Quaternionic representation of the moving frame for surfaces in Euclidean three-space and Lax pair
type: Item
instance of: scholarly article
title: Quaternionic representation of the moving frame for surfaces in Euclidean three-space and Lax pair
zbMATH Open document ID: 1087.53002
DOI: 10.1155/S0161171204310392
author: Paul Bracken
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2005-05-18T00:00:00Z
full work available at URL: https://eudml.org/doc/52347
review text: In the first part of this paper the author reworks the Gauss-Weingarten equations via the moving frames for a Riemann surface parametrized by means of the conformal parameters \((z, \bar z).\) Using suitable compatibility conditions, the Gauss-Codazzi equations are derived in terms of the conformal factor of the metric \(e^u,\) the mean curvature \(H,\) and the quantity \(Q = \langle F_{zz}, N \rangle ,\) with \(N\) being the unit normal and \(F\) the (parametrized) position vector in \(\mathbb R^3\). In the second part of the paper the author establishes a connection between quaternions and surfaces in \(\mathbb R^3,\) which gives rise to a certain identification of the Gauss-Weingarten equations with the Lax representation for Painlevé equations by means of an \(SU(2)\) transformation. As an example, the author considers a generalized Weierstraß representation for a constant-mean-curvature hypersurface and the Enneper minimal surface as special cases of solutions of the above pairing. Several typos, ambiguities, and reliance on other papers make the exposition hard to follow.
reviewed by: Ivko Dimitrić
Mathematics Subject Classification ID: 53A05
Mathematics Subject Classification ID: 35A99
zbMATH DE Number: 2168642
zbMATH Keywords: moving frame of a surface
zbMATH Keywords: conformal parametrization
zbMATH Keywords: Lax representation
zbMATH Keywords: surface with constant mean curvature
zbMATH Keywords: generalized Weierstraß representation
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2168642