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On the lengths of closed geodesics
on a two-sphere

Author: Nancy Hingston
Journal: Proc. Amer. Math. Soc. 125 (1997), 3099-3106
MSC (1991): Primary 58E10; Secondary 53C22
MathSciNet review: 1443831
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Abstract: Let $c$ be an isolated closed geodesic of length $L$ on a compact Riemannian manifold $M$ which is homologically visible in the dimension of its index, and for which the index of the iterates has the maximal possible growth rate. We show that $M$ has a sequence $\{c_n\}$, $n\in \mathbb {Z}^+$, of prime closed geodesics of length $m_nL-\varepsilon _n$ where $m_n\in \mathbb {Z}$ and $\varepsilon _n\downarrow 0$. The hypotheses hold in particular when $M$ is a two-sphere and the ``shortest'' Lusternik-Schnirelmann closed geodesic $c$ is isolated and ``nonrotating''.

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  • 1. W. Ballmann, G. Thorbergsson, and W. Ziller, Existence of closed geodesics on possibly curved manifolds, J. Differential Geom. 18 (1983), 221-252. MR 84i:58032
  • 2. V. Bangert, On the existence of closed geodesics on two-spheres, International J. of Math. Vol. 4, No. 1 (1993), 1-10. MR 94d:58036
  • 3. G. D. Birkhoff, Dynamical Systems, Amer. Math. Soc. Colloq. Publ. 9, Amer. Math. Soc., Providence, 1927.
  • 4. R. Bott, On the iteration of closed geodesics and the Stern intersection theory, Comm. Pure Appl. Math. 9 (1956), 171-206. MR 19:859f
  • 5. J. Franks, Geodesics on $S^2$ and periodic points of annulus diffeomorphisms, Invent. Math. 108 (1992), 403-418. MR 93f:58192
  • 6. M. Grayson, Shortening embedded curves, Ann. of Math. (2) 129 (1989), 71-111. MR 90a:53050
  • 7. D. Gromoll and W. Meyer, On differentiable functions with isolated critical points, Topology 8 (1969), 361-369. MR 39:7633
  • 8. N. Hingston, On the equivariant Morse complex of the free loop space of a surface, preprint, 1991.
  • 9. -, On the growth of the number of closed geodesics on the two-sphere, Internat. Math. Res. Notices 1993, no. 9, 253-262. MR 94m:58044
  • 10. W. Klingenberg, Riemannian Geometry, de Gruyter Stud. Math. 1, de Gruyter, Berlin, 1982. MR 84j:53001
  • 11. L. Lusternik and L. Schnirelmann, Sur le problème de trois géodésiques fermées sur les surfaces de Genre O, C. R. Acad. Sci. Sér. I Math 189 (1929), 269-271.
  • 12. J. Milnor, Morse Theory, Annals of Math. Studies, Princeton Univ. Press, Princeton, 1969. MR 29:634
  • 13. W. D. Neumann, Generalizations of the Poincaré Birkhoff fixed point theorem, Bull. Austral. Math. Soc. 17 (1977), 375-389. MR 58:28435

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Additional Information

Nancy Hingston
Affiliation: Department of Mathematics, The College of New Jersey, Trenton, New Jersey 08650

Received by editor(s): April 2, 1996
Communicated by: Christopher Croke
Article copyright: © Copyright 1997 American Mathematical Society

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