International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 15, Pages 755-762

Quaternionic representation of the moving frame for surfaces in Euclidean three-space and Lax pair

Paul Bracken

Department of Mathematics, University of Texas – Pan American, Edinburg 78541-2999, TX, USA

Received 5 October 2003; Revised 20 October 2003

Copyright © 2004 Paul Bracken. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The moving frame and associated Gauss-Codazzi equations for surfaces in three-space are introduced. A quaternionic representation is used to identify the Gauss-Weingarten equation with a particular Lax representation. Several examples are given, such as the case of constant mean curvature.