On the lengths of closed geodesics on almost round spheres. (English) Zbl 0595.53049
The following theorem is proved: For every \(\epsilon >0\) there is a \(\delta =(\epsilon,n)>0\) such that every prime closed geodesic on a Riemannian sphere \((S^ n,g)\) with curvature between 1-\(\delta\) and \(1+\delta\) has length between \(2\pi\)-\(\epsilon\) and \(2\pi +\epsilon\) or else larger than 1/\(\epsilon\). This was proved for \(n=2\) in [W. Ballmann, Invent. Math. 71, 593-597 (1983; Zbl 0505.53020)] and for general n in the special case of ellipsoids [M. Morse, ”The calculus of variations in the large” (1934; Zbl 0011.02802)]. The proof is based on a result for general flows of independent interest.
Reviewer: K.Grove
MSC:
| 53C22 | Geodesics in global differential geometry |
| 37D40 | Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) |
| 53D25 | Geodesic flows in symplectic geometry and contact geometry |
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