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

DOI:
https://doi.org/10.1090/S0002-9939-97-04235-4

MathSciNet review:
1443831

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Abstract: Let be an isolated closed geodesic of length on a compact Riemannian manifold 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 has a sequence , , of prime closed geodesics of length where and . The hypotheses hold in particular when is a two-sphere and the ``shortest'' Lusternik-Schnirelmann closed geodesic is isolated and ``nonrotating''.

**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 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

Email:
hingston@tcnj.edu

DOI:
https://doi.org/10.1090/S0002-9939-97-04235-4

Received by editor(s):
April 2, 1996

Communicated by:
Christopher Croke

Article copyright:
© Copyright 1997
American Mathematical Society