Ribaucour transformations and their singularities
type: Item
instance of: scholarly article
title: Ribaucour transformations and their singularities
zbMATH Open document ID: 1506.57018
author: Yuta Ogata
published in: Journal of Geometry
publication date: 2021-12-15T00:00:00Z
review text: The Ribaucour transformation generalises the notion of parallel surfaces and Darboux transformations. This paper introduces the construction of Ribaucour transforms via the Cauchy-Kovalevskaya theorem. The paper gives an explicit integral formula of Ribaucour transformations for surfaces of revolution. The paper also studies the singularities appearing on Ribaucour transforms and gives the criteria for cuspidal edges, swallowtails, cuspidal cross caps, and \(D_4^\pm\) singularities. In Section 2, Theorem 1 states that any Ribaucour transformation can be obtained by solving the damped wave equation with the help of the Cauchy-Kovalevskaya theorem. Theorem 2 provides an explicit integral formula for Ribaucour transformations for surfaces of revolution, which is used to create examples in later sections. The examples demonstrate the ability to construct non-linear Weingarten and non-isothermic Ribaucour transformations from surfaces of revolution. The Ribaucour transformations can be applied locally to any regular surface immersed in a space form, specifically rotational surfaces, without imposing any other geometric property on the transformed surface. In previous articles, the transformations were required to preserve the geometric property of being linear Weingarten (which includes surfaces of constant Gaussian or mean curvature), however in this paper the Ribaucour transformations are applied to rotational surfaces in \(\mathbb{R}^3\), without imposing any other geometric property on the transformed surface. In Section 3, the paper analyses singularities on Ribaucour transformations and establishes criteria for cuspidal edges, swallowtails, cuspidal cross caps, and \(D_4^\pm\) singularities. Parallel surfaces are noted as typical examples of Ribaucour transformations and the results of previous studies on singularities on parallel surfaces are generalized to Ribaucour transformations and some geometrical analysis is given.
Mathematics Subject Classification ID: 57R45
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 7445229
zbMATH Keywords: surface theory
zbMATH Keywords: singularity theory
zbMATH Keywords: transformation theory
reviewed by: Graham M. Reeve
MaRDI profile type: Publication
full work available at URL: s00022-021-00618-y
OpenAlex ID: W3217279966
cites work: The Ribaucour transformation in Lie sphere geometry
cites work: Ribaucour coordinates
cites work: Channel surfaces in Lie sphere geometry
cites work: Q4522321
cites work: Minimal surfaces obtained by Ribaucour transformations
cites work: Ribaucour transformations for constant mean curvature and linear Weingarten surfaces.
cites work: Ribaucour transformations revisited
cites work: Ribaucour transformations for flat surfaces in the hyperbolic 3-space
cites work: New minimal surfaces in the hyperbolic space
cites work: The vectorial Ribaucour transformation for submanifolds and applications
cites work: AN EXTENSION OF THE CLASSICAL RIBAUCOUR TRANSFORMATION
cites work: Commuting Codazzi tensors and the Ribaucour transformation for submanifolds
cites work: Singularities of maximal surfaces
cites work: Singularities of parallel surfaces
cites work: On Bäcklund and Ribaucour transformations for surfaces with constant negative curvature
cites work: Singularities of flat fronts in hyperbolic space
cites work: Lawson correspondence and Ribaucour transformations
cites work: Q5224873
cites work: Q5832281
cites work: Criteria for \(D_4\) singularities of wave fronts
cites work: Behavior of principal curvatures of frontals near non-front singular points and their applications
cites work: Ribaucour transformations for hypersurfaces in space forms
cites work: New constant mean curvature surfaces in the hyperbolic space
cites work: Parallel and dual surfaces of cuspidal edges
cites work: Principal curvatures and parallel surfaces of wave fronts
cites work: Maximal surfaces with singularities in Minkowski space
DOI: 10.1007/S00022-021-00618-Y
description: scientific article; zbMATH DE number 7445229