Symmetric pseudospherical surfaces. I: General theory (Q2655713)

The authors study symmetries of pseudospherical surfaces (ps-surfaces, i.e. surfaces with Gauss curvature \(K=-1\)) in \(\mathbb{R}^{3}\) via the loop group method developed by Zakharov-Shabat, Terng-Uhlenbeck and Toda (exact citations can be found in the references of the paper). One of the motivations for studying symmetries is to develop a theory for non-finite-type ps-surfaces. In section 2 of the paper the authors review the main results of Toda's algorithm as it has been used computationally for several years and they discuss the \(1:1\) correspondence (up to rigid motions) between ps-surfaces parametrized by asymptotic lines and pairs of normalized potentials. In section 3, they study symmetries of ps-surfaces, frames and potentials. In particular, they address the issue of how group actions on the surfaces relate to group actions on the space of general potentials.