An analog of bianchi transformations for two-dimensional surfaces in the space S 3 × ℝ1

Generalization of the Bianchi–Bäcklund Transformation of Pseudo-Spherical Surfaces

“…The following question is posed: In what cases does the constructed β-transformation φ : F 2 →F 2 satisfy conditions (B 1 )-(B 3 ) from Definition 1 * , i.e., has the properties of a pseudo-spherical congruence? In [15], the following theorem is proved.…”

“…And conversely, if any pseudo-spherical surface in S n , parametrized by the horocycic coordinates, is supplemented by the corresponding linear height function, then we obtain a standard pseudo-spherical surface in S n × R 1 ; such a construction is called the expansion procedure in [15].…”

“…Below, we will give results from [12,13,15] on a particular realization of this program for twodimensional surfaces in product-spaces S n × R 1 and H n × R 1 . The space S n × R 1 is represented as a hyper-surface in the Euclidean space E n+2 given in the Cartesian coordinates x 1 , .…”

“…Under projecting S n × R 1 → S n , each standard pseudo-spherical surface in S n × R 1 is mapped to a pseudo-spherical in S n ; in this case, the horocyclicity of the coordinate net is preserved; such transformation is called the contraction procedure in [15]. And conversely, if any pseudo-spherical surface in S n , parametrized by the horocycic coordinates, is supplemented by the corresponding linear height function, then we obtain a standard pseudo-spherical surface in S n × R 1 ; such a construction is called the expansion procedure in [15].…”

“…In [15] (see also [13]) a special class of surfaces of constant negative intrinsical curvature in S N × R 1 was separated.…”