Two-dimensional pseudospherical surfaces with degenerate Bianchi transformation
Given a smooth surface \(F^2\subset{\mathbb R}^3\) endowed with a parametrization \({\mathbf r}(u,v)\) such that the induced metric has the form \(ds^2=du^2+e^{-2v}dv^2\) (oricyclic coordinates), the classical Bianchi transformation assigns to \({\mathbf r}(u,v)\) a new parametrized surface \({\widetilde{F}}^2\) parametrized by \(\widetilde{{\mathbf r}}={\mathbf r}+\frac{\partial{\mathbf r}}{\partial{u}}\). If \(F^2\) has constant Gaussian curvature \(-1\), then the same is true for \({\widetilde{F}}^2\). This is a simple formula for a special case of Bäcklund transformations between pseudospherical surfaces offering direct generalizations to the case of submanifolds of higher dimensional Euclidian spaces (see for instance [\textit{Yu. Aminov}, The geometry of submanifolds. Amsterdam: Gordon and Breach Science Publishers (2001; Zbl 0978.53001); \textit{Yu. Aminov} and \textit{A. Sym}, CRM Proc. Lect. Notes 29, 91--93 (2001; Zbl 0999.53004)]). For certain surfaces \(F^2\) in \({\mathbb R}^{3}\), the transformed surface \({\widetilde{F}}^2\) degenerates to a curve, so for instance, in the case of the Beltrami surface given by \({\mathbf r}(u,v)=\left(\Phi(u),e^{-u}\cos\,v,e^{-u}\sin\,v\right), \left|\Phi'(u)\right|=\sqrt{1-e^{-2u}}\), obtained by rotating the tractrix around the \(x\)-axis. The main aim of this paper is the complete description of pseudospherical surfaces \(F^2\subset{\mathbb R}^{N}\), \(N\geq3\), with degenerate Bianchi transform. To construct these surfaces, the authors introduce so-called generalized tractrices. These are curves in \({\mathbb R}^{N}\) that move by certain 1-parameter families of motions of \({\mathbb R}^{N}\), in this way sweeping out the surfaces in question.
scientific article; zbMATH DE number 6110055