The parallel group of a plane curve

Abstract: For any smooth immersion $f$ of the circle in the plane, the parallel group $P(f)$ consists of all self-diffeomorphisms of the circle such that the normal lines at points of each orbit are parallel. The action of $P(f)$ on $S^1$ cannot be transitive. Thus, for example, $P(f)\neq SO(2)$. We construct examples where $P(f)$ contains a subgroup isomorphic to the group of self-diffeomorphisms of a closed interval (fixing the end-points), is isomorphic to the cyclic group $ Z_n$ for any $n\epsilon N$, and to the dihedral group $D_{n}$, for any $n\epsilon N$. If the curvature of $f$ is nowhere zero, however, then $P(f)$ is cyclic of even order.

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