Bäcklund transformations of constant torsion curves in 3-dimensional constant curvature spaces (Q2706755)

The classical Bäcklund transformation between two pseudospherical surfaces in the Euclidean space \(E^3\) can be restricted to a correspondence between their asymptotic lines, which are by Enneper curves of constant torsion [\textit{A.Calini} and \textit{T. Ivey}, J. Knot Theory Ramifications 7, 719-746 (1998; Zbl 0912.53002)]. In this paper the author deals with such transformations between curves in the hyperbolic space \(H^3\). The main result states: Let \(\nu\) be an arc length preserving correspondence between two curves \(\gamma\) and \(\gamma^*\) of \(H^3\) such that the tangent vectors of the geodesic form \(x\in\gamma\) to \(\nu(x)\in \gamma^*\) in \(x\) resp. \(\nu(x)\) lie in the osculating planes of \(\gamma\) resp. \(\gamma^*\), the geodesic distance \(r\) between \(x\) and \(\nu(x)\) is constant and the angle \(\theta\) between the binormals of \(\gamma\) and \(\gamma^*\) in \(x\) and \(\nu(x)\) is also constant. Then \(\gamma\) and \(\gamma^*\) have the same constant torsion \(\tau=\frac{\sin\theta}{\sinh r}\).