An analog of Bianchi transformations for two-dimensional surfaces in the space \(S^{3} \times \mathbb R^{1}\)
type: Item
instance of: scholarly article
title: An analog of Bianchi transformations for two-dimensional surfaces in the space \(S^{3} \times \mathbb R^{1}\)
zbMATH Open document ID: 1252.53069
DOI: 10.1134/S000143461105021X
author: E. N. Nevmerzhitskaya
author: Vasyl Gorkaviy
published in: Mathematical Notes
publication date: 2011-11-25T00:00:00Z
review text: 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 \(n\geq 4\).
reviewed by: Kaarin Riives
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 37K35
Mathematics Subject Classification ID: 53A07
zbMATH DE Number: 5980828
zbMATH Keywords: Bianchi transformation
zbMATH Keywords: 2-surface negative intrinsic curvature
zbMATH Keywords: pseudospherical surface
zbMATH Keywords: pseudospherical congruence
zbMATH Keywords: Gaussian curvature
zbMATH Keywords: Bäcklund transformation
MaRDI profile type: Publication
cites work: Q4433581
cites work: Q4857713
cites work: Bäcklund's theorem for \(n\)-dimensional submanifolds of \(R^{2n-1}\)
cites work: On Bianchi and Bäcklund transformations of two-dimensional surfaces in \(E^4\)
cites work: Q4653322
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description: scientific article; zbMATH DE number 5980828