# American Institute of Mathematical Sciences

May  2011, 30(2): 379-426. doi: 10.3934/dcds.2011.30.379

## Attaching maps in the standard geodesics problem on $S^2$

 1 Hill Center for the Mathematical Sciences, Rutgers University, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019

Received  March 2010 Revised  April 2010 Published  February 2011

Unstable manifolds of critical points at infinity in the variational problems relating to periodic orbits of Reeb vector-fields in Contact Form Geometry are viewed in this paper as part of the attaching maps along which these variational problems attach themselves to natural generalizations that they have. The specific periodic orbit problem for the Reeb vector-field $\xi_0$ of the standard contact structure/form of $S^3$ is studied; the extended variational problem is the closed geodesics problem on $S^2$. The attaching maps are studied for low-dimensional (at most $4$) cells. Some circle and ''loop" actions on the loop space of $S^3$, that are lifts (via Hopf-fibration map) of the standard $S^1$-action on the free loop space of $S^2$, are also defined. ''Conjugacy" relations relating these actions are established.
Citation: Abbas Bahri. Attaching maps in the standard geodesics problem on $S^2$. Discrete and Continuous Dynamical Systems, 2011, 30 (2) : 379-426. doi: 10.3934/dcds.2011.30.379
##### References:
 [1] A. Bahri, "Pseudo-Orbits of Contact Forms," Pitman Research Notes in Mathematics Series No. 173, Longman Scientific and Technical, Longman, London, 1988 [2] A. Bahri, "Flow-lines and Algebraic invariants in Contact Form Geometry PNLDE," Birkhauser, Boston, 53, 2003. [3] A. Bahri, Compactness, Advanced Nonlinear Stud., 8 (2008), 465-568. [4] A. Bahri, Topological remarks-critical points at infinity and string theory, Advanced Nonlinear Studies, 9 (2009), 499-512. [5] M. Chas and D. Sullivan, String topology, preprint, Math. GT /9911159, 1 (1999). [6] Y. Eliashberg, Contact 3-manifolds twenty years since J. Martinet's work, Ann. Inst. Fourier, Grenoble, 42 (1992), 165-192. [7] H. Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Inventiones Mathematicae, 114 (1993), 515-563. doi: doi:10.1007/BF01232679. [8] W. Klingenberg, Closed geodesics on surfaces of genus 0, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Sr.4, 6 (1979), 19-38. [9] L. Menichi, String topology for spheres, Comment. Math. Helv, 84 (2009), 135-157. doi: doi:10.4171/CMH/155. [10] P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure. Appl. Math., 31 (1978), 157-184. doi: doi:10.1002/cpa.3160310203. [11] D. Sullivan, Infinitesimal computations in topology, I.H.E.S., 47 (1977), 269-331. [12] C. H. Taubes, The Seiberg-Witten equations and the Weinstein conjecture, Geom. Topol., 11 (2007), 2117-2202. doi: doi:10.2140/gt.2007.11.2117. [13] A. S. Zvarc, Homology of the space of closed curves, Trudy Moskov, Mat. Obsv., 9 (1960), 3-44.

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##### References:
 [1] A. Bahri, "Pseudo-Orbits of Contact Forms," Pitman Research Notes in Mathematics Series No. 173, Longman Scientific and Technical, Longman, London, 1988 [2] A. Bahri, "Flow-lines and Algebraic invariants in Contact Form Geometry PNLDE," Birkhauser, Boston, 53, 2003. [3] A. Bahri, Compactness, Advanced Nonlinear Stud., 8 (2008), 465-568. [4] A. Bahri, Topological remarks-critical points at infinity and string theory, Advanced Nonlinear Studies, 9 (2009), 499-512. [5] M. Chas and D. Sullivan, String topology, preprint, Math. GT /9911159, 1 (1999). [6] Y. Eliashberg, Contact 3-manifolds twenty years since J. Martinet's work, Ann. Inst. Fourier, Grenoble, 42 (1992), 165-192. [7] H. Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Inventiones Mathematicae, 114 (1993), 515-563. doi: doi:10.1007/BF01232679. [8] W. Klingenberg, Closed geodesics on surfaces of genus 0, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Sr.4, 6 (1979), 19-38. [9] L. Menichi, String topology for spheres, Comment. Math. Helv, 84 (2009), 135-157. doi: doi:10.4171/CMH/155. [10] P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure. Appl. Math., 31 (1978), 157-184. doi: doi:10.1002/cpa.3160310203. [11] D. Sullivan, Infinitesimal computations in topology, I.H.E.S., 47 (1977), 269-331. [12] C. H. Taubes, The Seiberg-Witten equations and the Weinstein conjecture, Geom. Topol., 11 (2007), 2117-2202. doi: doi:10.2140/gt.2007.11.2117. [13] A. S. Zvarc, Homology of the space of closed curves, Trudy Moskov, Mat. Obsv., 9 (1960), 3-44.
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