Darboux transformations on timelike constant mean curvature surfaces
type: Item
instance of: scholarly article
title: Darboux transformations on timelike constant mean curvature surfaces
zbMATH Open document ID: 0980.53009
DOI: 10.1016/S0393-0440(99)00011-X
published in: Journal of Geometry and Physics
publication date: 2002-02-25T00:00:00Z
review text: One of the central topics of classical differential geometry in the 19th century was the transformation theory of surfaces, the best known example being the Bäcklund transformation of constant negative curvature surfaces in Euclidean space, developed by Bianchi and Bäcklund. Nowadays transformations for solutions to a soliton equation which add solitons to a given solution are generally called Bäcklund transformations. The construction of multisoliton solutions from the vacuum solution by using Bäcklund transformations are called the direct method. Recently \textit{I. Sterling} and \textit{H. Wente} [Indiana J. Math. 42, 1239-1266 (1993; Zbl 0803.53009)] studied Bäcklund transformations of constant mean curvature (CMC) surfaces. Further, \textit{H. Muto} [``A dressing up procedure for a constant mean curvature surface'', in Ohnita (Ed.) Workshop on Harmonic Maps and Related Topics, Tokyo Metropolitan University, 41-48 (1995)] obtained another formulation of such Bäcklund transformations, by reformulating the direct method as a transformation on framings and obtaining what is refered to as the Darboux transformation of CMC surfaces. In the paper under review, the author studies Bäcklund and Darboux transformations for timelike CMC surfaces in Minkowski 3-space with real distinct principal curvatures. Since the induced metric of a timelike surface is indefinite, Bäcklund and Darboux transformations are more complicated than in the case of CMC surfaces in Euclidean space. In particular, there exist four kinds of Bäcklund transformations. The paper is organized as follows. In Section 1 the author deals with extended framings in order to find special framings for timelike CMC surfaces, which will be useful to study such immersions via the Darboux transformation theory. This is done by using the \(2\times 2\) matrix formalism of the Gauss-Codazzi equations introduced by the author in his previous paper [Tokyo J. Math. 21, 141-152 (1998; Zbl 0930.53042)]. Section 2 is devo…
reviewed by: Luis J. Alías
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 37K35
Mathematics Subject Classification ID: 53B30
zbMATH DE Number: 1369698
zbMATH Keywords: timelike surface
zbMATH Keywords: Bäcklund transformations
zbMATH Keywords: Darboux matrix
zbMATH Keywords: loop group
zbMATH Keywords: Darboux transformation
author: Jun-ichi Inoguchi
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1369698