Bäcklund transformation for the Liouville equation and gauge conditions in the theory of a relativistic string

It is shown that the gauge conditions in the theory of a relativistic string, which make it possible to replace the nonlinear Liouville equation by the d'Alembert equation, are a direct consequence of the Bäcklund transformation relating the solutions of these equations. A purely geometrical derivation is given of the Bäcklund transformations for the Liouville equation. A classical theory of a relativistic string is constructed in the $t=\tau$ gauge using the moving frame formalism and exterior differential forms in the theory of surfaces. The moving frame on the string trajectory is chosen in a special way. As a result, the theory of a string in fourdimensional space-time reduces to the d'Alembert equation for a single scalar function.