1(M) and since for any p M the based loop space ΩpM can be seen as a subspace of the space∈ ΛM we obtain for all p M : ∈ (5) crl (M, g) crl (M, g) . ≤ p 2 In Example 1 we construct a one-parameter family (S , gr), 0 < r 1 2 2 ≤ of convex surfaces of revolution with crlp(S , gr) π for all p S , but 2 ≥ ∈ limr→0 crl(S , gr) = 0. For a compact Riemannian manifold (M, g) we denote by = (M, g) the length of a shortest (non-trivial) closed geodesic. For a pointL Lp M we denote by (p) the length of a shortest (non-trivial) geodesic loop∈ with initial point p.L Then we have the obvious estimates: crl (M, g) and (p) crl (M, g) . In particular we obtain as L ≤ L ≤ p Corollary 1. Let (M n, g) be a simply–connected Riemannian manifold of positive sectional curvature K 1 and p M. (a) 2π, and (p) 2π for≥ all p ∈M. (b)L≤ If = 2π orL if ≤(p) = 2π for∈ some p M then the manifold is isometricL to the sphere ofL constant curvature 1. ∈ Toponogov proves in [21] that a simple closed geodesic (i.e. a closed ge- odesic with no self-intersection) on a convex surface with K 1 has length 2π with equality if and only if the sectional curvature is≥ constant. He also≤ shows the analogous statement for a geodesic loop. Statement (b) for is claimed in several preprints (unpublished) by Itokawa & Kobayashi [12],L [13], [14]. In case of quaterly pinched metrics there are stronger results about a gap in the length spectrum obtained using the Toponogov triangle comparison results: Theorem 2. Let M be a manifold homeomorphic to Sn carrying a Rie- mannian metric g with sectional curvature 1 K ∆ 4. (a) (Ballmann [2, Teil III]) The critical length≤ ≤crl (M,≤ g) and the length of a shortest closed geodesic coincide and satisfy: 2π/√∆ crl (M, g) = L 2π. ≤ L≤(b) (Ballmann,Thorbersson & Ziller [4, Thm.1.7]) If there is a closed ge- odesic c with length 2π L(c) 4π/√∆ then the metric has constant sectional curvature. ≤ ≤ Remark 2. (a) The case n = 2 in Part (a) is also contained in [4, Thm. 4.2]. For even dimension Part (b) was shown by Sugimoto [19, Thm. B,C]. compare also the Remark following [4, Thm.1.7]. (b) In [2, Teil III] the following stronger statement is shown: For any shortest closed geodesic c there is a homotopically non-trivial map 2 2 G : Dn−1, ∂Dn−1 c Λ

(c) In [3] the existence of g(n) geometrically distinct closed geodesics on Sn is shown under the assumptions of Theorem 2. The lengths of these closed geodesics lie in the interval 2π/√∆, 2π and g(n) is the cup-length n+1 h i n+1 of the Grassmannian G2(R ) of unoriented 2-planes in R . Instead of an assumption about the sectional curvature, which is an assumption depending continously on the metric and its first and second derivatives one can also investigate assumptions depending continuously only on the metric. For example Ballmann, Thorbergsson & Ziller consider in [3, Sec.3] the so-called Morse-condition. Hingston shows in [10, Sec.5] a similar result about the existence of g(n) closed geodesics. These results motivate Proposition 2, n where we consider Riemannian metrics g on S satisfying g g1. Here g1 is the round metric of constant curvature 1. ≤

2. Rauch comparison result Let c : R M be a geodesic parametrized by arc length, then c(t ) is −→ 1 called focal point of p = c(0) (along c [0,t1]) if there is a non-trivial normal Jacobi field Y = Y (t) along c with | Y (t), c′(t) = 0 for all t and vanish- h i ing covariant derivative Y (0)/dt = 0 (along the ) and Y (t1) = 0, cf. [16, Def. 1.12.20]. The∇focal radius ρ = ρ(M, g) of a compact Riemann- ian manifold (M, g) equals the minimal positive number t1 with the above property. The following statement is also called second Rauch comparison theorem (Rauch II), cf. [7, Thm.1.34], which is due to Berger. But note that we also include a rigidity statement, see for example [8, Thm.3.3]. Proposition 1. Let c : [0, a] M be a geodesic parametrized by arc length on a Riemannian manifold−→ with positive sectional curvature K 1 ≥ and let c(t1) be the first focal point. Let Y = Y (t) be a Jacobi field along c = c(t) with covariant derivative along c: Y ′(0) = Y (0)/dt = 0. Then Y (t) Y (0) cos(2πt) for all 0 t t . If Y (t )∇ = Y (0) cos(2πt ) k k ≤ k k ≤ ≤ 1 k 0 k k k 0 for some t0 (0,t1], then Y (t) = Y (0) cos(2πt) for all t [0,t0] and Y/ Y is parallel∈ along c [0k,t ] .k k k ∈ k k | 0 Using this Rauch comparison result one obtains the following consequence. This result can be found for example in [7, Cor.1.36] or in [16, Cor. 2.7.10]. But note that we also include a rigidity statement. We denote by exp : T M M the exponential map, i.e. for v T M the curve t R −→ ∈ p ∈ 7−→ exp(tv) equals the geodesic γv = γv(t) with initial point p = γv(0) and direction v = γ′ (0). And exp : T M M denotes the restriction to the v p p −→ tangent space TpM at p. Corollary 2. Let c : [0, a] M be a geodesic on a compact Riemannian manifold of positive sectional−→ curvature K 1 and focal radius ρ > 0. Let n ≥ n c1 : [0, a] S1 be a great circle of length L(c) on the standard sphere S1 of constant−→ sectional curvature 1. Choose parallel unit vector fields E along c resp. E1 along c1 which are orthogonal to the geodesic c resp. c1. For a smooth non-negative function f : [0, a] R with 0 < f(t) ρ for all −→ ≤ t (0, a) we define the smooth curve b(t) = expc(t) (f(t)E(t)) M,t [0, a] ∈ n ∈ ∈ on M resp. the curve b1(t) = expc1(t) (f(t)E1(t)) S1 ,t [0, a] on the n ∈ ∈ round sphere S1 of constant sectional curvature 1. CRITICALVALUESANDPOSITIVECURVATURE 5

Then the following statements hold:

(a) L(b) L(b1). ≤ ∗ (b) Assume L(b) = L(b1). Let f : [0, a] R be a non-negative func- tion with 0 < f ∗(t) f(t) for all t (0, a)−→and define the curve b∗(t) = ∗ ≤ ∈ ∗ expc(t) (f (t)E(t)) M,t [0, a] on M resp. the following curve b1(t) = ∗ ∈ n ∈ n expc1(t) (f (t)E1(t)) S1 ,t [0, a] on the round sphere S1 . ∗ ∗∈ ∈ Then L(b )= L(b1). Proof. The proof of (a) is given in [7, Cor. 1.36] or [16, Cor. 2.7.10]. Assume L(b) = L(b1). Let for t [0, a] : s [0,f(t)] γt(s) = ∈ ∈ ′ 7−→ expc(t)(sE(t)) denote the geodesic with γt(0) = c(t) and γt(0) = E(t) and let s [0,f(t)] Wt(s) be the parallel unit vector field along γt with Wt(0) = c′(∈t) for all t7−→[0, a]. We denote by Y (s)= ∂γ (s)/∂s the Jacobi field defined ∈ t t by the geodesic variation t γt, which satisfies Yt(f(t)) = cos(2πf(t)). It follows from the Proof of part7−→ (a) that Y (s) =k cos(2πs) kfor all s [0,f(t)]. k t k ∈ Then we conclude from Proposition 1 that Wt(s) = Yt(s)/ cos(2πs) is the parallel unit vector field along c : [0,f(t)] M with W (0) = c′(t). Then t −→ t U := exp (sE (t)) Sn ; 0 s f(t) is a totally geodesic surface in c1(t) 1 ∈ 1 ≤ ≤ the roundn sphere. Hence the map o exp (sE (t)) U Sn exp (sE(t)) M c1(t) 1 ∈ ⊂ 1 7−→ c(t) ∈ defines a totally geodesic isometric immersion of the totally geodesic sur- n face (U, g1) in S1 into the Riemannian manifold (M, g). Therefore the claim follows. 

3. Morse Theory on spaces The energy functional E :Ω M R is differentiable and its critical pq −→ points are geodesics c ΩpqM. The maximal dimension of a subspace of the ∈ 2 tangent space TcΩpqM on which the hessian d E(c) is negative definite, is called index ind0(c). If p = q and if c is a closed geodesic, then its index ind(c) as critical point of E : ΛM R is analogously defined. In general −→ the invariants ind0(c) and ind(c) differ, the difference 0 conc(c) := ind(c) ind (c) n 1 is called concavity, cf. [16, Thm.2.5.14].≤ The nullity null (c−) 0 ≤ − 0 of a geodesic c ΩpqM is defined as the dimension of the kernel of the hessian. It coincides∈ with the dimension of Jacobi fields Y = Y (t) along c = c(t),t [0, 1] which vanish at the end points, i.e. Y (0) = Y (1) = 0. It follows that∈ 0 null (c) n 1. If c is a closed geodesic c Ω M ΛM ≤ 0 ≤ − ∈ p ⊂ then the nullity null(c) as critical point of the S1-invariant energy functional E : ΛM R equals the dimension of periodic and orthogonal Jacobi fields. Hence 0 −→null(c) 2n 2. The energy≤ functional≤ −E :Ω M R is a Morse function, if all geodesics pq −→ c ΩpqM are non-degenerate, i.e. null0(c) = 0. This is the case if and only if ∈q M is a regular point of the exponential mapping exp : T M M, ∈ p p −→ cf. [18, §18]. The Morse lemma implies that around any geodesic c there 2 2 are coordinates (x,y) V− V with E(x,y) = x + y . Here V− ∈ ⊕ + −k k k k is a finite-dimensional subspace of dimension ind0(c) on which the Hessian 2 d E(c) is negative definite and V+ is a closed complement in TcΩpqM. Let the energy functional E :Ω M R be a Morse function, l N and let a pq −→ ∈ 6 HANS-BERTRADEMACHER be a critical value such that for some ǫ > 0 the value a is the only critical value in the interval (a 2ǫ, a + 2ǫ). Then the following holds: There are − geodesics c1,...,cr ΩpqM with E(c1) = . . . = E(cr) = a and ind0(c1) = . . . = ind (c )= l that∈ for all δ (0,ǫ] : 0 r ∈ r (6) H Ω≤a+δM, Ω≤a−δM; Z Z [F ] . l pq pq 2 ∼= 2 ck   Mk=1 The relative homology class [F ] H Ω≤a+δM, Ω≤a−δM; Z can be rep- ck ∈ l pq pq 2 resented by the set (x, 0); x V− in a neigborhood of c in which Morse { ∈ } k coordinates (x,y) V− V exist. The Morse Lemma implies that there is a ∈ ⊕ + sufficiently small neighborhood U(ck) ΩpqM of ck such that the homology ⊂ ≤a−δ class [Fck ] is the unique generator of the critical group Hl Ωpq M U(ck), ≤a−δ Z Z ∪ Ωpq M; 2 ∼= 2. In the sequel we use an approximation argument. If E :ΩpM R −→ is not a Morse function, we choose a sequence pj of regular points for the exponential map expp converging to p. This is possible since the regular points of the exponential map are dense, cf. [18, Cor. 18.2].

4. Critical values of homology classes of loops Let inj = inj(M, g) > 0 be the injectivity radius of the Riemannian man- ifold (M, g). Then between two points p,q of distance d(p,q) < inj there is a unique minimal geodesic c : [0, 1] M with p = c(0),q = c(1) and L(c)= d(p,q). Here d(p,q) is the distance−→of the points p and q. Lemma 1. Let p, q, r M with d(q, r) min inj/2, 1/2 and let c : ∈ ≤ { } rq [0, d] M be the unique minimal geodesic joining r = crq(0) = r, q = crq(1) with L−→(c )= d = d(q, r). Let ζ :Ω M Ω M be defined by rq rq pq −→ pr σ (t/(1 d(q, r)) ; t [0, 1 d(q, r)] ; − ∈ − ζrq(σ)(t) :=  .  c (t (1 d(q, r))) ; t [1 d(q, r), 1] rq − − ∈ − Z Z Z Let h Hj (ΩpqM; s) and let (ζrq)∗ : Hj (ΩpqM; s) Hj (ΩprM; s) be the induced∈ homomorphism. Then we obtain: −→ 1 cr(h) 1 (7) (1 d(q, r))cr(h) d(q, r) cr (ζ ) (h) + d(q, r). − − 2 ≤ rq ∗ ≤ 1 d(q, r) 2  − In particular the function r M cr (ζ ) (h) R+ is continuous at q. ∈ 7−→ rq ∗ ∈  Proof. For σ : [0, 1] M, with p = σ(0),q = σ(1) and r M with d(q, r) inj we obtain−→ ∈ ≤ E (σ) 1 (8) E (ζ (σ)) = + d(q, r) . rq 1 d(q, r) 2 − We conclude: cr(h) d(q, r) cr (ζ ) (h) + . qr ∗ ≤ 1 d(q, r) 2  − Applying this inequality to (ζqr)∗ (ζrq)∗ (h) and using the fact that  cr(h)= cr (ζ ) (ζ ) (h) qr ∗ ◦ rq ∗  CRITICALVALUESANDPOSITIVECURVATURE 7 yields also the left inequality. Here we use that ζqr ζrq :ΩpM ΩpM is homotopic to the identity map. ◦ −→  Lemma 2. Let (M, g) be a compact Riemannian manifold of positive sec- tional curvature K 1, and p M : Let hp Hk (ΩpM; Zs) be a non-zero homology class. ≥ ∈ ∈ (a) If k = n 1 then 2cr(h ) 2π. − p ≤ (b) If k 1, 2,...,n p2 then 2cr(h ) π. ∈ { − } p ≤ p Proof. Choose a sequence pj of regular points of expp with limj→∞ d(p,pj)= R 0, cf. for example [18, Cor. 18.2]. Then E :Ωppj M is a Morse function for any j 1. Morse theory implies that there are−→ finitely many geodesics c ,...,c ≥ Ω (M) such that there exists one geodesic c with j,1 j,r(j) ∈ ppj j,1 cr 2 ζpj p ∗ (hp) = L (cj,1) = max L (cj,1) ,...,L cj,r(j) and ind0(cj)= kq , cf. Section 3.    (a) Since K 1 and k = n 1 we conclude from Rauch comparison: ≥ − 2 L(c ) 2π, cf. [16, ch. 2.6]. Hence cr(h) = lim →∞ cr f (h ) 2π . j,1 ≤ j pj p ∗ p ≤ (b) Since K 1 and 1 k n 2 we conclude from Rauch comparison: ≥ ≤ ≤ − L(c ) π, cf. [16, ch. 2.6]. Hence cr(h) = lim →∞ cr f (h ) j,1 ≤ j pjp ∗ p ≤ π2/2.   

For a geodesic c ΩpqM with L(c) > π on a Riemannian manifold with ∈ n−1 ≤a K 1 we construct a mapping fc : D Ωpq M which defines a relative ≥

(13) E (fc(v)) < E (fc(0)) = E(c) for all v Dn−1(ρ), v = 0 and ∈ 6 (14) E (f (v)) E(c) ǫ c ≤ − n−1 n−1 for all v ∂D (ρ)= v D (ρ) ; v = ρ. . Hence fc defines a relative homology∈ class: { ∈ k k } ≤E(c)−ǫ Z (15) [fc] Hn−1 ΩpqM, Ωpq M; 2 . ∈   (b) If L(c) (π, 2π) and if q is a regular point for exp and if ind (c) = ∈ p 0 n 1 then there is an ǫ> 0 such that for all δ (0,ǫ] : − ∈ ≤E(c)+δ ≤E(c)−δ Z (16) 0 = [fc] Hn−1 Ωpq M, Ωpq M; 2 . 6 ∈   (c) Let q be a regular point for exp and let 0 = h H − (Ω M; Z ) with p 6 ∈ n 1 pq 2 critical value a := cr(h) (π2/2, 2π2]. Then there is a geodesic c Ω M ∈ ∈ pq CRITICALVALUESANDPOSITIVECURVATURE 9 with E(c)= a, ind0(c)= n 1 and an ǫ> 0 such that the mapping fc defines a non-trivial relative homology− class ≤a−δ Z (17) 0 = [fc] Hn−1 ΩpqM, Ωpq M; 2 6 ∈   for all δ (0,ǫ]. ∈ Proof. (a) We denote by f , f , f : Dn−1(ρ) Ω M the mappings intro- c c c −→ pq duced in Definition 1 for thee Riemannian manifold (M, g) and by φc, φc, φc : Dn−1(π/2) Ω Sn the mappings associated to the standard round metric pq e of constant sectional−→ curvature 1 on the n-sphere Sn. Hence t [0,π/L(c)] ∈ 7−→ φ (v)(t) Sn is a parametrization of a half great circle and t [0,π/L(c)] c ∈ ∈ 7−→ φ (v)(t) Sn is its parametrization proportional to arc length. Hence ec ∈ L φc(v) = L φc(v) = L(c). Then Corollary 2 implies:    e L f (v) [0,π/L(c)] = L f (v) [0,π/L(c)] c | c |    L φ (v)e[0,π/L(c)] = L φ (v) [0,π/L(c)] = π ≤ c | c |    since K 1. Thise shows Equation (12). Now assume L(c) (π, 2π]. Then ≥ ∈ for v = 0 the curves f (v) are not smooth at t = π/L(c), hence 6 c 1 E Φ1 f (v) < E f (v) E(c) . c c ≤ This proves Equation (13) and Equation (15). (b) We define homotopies f , f , f : Dn−1(ρ) Ω M with u c,u c,u c,u −→ pq ≥ π/L(c) of the mappings fc = fc,π/Le (c), fc = fc,π/L(c), f c = f c,π/L(c) resp. homotopies φ , φ , φ : Dn−1(ρ) Ω M with u π/L(c) of the c,u c,u c,u −→e pq e ≥ mappings φ = φ , φ = φ , φ = f as follows: For v c ec,π/L(c) c c,π/L(c) c c,π/L(c) ∈ Dn−1(ρ), u (π/L(c), 2π/L(c)) let ∈ e e tan v (18) g (t) = arctan sin (L(c)t) k k . v,u  sin (L(c)u/2)  Let exp (g (t)V (t)) ; 0 t u/2 c(t) v,u v ≤ ≤ f (t)=  exp (g (u t)V (t)) ; u/2 t u . c,u c(t) v,u − v ≤ ≤  c(t) ; u t 1 e ≤ ≤ Let t [0, u] f (v)(t) be the parametrization of t [0, u] f (v)(t) ∈ −→ c,u ∈ −→ c,u proportional to arc length. We denote by φc,u(v), φc,u(v) the correspond-e ing mappings for the standard round metric on Sn. Using the formulae e from spherical trigonometry one can check that the function gv,u(t) is the uniquely defined smooth function such that t [0, u] φc,u(v)(t) = = exp (g (t)E (t)) Sn is a parametrization∈ of the arc7−→ of a great circle c(t) v,u v ∈ e joining p = c(0) and exp ( v V (u/2)) . From spherical trigonometry we c(u/2) k k v conclude for the length L(u, v)= L φ (v) [0, u/2] = L φ (v) [u/2, u] c,u | c,u | of the great circle arcs:   u L(u, v) = arccos cos L(c) cos v .   2  k k 10 HANS-BERTRADEMACHER

We compute:

∂2L φ (v) cos (L(c)u/2) c,u = . 2  2 ∂vi 1 cos (L(c)u/2) vi=0 − p We conclude that for u > π/L (c) the Hessian d2 E φ (0) at the point ◦ c,u 0 Dn−1 (π/2) is negative definite, i.e. c = φ (0)is a non-degenerate maxi- ∈ c,u mum point of the restriction E : φ Dn−1(π/2) R. Using again Corol- c,u −→ lary 2 we obtain, that E f (v) E φ (v) for all u (π/L(c), 2π/L(c)), c,u ≤ c,u ∈ v Dn−1(ρ). Therefore c = f (0) is also a non-degenerate maximum ∈ c,u point of the restriction of the energy functional E : f Dn−1(ρ) c,u −→ R to the local (n 1)-dimensional submanifold f Dn−1(ρ) . Consider − c,u 1 n−1 ≤E(c) Φ f : D (ρ) Ωpq M, u π/L(c). Then there is ǫ>0 such that ◦ c,u −→ ≥ 1 n−1 ≤E(c)−ǫ Φ f ∂D (ρ) Ωpq M for all u π/L(c). c,u ⊂ ≥ This shows that fc,u defines a non-zero homology class

≤E(c)+δ ≤E(c)−δ Z (19) 0 = [fc,u] Hn−1 Ωpq M, Ωpq M; 2 6 ∈   for all δ (0,ǫ] and u > π/L(c). This holds by the Morse-Lemma since c is ∈ ≤E(c)+ǫ a non-degenerate critical point of E :Ωpq M R with ind0c = n 1, cf. Section 3. Define −→ −

f := Φ1 f : Dn−1(ρ), ∂Dn−1(ρ) Ω≤E(c)+δM, Ω≤E(c)−δM . c,u c,u −→ pq pq     for u π/L(c). The clue here is that by introducing the flow Φ1 this homo- topy is≥ also defined for u = π/L(c). Then Equation (19) implies:

≤E(c)+δ ≤E(c)−δ Z (20) 0 = [fc] = [fc,u] Hn−1 Ωpq M, Ωpq M; 2 6 ∈   for some u > π/L(c). (c) We conclude from Part (b) and Section 3: There are finitely many geodesics c ,...,c Ω M with ind (c ) = . . . = ind (c ) = n 1 and 1 r ∈ pq 0 1 0 r − E(c1) = E(c2) = . . . = E(cr) = a such that the class h as relative class in ≤a+δ ≤a−δ Z Hn−1 Ωpq M, Ωpq M; 2 for some δ > 0 has the following representa- n−1 n−1 tion: For sufficiently small ǫ>0 the mappings fcj : D (ρ), ∂D (ρ) ≤a+δ ≤a−δ −→ Ωpq M, Ωpq M , j = 1, 2, . . . , r represent non-trivial relative homology ≤a+ǫ ≤a−ǫ Z classes 0 = fcj = Fcj Hn−1 Ωpq M, Ωpq M; 2 , j = 1, 2, . . . , r 6 ∈ ≤a+δ ≤a−δ for all δ (0,ǫ].Hence in H − Ω M, Ω M; Z we have the follow- ∈ n 1 pq pq 2 ing representation: h = [fc1 ]+ . . . + [fcr ]. Hence the image of h under the homomorphism

≤a+δ Z ≤a−δ Z Hn−1 Ωpq M; 2 Hn−1 ΩpqM, Ωpq M; 2   −→   r is mapped onto α∗ j=1 fcj . Here P   ≤a+δ ≤a−δ Z ≤a−δ Z α∗ : Hn−1 Ωpq M, Ωpq M; 2 Hn−1 ΩpqM, Ωpq M; 2   −→   CRITICAL VALUES AND POSITIVE CURVATURE 11

is the homomorphism induced by the inclusion. If α∗ fcj = 0 for all j = 1, 2, . . . , r then h lies in the image of the homomorphism  ≤a−δ Z Z Hn−1 Ωpq M; 2 Hn−1 (ΩpqM; 2) ,   −→ i.e. cr(h) a δ. This is a contradiction.  ≤ − Lemma 4. Let (M, g) be a compact and simply-connected Riemannian man- ifold of positive sectional curvature K 1 : ≥ (a) Let γ ΩqpM be a geodesic of length L(γ)= π joining q = γ(0) and p = γ(1). If ∈γ is not minimal, i.e. if d(p,q) < π then there is a continuous path s [0, 1] γ Ω M with γ = γ, E (γ ) π2/2 for s (0, 1) and ∈ 7−→ s ∈ qp 0 s ≤ ∈ E(γ ) <π2/2. If E(γ )= π2/2, then E(γ )= π2/2 for all s [0,s ]. 1 s1 s ∈ 1 (b) If c ΩpM is a geodesic loop with p = c(0),q = c(1/2) and length ∈ 2 L(c) = 2π then there is a path s [0, 1] c Ω≤2π M such that the ∈ 7−→ s ∈ p following holds: For s [0, 1/2] the curves cs are geodesics with L(cs) = 2π, c (1/2) = q. And E(∈c ) < 2π2 for all s (1/2, 1]. s s ∈ Proof. (a) Define the subset A := y T M; y = π, exp (y)= p . ∈ q k k q π  of the set Tq M := y TqM; y = π, of tangent vectors at q of norm π. For y A denote by{ ∈ k k } ∈ c : [0, 1] M, c (t) = exp (ty) y −→ y q the geodesic of length π joining q = cy(0) and p = cy(1) = expq(y). For 2 n−1 ≤π /2 y A one can consider the mapping fcy : D (ρ) Ωqp M defined in Definition∈ 1, compare Lemma 3 resp. Equation (12).−→ For y A define ∈ η := sup η [0, ρ] ; E f Dn−1(η) = κ2/2 . y ∈ cy   If ηy > 0 for y A then the curves t [0, 1] fcy (v)(t) M with v ηy ∈2 ∈ 7−→ ∈ k k≤ have energy π /2 and are therefore by construction of fcy geodesics. This implies that there is a neigborhood U of y in A such that for z U : η > 0. ∈ z Hence y is a point in the interior A˚ of the set A. By assumption c′(0) A, let B A be the path-component of A containing c′(0).B is a closed subset∈ of T πM⊂ = v T M; v = π . Now assume that for all y B : η > 0. q { ∈ q k k } ∈ y Then the set B T πM is also open, i.e. B = T πM. Hence d(q,p) = π ⊂ q q in contradiction to the assumption. Therefore there is y B with ηy = 0. 2 ∈ Then one can construct a path s [0, 1] Ω M ≤π /2M with the following ∈ −→ qp properties: The restriction s [0, 1/2] αs ΩqpM is a path consisting of ∈ 7→ ∈ ′ geodesics of length π joining α0 = c and α1/2 = cy, i.e. αs(0) B. And s 2 ∈ ∈ ≤π /2 2 [1/2, 1] α (v) = f (v(s)) Ωqp M is a path satisfying E (α ) < π /2 7→ s cy ∈ s for all s (1/2, 1] and a path s [1/2, 1] v(s) Dn−1(ρ). This is possible ∈ ∈ 7→ ∈ since ηy = 0 and by the equality discussion in Corollary 2. π π R (b) Let δ : Tq M Tq M be the distance induced by the Riemann- 2 × −→ π ian metric gq/π on the sphere Tq M. This is the standard round metric of π sectional curvature 1 on Tq M. We define the numbers π ′ (21) θ± := sup θ 0; w T M, δ( c (1/2), w) θ : exp (w)= p . ≥ ∀ ∈ q ± ≤ q  12 HANS-BERTRADEMACHER

π ′ Then it follows that there are y± T M with δ( c (1/2),y±) = θ± and ∈ q ± η ± = 0. Then 0 θ± < π since d(p,q) < π. Without restriction we y ≤ can assume that θ− θ+. Then there exists a geodesic c ΩpM with ′ ′ ≥ ′ ∈ 2π2 δ(c (1/2), c (1/2) = θ+ and η(c (1/2) = 0. Then s [0, 1/2] cs Ω M ∈ e7→ ∈ p is a path of geodesics of length 2π joining c0 = c. In addition c1/2 = c e e and cs ΩpM for s [1/2, 1] is a path with cs(t) = c(t) for t [0, 1/2] ∈ ∈ 2 ∈ e and γs = cs [1/2, 1] is a curve with E(γs) < π /2 for all s (1/2, 1] as constructed in| part (a). e ∈  Lemma 5. Let (M, g) be a compact and simply-connected Riemannian man- ifold with positive sectional curvature K 1 and let c Ω M be a geodesic ≥ ∈ p loop based at p of length 2π, resp. energy 2π2. For sufficiently small positive ǫ > 0 the mapping f : Dn−1(ρ) Ω M introduced in Definition 1 with c −→ p E (f (v)) < 2π2 for all v = 0 defines a relative homology class c 6 ≤2π−ǫ (22) [f ] H − Ω M, Ω M; Z . c ∈ n 1 p p 2  If its critical value satisfies cr ([fc]) = 2π then the Riemannian manifold is a round sphere of constant sectional curvature 1. Proof. If the sectional curvature is not constant, the diameter diam(M, g) satisfies diamM <π. This is a consequence of Toponogov’s maximal diame- ter theorem [7, Thm. 6.5] resp. [20]. Let q = c(1/2),p = c(0) = c(1). Then d(q,p) diam(M, g) < π. We ≤ conclude from Lemma 4 that there is a path s [0, 1] cs ΩpM with ∈ 7−→ ∈ 2 c0 = c, c1/2 = c ΩpM a geodesic with L(c) = 2π resp. E(c) = 2π and 2 ∈ E(cs) < 2π for s (1/2, 1]. It follows that for sufficiently small ǫ> 0 e ∈ e e ≤2π2−ǫ Z (23) [fc] = [fce] Hn−1 ΩpM, Ωp M; 2 ∈   and cs(t) = c(t) for all s 1/2 and all t [0, 1/2]. Then we define a 2 ≥ n−1 ∈ ≤2π /2 homotopy (fce)s of the mapping fce : D (ρ) Ωp M. At first we e n−1 ≤2π2 −→ define a homotopy f ce s : D (ρ) Ωp M,s [0, 1] of the mapping n−1 ≤2π2 −→ ∈ f e : D Ω M: c −→ p f ce(v)(t) ; 0 t 1/2 f e (v)(t)= ≤ ≤ . c,s  c (1/2 + 2t) ; 1/2 t 1 s ≤ ≤ 2 2 Then E f e (v) 2π for all s [0, 1] and E f e (v) < 2π for all c,s ≤ ∈ c,s v Dn−1(ρ),s > 1/2. We use the negative gradient flowΦs :Ω M ∈ p −→ Ω M,s 0 of the energy functional to define: f (v):=Φ1 f (v) . Then p ≥ c,s c,s E (f (v)) 2π2 for all v Dn−1 and s [0, 1] and E (f (v)) <2π2 for c,s ≤ ∈ ∈ c,s all v Dn−1,s > 1/2. Hence the homotopy f ,s [0, 1] shows cr ([f ]) = ∈ c,s ∈ c cr ([fce]) < 2π. 

5. Proof of Theorem 1

(a) Lemma 2 shows that crlp (M, g) 2π for all p M. Hence also crl (M, g) 2π by Equation (5). ≤ ∈ ≤ (b) Let hp Hk (ΩpM,p; Z2) with crlp (M, g) = cr(h) > π. Lemma 2(b) implies k = n ∈ 1. Hence the manifold M is (n 1)-connected and hence by − − CRITICAL VALUES AND POSITIVE CURVATURE 13 the solution of the Poincaré conjecture homeomorphic to Sn . If crl (M, g) >π then crlp (M, g) >π by Equation (5).

(c) If crl (M, g) = 2π Equation (5) and part (a) imply that crlp (M, g) = 2π for all p M. Hence we assume now crlp (M, g) = 2π for some p M. ∈ n ∈ Part (b) implies that M is homeomorphic to S and crlp (M, g) = cr (hp) , where h H − (Ω M; Z ) = Z is non-zero. Choose a sequence p of p ∈ n 1 pq 2 ∼ 2 j regular points of expp with limj→∞ d(p,pj) = 0, cf. for example [18, Cor. 18.2]. Then E :Ω M R is a Morse function for any j 1, cf. ppj −→ ≥ Section 3. This holds since pj as a regular point of expp is not conjugate to p along any geodesic joining p and q. Using the homotopy equivalence ζ :Ω M Ω M introduced in Lemma 1 we define pj p p −→ ppj cr (24) aj := ζpjp ∗ (hp) ,   Z here ζpj p ∗ (hp) Hn−1 Ωppj M; 2 . ∈ 2 Then Equation (7) implies that lim j→∞ aj = 2π since p = limj→∞ pj.

Morse theory implies that for any j 1 there is a geodesic cj Ωppj M and ǫ > 0 with a = E(c ), ind (c )= n≥ 1 such that the mapping∈ j j j 0 j − (25) f = f : Dn−1(ρ) Ω M j cj −→ ppj introduced in Definition 1 resp. Equation (9) satisfies: a ǫ = j − j max F (f (v)) ; v ∂Dn−1 , cf. Lemma 3. In addition we can assume that j ∈ for any δ (0,ǫj] the relative homology class ∈ ≤aj −δ Z (26) 0 = [fj] Hn−1 Ωppj M, Ωppj M; 2 6 ∈   is non-zero. This follows from Lemma 3 (b) and (c). A subsequence (c ) converges to a geodesic loop c Ω M, which we j j ∈ p also denote by (c ) . Then we obtain for the maps f , j N and the map j j j ∈ f : Dn−1(ρ) Ω M the following statements: c −→ p (27) lim sup d (fj(v)(t),fc(v)(t)) ,t [0, 1] = 0 j→∞ { ∈ } and

(28) E (fc(v)) = lim E (fj(v)) . j→∞

′ n−1 ′ Since ǫ := 2π max F (fc(v)) ; v ∂D > 0 by Lemma 3 and ǫ = − ′ ∈ limj→∞ ǫj we obtain that ǫj ǫ /2 for all j j1 for some j1 N. Then ≥ ≥ ∈

≤2π−δ/4 (29) f = [f ]= ζ f H − Ω M, Ω M = 0 ǫ c ppj ◦ j ∈ n 1 p p 6     ′ for all δ (0,ǫ ]. Equation (27) implies that fc and ζpj p fj are homotopic for sufficiently∈ large j, which implies the second equality◦ of Equation (29). Since Equation (29) holds for any δ (0,ǫ′] we conclude for the critical value cr ∈ f ǫ = 2π. Lemma 5 implies that the Riemannian metric g has constant sectional  curvature. This finishes the proof of Theorem 1. 14 HANS-BERTRADEMACHER

6. Morse condition and critical length Instead of comparing the sectional curvature K of the Riemannian metric n g with the sectional curvature 1 of the standard metric g1 on S we can n compare the metrics g and g1 on S . For existence results of closed geodesics on spheres instead of the assumption 1 K 4 several authors considered ≤ ≤ the assumption g1/4

2 Example 1. One can write the standard metric g1 on S as warped product dt2 + sin2 tdx2 in the coordinates (t,x) [0,π] [0, 2π]/ 0, 2π = [0,π] Z/(2πZ). Denote by p = (0,x) resp. p′ =∈ (π,x) the× coordinate{ singularities.} × Choose a one-parameter family of smooth functions fr : [ π,π] R, r ′ − −→ ∈ (0, 1] with fr(0) = 1,fr(π/2) = r and f1(t) = sin t,fr(t) = fr( t), 0 < ′′ − − fr(t) sin t,fr(π/2+ t)= fr(π/2 t) and fr (t) < 0 for all t (0,π). ≤ − 2 2 2 ∈ Then the warped product metric dt + fr (t)dx on (0,π) (0, 2π)/ 0, 2π 2 × { } extends smoothly to a Riemannian metric g = gr on S . ′′ The Gauss curvature K(t,x)= fr (t)/fr(t) is positive everywhere. The surfaces can be seen as surfaces of− revolution generated by a one-parameter family of convex curves t [0, 2π] cr(t) = (x(t), z(t)) [0, r] [ π,π] parametrized by arc length,∈ which7−→ intersect at t = 0 and t∈= π the× axis− of revolution orthogonally, having distance r from the axis of revolution and ≤ such that c1 is a half-circle. For r < 1 we have g g1 and g = g1. Since all geodesics starting from ≤ 6 n the point p are closed and have length 2π we obtain crlp (S , g) = 2π. For any other point q S2 the distance to p or to p′ is at least π/2. There- 2 ∈ fore crlq(S , gr) π. For t [0,π] the curve s [0, 1] γt(s) = (t, 2πs) is a circle of length≥ 2πf(t)∈on the surface of revolution.∈ 7−→ Hence the map 2 0 2 t ([0,π], 0,π ) γt ΛS , Λ S represents the generator h ∈ 2 0{ } 7−→ ∈ n ∈ H1 ΛS , Λ ; Z2 . Therefore crl(S , g) 2πf (π/2) = 2πr. Hence we obtain ≤ 2 a one-parameter  family gr, r (0, 1) of convex surfaces (S , gr) of revolution 2 ∈ 2 2 with crq(S , gr) π for all q, crp(S , gr) = crp′ (S , gr) = 2π, g g1 and 2 ≥ ≤ limr→0 cr(S , gr) = 0. This example shows that in Part (b) of Proposition 2 it is not sufficient to assume that crlp (M, g) = 2π.

References

[1] F. Balacheff, C. Croke, and M. Katz. A Zoll counterexample to a geodesic length conjecture. Geom. Funct. Anal., 19(1):1–10, 2009. [2] W. Ballmann. Über Geschlossene Geodätische, Habilitationsschrift Math. Naturwiss. Fak. Univ. Bonn. 1983. [3] W. Ballmann, G. Thorbergsson, and W. Ziller. Existence of closed geodesics on pos- itively curved manifolds. J. Differential Geom., 18(2):221–252, 1983. [4] W. Ballmann, G. Thorbergsson, and W. Ziller. Some existence theorems for closed geodesics. Comment. Math. Helv., 58(3):416–432, 1983. [5] G. D. Birkhoff. Dynamical systems with two degrees of freedom. Trans. Amer. Math. Soc., 18(2):199–300, 1917. [6] R. Bott and H. Samelson. Applications of the theory of Morse to symmetric spaces. Amer. J. Math., 80:964–1029, 1958. [7] J. Cheeger and D. Ebin. Comparison theorems in Riemannian geometry, volume 9 of Math. Libr. North-Holland Publ. Comp.„ Amsterdam, Oxford, 1975, Reprint Amer. Math. Soc. 2008. [8] J.-H. Eschenburg. Comparison theorems and hypersurfaces. Manuscripta Math., 59(3):295–323, 1987. [9] K. Grove and K. Shiohama. A generalized sphere theorem. Ann. of Math. (2), 106(2):201–211, 1977. [10] N. Hingston. Equivariant Morse theory and closed geodesics. J. Differential Geom., 19(1):85–116, 1984. [11] N. Hingston and H.-B. Rademacher. Resonance for loop homology of spheres. J. Differential Geom., 93(1):133–174, 2013. 16 HANS-BERTRADEMACHER

[12] Y. Itokawa and R. Kobayashi. The length of the shortest closed geodesics on a posi- tively curved manifold. Max-Planck-Institute for Mathematics Bonn, Preprint 1991- 43, 1991. [13] Y. Itokawa and R. Kobayashi. The length of the shortest closed geodesics on a posi- tively curved manifold. arXiv:math/0507489, July 2005. [14] Y. Itokawa and R. Kobayashi. The length of the shortest closed geodesics on a posi- tively curved manifold. arXiv:math/0805.2793, May 2008. [15] W. Klingenberg. Lectures on closed geodesics. Springer-Verlag, Berlin-New York, 1978. Grundlehren der Mathematischen Wissenschaften, Vol. 230. [16] W. Klingenberg. Riemannian geometry, volume 1 of De Gruyter Studies in Mathe- matics. Walter de Gruyter & Co., Berlin, second edition, 1995. [17] L. A. Lyusternik and A. I. Fet. Variational problems on closed manifolds. Doklady Akad. Nauk SSSR (N.S.), 81:17–18, 1951 (Russian). [18] J. Milnor. Morse theory. Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51. Princeton University Press, Princeton, N.J., 1963. [19] M. Sugimoto. On Riemannian manifolds with a certain closed geodesic. Tôhoku Math. J. (2), 22:56–64, 1970. [20] V. A. Toponogov. Riemannian spaces having their curvature bounded below by a positive number. Dokl. Akad. Nauk SSSR, 120:719–721, 1958 (Russian). [21] V. A. Toponogov. Evaluation of the length of a closed geodesic on a convex surface. Dokl. Akad. Nauk SSSR, 124:282–284, 1959 (Russian).

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