On the construction of integrable surfaces on Lie groups

The problem of the immersion of a two-dimensional surface into a three-dimensional Euclidean space can be formulated in terms of the immersion of surfaces in Lie groups and Lie algebras. A general formalism for this problem is developed, as well as an equivalent Mauer-Cartan system of differential forms. The particular case of the Lie group SU(2) is examined, and it is shown to be useful for studying integrable surfaces. Some examples of such surfaces and their equations are presented at the end, in particular, the cases of constant mean curvature and of zero Gaussian curvature.