Attaching maps in the standard geodesics problem on \(S^2\) (Q541543)

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.