An analog of Bianchi transformations for two-dimensional surfaces in the space \(S^{3} \times \mathbb R^{1}\) (Q650502)

A class of surfaces in \(S^3\times\mathbb{R}^1\) called standard pseudospherical surfaces are introduced. They are closely related to pseudospherical surfaces in \(S^3\) through the procedure which is referred to as contraction operation. Its reverse is dilation operation. For each standard pseudospherical surface \(F\) in \(S^3\times\mathbb{R}^1\) the classical construction of a Bianchi transformation can be implemented. Some properties of the transformed surface \(F^*\) are pointed out. The theory of Bianchi transformations of standard pseudospherical surfaces \(F\) in \(S^3\times\mathbb{R}^1\) is described. Any such transformation is generated by a Bianchi transformation of a pseudospherical surface in \(S^3\) combined with a contraction and dilation. Under certain conditions on the extrinsic geometry of surfaces, the results obtained can be generalized to the case of pseudospherical 2-surfaces in \(S^n\times \mathbb{R}^1\) and \(H^n\times \mathbb{R}^1\), where \(n\geq 4\).