Attaching maps in the standard geodesics problem on S2

Abstract

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 ε0 of the standard contact structure/form of S3 is studied; the extended variational problem is the closed geodesics problem on S2. The attaching maps are studied for low-dimensional (at most 4) cells. Some circle and "loop" actions on the loop space of S3, that are lifts (via Hopf-fibration map) of the standard S1-action on the free loop space of S2, are also defined. "Conjugacy" relations relating these actions are established.