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

Abstract

In this paper, we discuss generalizations of the classical Bianchi-Bäcklund transformation of two-dimensional pseudo-spherical surfaces in three-dimensional spaces of constant curvature to the case of submanifolds of arbitrary codimension in spaces of constant curvature and in homogeneous Riemannian products.

Degenerate Bianchi transformations for three-dimensional pseudo-spherical submanifolds in $\mathbb R^5$

“…Geometrically, (10) means that null-curves situated in different coordinate horospheres u 1 = const are mapped to each other by translations along u 1 -coordinate curves in F 3 . Now let us discuss in more details how the coordinate system (v 1 , v 2 , v 3 ) is related to the original coordinates (u 1 , u 2 , u 3 ).…”

“…As for the case of p > n − 1, it quite differs from the classical one where p = n − 1. For instance, in this non-classical case the pseudo-sphericity is not obliged to be preserved under the Bianchi transformation, see [3], [9], [10].…”

On Circular Tractrices in $\mathbb R^3$

“…Thus, F consists of two symmetric parts sharing the common coordinate line t = 0 situated in the plane x 2 = 0. Besides, F is symmetric with respect to the coordinate plane x 3 = 0, its position vector satisfies (10).…”

“…Consequently, F is symmetric with respect to any plane x 1 sin ϕ 2 n + x 2 cos ϕ 2 n = 0, n ∈ Z, in R 3 . Besides, F is symmetric with respect to the coordinate plane x 3 = 0, its position vector satisfies (10).…”

Circular tractrices and generalized Dini surfaces

“…While exploring the question posed by Yu.Aminov and A. Sym, we was interested in finding surfaces in E n , n ≥ 4, which inherit geometric properties of the Beltrami and Dini surfaces related to the degeneracy of Bianchi-Backlund transformations. As result, a novel family of pseudo-spherical surfaces in E n , n ≥ 4, called generalized Beltrami surfaces was described in [6], where it was shown that every generalized Beltrami surface admits a degenerate Bianchi transformation like the classical Beltrami surface in E 3 do, see also [7].…”