Integration of the two-dimensional Heisenberg model by methods of differential geometry

The methods of classical differential geometry are used to integrate the two-dimensional Heisenberg model. After the hodograph transformation, the model equations are written in terms of the metric tensor associated with a curvilinear coordinate system and its derivatives. It is shown that their general solution describes all previously known exact solutions except a flat vortex. A new type of vortex structure, a “vortex strip”, is predicted and analyzed in two-dimensional ferromagnets. Its typical properties are the finite dimensions of the domain of definition, the finiteness of the total energy, and the absence of a vortex core in the presence of a vortex structure.