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

摘要

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