  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.
