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

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

Abbas Bahri 1,

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.
Keywords: Attaching maps, Circle and Loop actions, Critical points at Infinity, Contact form Geometry, Loop spaces..
Mathematics Subject Classification: 37J, 53D, 70.
Citation: Abbas Bahri. Attaching maps in the standard geodesics problem on $S^2$. Discrete & Continuous Dynamical Systems - A, 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, (1988).   Google Scholar

[2]

A. Bahri, "Flow-lines and Algebraic invariants in Contact Form Geometry PNLDE,", "Flow-lines and Algebraic invariants in Contact Form Geometry PNLDE,", 53 (2003).   Google Scholar

[3]

A. Bahri, Compactness,, Advanced Nonlinear Stud., 8 (2008), 465.   Google Scholar

[4]

A. Bahri, Topological remarks-critical points at infinity and string theory,, Advanced Nonlinear Studies, 9 (2009), 499.   Google Scholar

[5]

M. Chas and D. Sullivan, String topology,, preprint, 1 (1999).   Google Scholar

[6]

Y. Eliashberg, Contact 3-manifolds twenty years since J. Martinet's work,, Ann. Inst. Fourier, 42 (1992), 165.   Google Scholar

[7]

H. Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three,, Inventiones Mathematicae, 114 (1993), 515.  doi: doi:10.1007/BF01232679.  Google Scholar

[8]

W. Klingenberg, Closed geodesics on surfaces of genus 0,, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, 6 (1979), 19.   Google Scholar

[9]

L. Menichi, String topology for spheres,, Comment. Math. Helv, 84 (2009), 135.  doi: doi:10.4171/CMH/155.  Google Scholar

[10]

P. H. Rabinowitz, Periodic solutions of Hamiltonian systems,, Comm. Pure. Appl. Math., 31 (1978), 157.  doi: doi:10.1002/cpa.3160310203.  Google Scholar

[11]

D. Sullivan, Infinitesimal computations in topology,, I.H.E.S., 47 (1977), 269.   Google Scholar

[12]

C. H. Taubes, The Seiberg-Witten equations and the Weinstein conjecture,, Geom. Topol., 11 (2007), 2117.  doi: doi:10.2140/gt.2007.11.2117.  Google Scholar

[13]

A. S. Zvarc, Homology of the space of closed curves,, Trudy Moskov, 9 (1960), 3.   Google Scholar

show all references

References:
[1]

A. Bahri, "Pseudo-Orbits of Contact Forms,", Pitman Research Notes in Mathematics Series No. 173, (1988).   Google Scholar

[2]

A. Bahri, "Flow-lines and Algebraic invariants in Contact Form Geometry PNLDE,", "Flow-lines and Algebraic invariants in Contact Form Geometry PNLDE,", 53 (2003).   Google Scholar

[3]

A. Bahri, Compactness,, Advanced Nonlinear Stud., 8 (2008), 465.   Google Scholar

[4]

A. Bahri, Topological remarks-critical points at infinity and string theory,, Advanced Nonlinear Studies, 9 (2009), 499.   Google Scholar

[5]

M. Chas and D. Sullivan, String topology,, preprint, 1 (1999).   Google Scholar

[6]

Y. Eliashberg, Contact 3-manifolds twenty years since J. Martinet's work,, Ann. Inst. Fourier, 42 (1992), 165.   Google Scholar

[7]

H. Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three,, Inventiones Mathematicae, 114 (1993), 515.  doi: doi:10.1007/BF01232679.  Google Scholar

[8]

W. Klingenberg, Closed geodesics on surfaces of genus 0,, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, 6 (1979), 19.   Google Scholar

[9]

L. Menichi, String topology for spheres,, Comment. Math. Helv, 84 (2009), 135.  doi: doi:10.4171/CMH/155.  Google Scholar

[10]

P. H. Rabinowitz, Periodic solutions of Hamiltonian systems,, Comm. Pure. Appl. Math., 31 (1978), 157.  doi: doi:10.1002/cpa.3160310203.  Google Scholar

[11]

D. Sullivan, Infinitesimal computations in topology,, I.H.E.S., 47 (1977), 269.   Google Scholar

[12]

C. H. Taubes, The Seiberg-Witten equations and the Weinstein conjecture,, Geom. Topol., 11 (2007), 2117.  doi: doi:10.2140/gt.2007.11.2117.  Google Scholar

[13]

A. S. Zvarc, Homology of the space of closed curves,, Trudy Moskov, 9 (1960), 3.   Google Scholar

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