JournalsrlmVol. 17, No. 3pp. 227–242

Closed curves in R3\mathbb{R}^{3} with prescribed curvature and torsion in perturbative cases. Part 1: Necessary condition and study of the unperturbed problem

  • Paolo Caldiroli

    Università degli Studi di Torino, Italy
  • Michela Guida

    Università degli Studi di Torino, Italy
Closed curves in $\mathbb{R}^{3}$ with prescribed curvature and torsion in perturbative cases. Part 1: Necessary condition and study of the unperturbed problem cover
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Abstract

We study the problem of (κ,τ)(\kappa,\tau)-loops, namely closed curves in the three-dimensional Euclidean space, with prescribed curvature κ\kappa and torsion τ\tau. We state a necessary condition for the existence of a bounded sequence of (κn,τn)(\kappa_{n},\tau_{n})-loops when the functions κn\kappa_{n} and τn\tau_{n} converge to the constants 1 and 0, respectively. Moreover we prove some Fredholm-type properties for the ``unperturbed'' problem, with κ1\kappa\equiv 1 and τ0\tau\equiv 0.

Cite this article

Paolo Caldiroli, Michela Guida, Closed curves in R3\mathbb{R}^{3} with prescribed curvature and torsion in perturbative cases. Part 1: Necessary condition and study of the unperturbed problem. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 17 (2006), no. 3, pp. 227–242

DOI 10.4171/RLM/466