Degenerate Bäcklund Transformation

Abstract

A concept of degenerate Backlund transformation is introduced for two-dimensional surfaces in many-dimensional Euclidean spaces. It is shown that if a surface in $$ {\mathbb{R}}_n $$ , n ≥ 4, admits a degenerate Backlund transformation, then this surface is pseudospherical, i.e., its Gauss curvature is constant and negative. The complete classification of the pseudospherical surfaces in $$ {\mathbb{R}}_n $$ , n ≥ 4, admitting the degenerate Bianchi transformations is proposed. Moreover, we also obtain a complete classification of the pseudospherical surfaces in $$ {\mathbb{R}}_n $$ , n ≥ 4, that admit degenerate Backlund transformations into straight lines.