The second closed geodesic on Finsler spheres of dimension 𝑛>2

Rademacher, Hans-Bert

doi:10.1090/S0002-9947-09-04745-X

The second closed geodesic on Finsler spheres of dimension $n > 2$
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Trans. Amer. Math. Soc. 362 (2010), 1413-1421 Request permission

Abstract:

We show the existence of at least two geometrically distinct closed geodesics on an $n$-dimensional sphere with a bumpy and non-reversible Finsler metric for $n > 2.$

References

D. V. Anosov, Geodesics and Finsler geometry, Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974) Canad. Math. Congress, Montreal, Que., 1975, pp. 293–297 (Russian). MR 0426058
W. Ballmann, G. Thorbergsson, and W. Ziller, Closed geodesics on positively curved manifolds, Ann. of Math. (2) 116 (1982), no. 2, 213–247. MR 672836, DOI 10.2307/2007062
V. Bangert, Geodätische Linien auf Riemannschen Mannigfaltigkeiten, Jahresber. Deutsch. Math.-Verein. 87 (1985), no. 2, 39–66 (German). MR 789708
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
V. Bangert and Y. Long: The existence of two closed geodesics on every Finsler $2$-sphere. arXiv:0709.1243
D. Bao, S.-S. Chern, and Z. Shen, An introduction to Riemann-Finsler geometry, Graduate Texts in Mathematics, vol. 200, Springer-Verlag, New York, 2000. MR 1747675, DOI 10.1007/978-1-4612-1268-3
Raoul Bott, On the iteration of closed geodesics and the Sturm intersection theory, Comm. Pure Appl. Math. 9 (1956), 171–206. MR 90730, DOI 10.1002/cpa.3160090204
Shiing-Shen Chern and Zhongmin Shen, Riemann-Finsler geometry, Nankai Tracts in Mathematics, vol. 6, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. MR 2169595, DOI 10.1142/5263
A. I. Fet, On a periodicity problem in the calculus of variations, Dokl. Akad. Nauk SSSR 160 (1965), 287–289 (Russian). MR 0220316
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
N. Hingston, Equivariant Morse theory and closed geodesics, J. Differential Geom. 19 (1984), no. 1, 85–116. MR 739783, DOI 10.4310/jdg/1214438424
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
H. Hofer, K. Wysocki, and E. Zehnder, Finite energy foliations of tight three-spheres and Hamiltonian dynamics, Ann. of Math. (2) 157 (2003), no. 1, 125–255. MR 1954266, DOI 10.4007/annals.2003.157.125
Wilhelm Klingenberg, Lectures on closed geodesics, Grundlehren der Mathematischen Wissenschaften, Vol. 230, Springer-Verlag, Berlin-New York, 1978. MR 0478069, DOI 10.1007/978-3-642-61881-9
Yiming Long, Index theory for symplectic paths with applications, Progress in Mathematics, vol. 207, Birkhäuser Verlag, Basel, 2002. MR 1898560, DOI 10.1007/978-3-0348-8175-3
Yiming Long, Multiplicity and stability of closed geodesics on Finsler 2-spheres, J. Eur. Math. Soc. (JEMS) 8 (2006), no. 2, 341–353. MR 2239281, DOI 10.4171/JEMS/56
Yiming Long and Chaofeng Zhu, Closed characteristics on compact convex hypersurfaces in $\mathbf R^{2n}$, Ann. of Math. (2) 155 (2002), no. 2, 317–368. MR 1906590, DOI 10.2307/3062120
L. A. Lyusternik and A. I. Fet, Variational problems on closed manifolds, Doklady Akad. Nauk SSSR (N.S.) 81 (1951), 17–18 (Russian). MR 0044760
Hans-Bert Rademacher, On the average indices of closed geodesics, J. Differential Geom. 29 (1989), no. 1, 65–83. MR 978076
Hans-Bert Rademacher, Morse-Theorie und geschlossene Geodätische, Bonner Mathematische Schriften [Bonn Mathematical Publications], vol. 229, Universität Bonn, Mathematisches Institut, Bonn, 1992 (German). Habilitationsschrift, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, 1991. MR 1237191
Hans-Bert Rademacher, On a generic property of geodesic flows, Math. Ann. 298 (1994), no. 1, 101–116. MR 1252820, DOI 10.1007/BF01459728
Hans-Bert Rademacher, A sphere theorem for non-reversible Finsler metrics, Math. Ann. 328 (2004), no. 3, 373–387. MR 2036326, DOI 10.1007/s00208-003-0485-y
Hans-Bert Rademacher, Existence of closed geodesics on positively curved Finsler manifolds, Ergodic Theory Dynam. Systems 27 (2007), no. 3, 957–969. MR 2322187, DOI 10.1017/S0143385706001064
I. A. Taĭmanov, Closed extremals on two-dimensional manifolds, Uspekhi Mat. Nauk 47 (1992), no. 2(284), 143–185, 223 (Russian, with Russian summary); English transl., Russian Math. Surveys 47 (1992), no. 2, 163–211. MR 1185286, DOI 10.1070/RM1992v047n02ABEH000880
Wolfgang Ziller, Geometry of the Katok examples, Ergodic Theory Dynam. Systems 3 (1983), no. 1, 135–157. MR 743032, DOI 10.1017/S0143385700001851