Note on the geometry of the nonlinearσmodel in two dimensions

It is pointed out that, in two space-time dimensions, the classical nonlinear $\sigma$ model may be studied as the embedding of a two-dimensional surface in a three-dimensional sphere which is itself embedded in four-dimensional Euclidean space. The Gauss-Weingarten equations provide then a natural way of associating a linear eigenvalue problem in terms of representations of O(4) with a set of nonlinear evolution equations having solitary wave solutions.