Differential geometry and nonlinear field models

Usage of the differential geometry methods in the theory of two-dimensional nonlinear field models has been reviewed. Principal attention is drawn to the theorems of Rieman manifold inclusion and to construction of the Lax operators for nonlinear equations defining the quadratic shape of a surface. The geometric method is used in the theory of a two-dimensional sigma-model with O(3) and O(4) symmetry, and also in the relativistic string theory in the three- and four-dimensional space-time. A possibility of solving the arising nonlinear equations by the inverse scattering problem method is discussed. Considered are soliton solutions in these models and their contribution to the mass spectrum in the quasiclassical approximation. Generalization of the string model in the framework of the geometric approach is discussed