Surfaces of constant negative curvature in Galilean space (Q1381662)

Simulating the well-known geometrical theory of the sine-Gordon equation, the article is concerned with an analogous interpretation of the Klein-Gordon equation \[ \psi_{tt}- \psi_{vv}= m^2\psi\tag{\(*\)} \] by means of the curvature radius of certain special curves lying on surfaces of constant curvature in the Galilean space. In more detail, the three-dimensional projective space equipped with a plane, a line \(T\) in this plane, and two conjugate points in \(T\) (which together induce a projective metric) are employed. Then canonical coordinates can be defined on a generic surface: curves \(t=\text{const.}\) are intersections with the planes of the pencil with the axis \(T\), and \(v=\text{const.}\) are determined by enveloping cones with vertices on \(T\). If the surface is of constant curvature, then the curvature radius of certain special curves on the surface (namely of the curves such that the tangents intersect \(T\)) satisfies the equation \((*)\). The main part of the article deals with the inverse problem: to obtain the constant curvature surface if a solution of \((*)\) is known. On this occasion, an analog of the Bianchi-Lie transformation is discovered.