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Homologically visible closed geodesics on complete surfaces

Simon Allais

https://orcid.org/0000-0001-7167-6866

Université de Paris, IMJ-PRG, 8 Place Aurélie de Nemours, 75013 Paris, France

E-mail Address: simon.allais@imj-prg.fr

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and

Tobias Soethe

RWTH Aachen, Fachgruppe Mathematik, Jakobstr 2, 52064 Aachen, Germany

E-mail Address: soethe@mathga.rwth-aachen.de

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https://doi.org/10.1142/S1793525321500692Cited by:0 (Source: Crossref)

Abstract

In this paper, we give multiple situations when having one or two geometrically distinct closed geodesics on a complete Riemannian cylinder, a complete Möbius band or a complete Riemannian plane leads to having infinitely many geometrically distinct closed geodesics. In particular, we prove that any complete cylinder with isolated closed geodesics has zero, one or infinitely many homologically visible closed geodesics; this answers a question of Alberto Abbondandolo.

Keywords:

AMSC: 53C22, 58E10

References

1. L. Asselle and M. Mazzucchelli , On the existence of infinitely many closed geodesics on non-compact manifolds, Proc. Amer. Math. Soc. 145(6) (2017) 2689–2697. Crossref, Google Scholar
2. L. Asselle and M. Mazzucchelli , Closed geodesics with local homology in maximal degree on non-compact manifolds, Diff. Geom. Appl. 58 (2018) 17–51. Crossref, Google Scholar
3. W. Ballmann, G. Thorbergsson and W. Ziller , Closed geodesics on positively curved manifolds, Ann. Math. 116(2) (1982) 213–247. Crossref, Google Scholar
4. V. Bangert , Closed geodesics on complete surfaces, Math. Ann. 251(1) (1980) 83–96. Crossref, Google Scholar
5. V. Bangert , On the existence of closed geodesics on two-spheres, Int. J. Math. 4(1) (1993) 1–10. Link, Google Scholar
6. V. Bangert and W. Klingenberg , Homology generated by iterated closed geodesics, Topology 22(4) (1983) 379–388. Crossref, Google Scholar
7. V. Benci and F. Giannoni , Closed geodesics on noncompact Riemannian manifolds, C. R. Acad. Sci. Paris Sér. I Math. 312(11) (1991) 857–861. Google Scholar
8. V. Benci and F. Giannoni , On the existence of closed geodesics on noncompact Riemannian manifolds, Duke Math. J. 68(2) (1992) 195–215. Crossref, Google Scholar
9. A. L. Besse , Manifolds all of whose geodesics are closed, Ergebnisse der Mathematik und ihrer Grenzgebiete, Results in Mathematics and Related Areas, Vol. 93, eds. D. B. A. Epstein, J.-P. Bourguignon, L. Bérard-Bergery, M. Berger and J. L. Kazdan (Springer-Verlag, Berlin-New York, 1978). Crossref, Google Scholar
10. G. D. Birkhoff , Dynamical Systems, American Mathematical Society Colloquium Publications, Vol. IX, ed. J. Moser (American Mathematical Society, Providence, 1966). Google Scholar
11. A. Borel , Seminar on Transformation Groups, Annals of Mathematics Studies, Vol. 46, eds. G. Bredon, E. E. Floyd, D. Montgomery, R. Palais (Princeton University Press, Princeton, 1960). Google Scholar
12. R. Bott , On the iteration of closed geodesics and the Sturm intersection theory, Comm. Pure Appl. Math. 9 (1956) 171–206. Crossref, Google Scholar
13. K. Burns and V. S. Matveev , Open problems and questions about geodesics, Ergodic Theory and Dynamical Systems (Cambridge University Press, 2019), pp. 1–44. Google Scholar
14. E. Caponio and M. Á. Javaloyes , A remark on the Morse theorem about infinitely many geodesics between two points, Int. Meeting on Differential Geometry (University of Córdoba, 2013), pp. 39–48. Google Scholar
15. E. Çineli and V. L. Ginzburg , On the iterated hamiltonian floer homology, Commun. Contemp. Math. 23(3) (2021) 2050026. Link, Google Scholar
16. K.-C. Chang , Infinite-Dimensional Morse Theory and Multiple Solution Problems, Progress in Nonlinear Differential Equations and their Applications, Vol. 6 (Birkhäuser Boston, Boston, MA, 1993). Crossref, Google Scholar
17. A. I. Fet , Variational problems on closed manifolds, Mat. Sbornik N. S. 30(72) (1952) 271–316. Google Scholar
18. J. Franks , Geodesics on $S^{2}$ $S^{2}$ and periodic points of annulus homeomorphisms, Invent. Math. 108(2) (1992) 403–418. Crossref, Google Scholar
19. M. A. Grayson , Shortening embedded curves, Ann. Math. 129(1) (1989) 71–111. Crossref, Google Scholar
20. D. Gromoll, W. Klingenberg and W. Meyer , Riemannsche Geometrie im Grossen, Lecture Notes in Mathematics, Vol. 55 (Springer-Verlag, Berlin-New York, 1968). Crossref, Google Scholar
21. D. Gromoll and W. Meyer , On differentiable functions with isolated critical points, Topology 8 (1969) 361–369. Crossref, Google Scholar
22. D. Gromoll and W. Meyer , Periodic geodesics on compact riemannian manifolds, J. Diff. Geom. 3 (1969) 493–510. Crossref, Google Scholar
23. J. Hadamard , Les surfaces à courbures opposées et leurs lignes géodésiques, J. Math. 4(5) (1898) 27–73 (French). Google Scholar
24. N. Hingston , On the growth of the number of closed geodesics on the two-sphere, Int. Math. Res. Notices 1993(3) (1993) 253–262. Crossref, Google Scholar
25. L. A. Lyusternik and A. I. Fet , Variational problems on closed manifolds, Doklady Akad. Nauk SSSR (N.S.) 81 (1951) 17–18. Google Scholar
26. J. A. Oaks , Singularities and self-intersections of curves evolving on surfaces, Indiana Univ. Math. J. 43(3) (1994) 959–981. Crossref, Google Scholar
27. H. Poincaré , Sur les lignes géodésiques des surfaces convexes, Trans. Amer. Math. Soc. 6(3) (1905) 237–274. Google Scholar
28. M. Struwe , Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems (Springer-Verlag, Berlin, 1990). Crossref, Google Scholar
29. G. Thorbergsson , Closed geodesics on non-compact Riemannian manifolds, Math. Z. 159(3) (1978) 249–258. Crossref, Google Scholar