On surfaces with a prescribed curvilinear projection of one field of principal directions

A class of surfaces-graphs in a Riemannian 3-space with a prescribed projection of one field of principal directions onto a surface $\Pi$ is considered. A problem of determination of such surfaces when both principal curvatures are given over a line in $\Pi$ is formulated and studied. The geometric problem is reduced to the Cauchy problem for quasilinear PDE's which, under certain conditions for data, are hyperbolic and admit a unique solution. It is shown that the parallel curved (PC) surfaces in space forms provide a special class of global solutions to the geometrical problem with weaker regularity assumptions. Such solutions may be found by an iteration function sequence.