  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.
