SURFACES ON LIE-GROUPS, ON LIE-ALGEBRAS, AND THEIR INTEGRABILITY

It is shown that the problem of the immersion of a 2-dimensional surfa ce into a 3-dimensional Euclidean space, as well as the n-dimensional generalization of this problem, is related to the problem of studying surfaces in Lie groups and surfaces in Lie algebras. A particular case of the general formalism presented here implies that any surface can be characterized in terms of 2 x 2 matrices using an arbitrary paramet rization. It is also shown that this generality of parametrization is useful for studying integrable surfaces, i.e. surfaces described by in tegrable equations. In particular starting from a suitable Lax pair (i .e. a suitable integrable equation), it is possible to construct expli citly large classes of integrable surfaces.