On the lengths of closed geodesics on a two-sphere
HTML articles powered by AMS MathViewer
- by Nancy Hingston
- Proc. Amer. Math. Soc. 125 (1997), 3099-3106
- DOI: https://doi.org/10.1090/S0002-9939-97-04235-4
- PDF | Request permission
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”.References
- W. Ballmann, G. Thorbergsson, and W. Ziller, Existence of closed geodesics on positively curved manifolds, J. Differential Geometry 18 (1983), no. 2, 221–252. MR 710053
- Victor Bangert, On the existence of closed geodesics on two-spheres, Internat. J. Math. 4 (1993), no. 1, 1–10. MR 1209957, DOI 10.1142/S0129167X93000029
- G. D. Birkhoff, Dynamical Systems, Amer. Math. Soc. Colloq. Publ. 9, Amer. Math. Soc., Providence, 1927.
- Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
- John Franks, Geodesics on $S^2$ and periodic points of annulus homeomorphisms, Invent. Math. 108 (1992), no. 2, 403–418. MR 1161099, DOI 10.1007/BF02100612
- Matthew A. Grayson, Shortening embedded curves, Ann. of Math. (2) 129 (1989), no. 1, 71–111. MR 979601, DOI 10.2307/1971486
- Detlef Gromoll and Wolfgang Meyer, On differentiable functions with isolated critical points, Topology 8 (1969), 361–369. MR 246329, DOI 10.1016/0040-9383(69)90022-6
- N. Hingston, On the equivariant Morse complex of the free loop space of a surface, preprint, 1991.
- Nancy Hingston, On the growth of the number of closed geodesics on the two-sphere, Internat. Math. Res. Notices 9 (1993), 253–262. MR 1240637, DOI 10.1155/S1073792893000285
- Wilhelm Klingenberg, Riemannian geometry, de Gruyter Studies in Mathematics, vol. 1, Walter de Gruyter & Co., Berlin-New York, 1982. MR 666697
- 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.
- J. Milnor, Morse theory, Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. Based on lecture notes by M. Spivak and R. Wells. MR 0163331
- Walter D. Neumann, Generalizations of the Poincaré Birkhoff fixed point theorem, Bull. Austral. Math. Soc. 17 (1977), no. 3, 375–389. MR 584597, DOI 10.1017/S0004972700010650
Bibliographic Information
- Nancy Hingston
- Affiliation: Department of Mathematics, The College of New Jersey, Trenton, New Jersey 08650
- Email: hingston@tcnj.edu
- Received by editor(s): April 2, 1996
- Communicated by: Christopher Croke
- © Copyright 1997 American Mathematical Society
- 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