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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) = π j+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 free loop 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
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