O(3) nonlinear σ-model and pseudospherical surface

Abstract

The authors relate the dual symmetry of the chiral field to the dual symmetry of the first and second forms on pseudospherical surfaces in asymptotic coordinates. Every given (up to conformal similarity) solution of chiral field on O(3)/O(2) sphere has been put into one-to-one correspondence with the Gauss image of a definite (up to homothetic transformations) pseudospherical surface. They establish a gauge covariant formulation which unifies the chiral field and the differential forms on the pseudospherical surface. Then they get the explicit geometrical pictures and the covariant relations for Backlund transformations, Riccati equations and an infinite number of conserved currents in both cases. The SO(3) and SO(2,1) sigma -model are also related by dual symmetry.