Surfaces of Constant negative Scalar Curvature and the Correpondence between the Liouville and the sine–Gordon Equations

By studying the 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– Bäcklund transformation interpolating between Liouville and sine–Gordon. Third, we use isometric immersions into the Lobachevskian plane to describe sine– Gordon N–solitons explicitly. Corresponding author E–mail address [email protected]