Shadow problem and isometric embeddings of pseudospherical surfaces

The shadow problem for horospheres is related to the problem of global isometric embedding of surfaces of revolution of constant negative curvature into the three-dimensional Euclidean space. Euclidean surfaces of revolution of constant negative curvature are globally isometric to parts of tangent cones of horospheres in the three-dimensional Lobachevsky space. In this work, meridians of Euclidean pseudospherical surfaces of revolution are expressed in terms of metric characteristics in the hyperbolic space, namely, in terms of the distance from the vertex of the tangent cone to the horosphere or through the distance from the polar of the vertex to the horosphere.