Surfaces of Constant negative Scalar Curvature and the Correpondence between the Liouvulle and the sine-Gordon Equations

By studying the {\it internal} Riemannian geometry of the surfaces of constant negative scalar curvature, we obtain a natural map between the Liouville, and the sine-Gordon equations. First, considering isometric immersions into the Lobachevskian plane, we obtain an uniform expression for the general (locally defined) solution of both the equations. Second, we prove that there is a Lie-B\"acklund transformation interpolating between Liouville and sine-Gordon. Third, we use isometric immersions into the Lobachevskian plane to describe sine-Gordon N-solitons explicitly.