# 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 & Continuous Dynamical Systems - A, 2011, 30 (2) : 379-426. doi: 10.3934/dcds.2011.30.379
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