Closed curves in R3 with prescribed curvature and torsion in perturbative cases - Part 1: Necessary condition and study of the unperturbed problem

We study the problem of $(\kappa,\tau)$-loops, namely closed curves in the three-dimensional Euclidean space, with prescribed curvature $\kappa$ and torsion $\tau$. We state a necessary condition for the existence of a bounded sequence of $(\kappa_{n},\tau_{n})$-loops when the functions $\kappa_{n}$ and $\tau_{n}$ converge to the constants 1 and 0, respectively. Moreover we prove some Fredholm-type properties for the "unperturbed" problem, with $\kappa\equiv 1$ and $\tau\equiv 0$.