Deformation of surfaces, integrable systems, and Chern–Simons theory

A few years ago, some of us devised a method to obtain integrable systems in (2+1)-dimensions from the classical non-Abelian pure Chern–Simons action via the reduction of the gauge connection in Hermitian symmetric spaces. In this article we show that the methods developed in studying classical non-Abelian pure Chern–Simons actions can be naturally implemented by means of a geometrical interpretation of such systems. The Chern–Simons equation of motion turns out to be related to time evolving two-dimensional surfaces in such a way that these deformations are both locally compatible with the Gauss–Mainardi–Codazzi equations and completely integrable. The properties of these relationships are investigated together with the most relevant consequences. Explicit examples of integrable surface deformations are displayed and discussed.