Degenerate Backlund transformation - Gor'kavyi, V. A.; Nevmerzhitskaya, E. N.; Горькавий, В. А.; Невмержицкая, Е. Н.; Горькавий, В. А.; Невмержицкая, Е. Н. (Institute of Mathematics, NAS of Ukraine, 2016)
A concept of degenerate B¨acklund transformation is introduced for two-dimensional surfaces in many-dimensional Euclidean spaces. It is shown that if a surface in $R^n, n ≥ 4$, admits a degenerate B¨acklund transformation, then this surface is pseudospherical, i.e., its Gauss curvature is constant and negative. The complete classification of pseudospherical surfaces in $R^n, n ≥ 4$ that admit degenerate Bianchi transformations is obtained. Moreover, we also obtain a complete classification of pseudospherical surfaces in $R^n, n ≥ 4$, admitting degenerate Backlund transformations into straight lines.