Closed curves in \((\mathbb R^3)\) with prescribed curvature and torsion in perturbative cases. I: Necessary condition and study of the unperturbed problem
Summary: We study the problem of \((\kappa, \tau)\)-loops, i.e. 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 \(\tau \equiv 0\).
scientific article; zbMATH DE number 5177498