Abstract
The moving frame and associated Gauss-Codazzi equations for surfaces in Minkowski three space are introduced. A split-quaternionic representation is used to identify the Gauss-Weingarten equations with a Lax pair representation. This Lax pair representaion is calculated and given explicitly.
Highlights
The study of surfaces in higher-dimensional spaces and in non-Euclidean spaces has been an active area of study recently due to the variety of applications of surfaces to the areas of integrable systems and mathematical physics (Bracken and Grundland, 1999; Bracken et al, 1999)
The Gauss-Codazzi equations for surfaces in Euclidean three-space were established from a three by three matrix representation and a quaternionic representation was introduced for the moving frame of the conformally parametrized surface (Konopelchenko and Taimanov, 1996)
It will be shown that a Lax pair can be derived for the Gauss-Codazzi equations by using a representation of the split-quaternions
Summary
The study of surfaces in higher-dimensional spaces and in non-Euclidean spaces has been an active area of study recently due to the variety of applications of surfaces to the areas of integrable systems and mathematical physics (Bracken and Grundland, 1999; Bracken et al, 1999). The Gauss-Codazzi equations for surfaces in Euclidean three-space were established from a three by three matrix representation and a quaternionic representation was introduced for the moving frame of the conformally parametrized surface (Konopelchenko and Taimanov, 1996). It will be shown that a Lax pair can be derived for the Gauss-Codazzi equations by using a representation of the split-quaternions
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