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

TL;DR

This paper studies surfaces in 3D space with a prescribed projection of one principal direction, formulating a PDE-based problem that admits unique solutions under certain conditions, and identifies a special class of solutions called PC surfaces.

Contribution

It formulates a geometric problem involving prescribed principal direction projections and reduces it to a solvable PDE Cauchy problem, highlighting a class of global solutions in space forms.

Findings

The problem reduces to hyperbolic quasilinear PDEs with unique solutions under certain data conditions.

Parallel curved (PC) surfaces form a special class of global solutions.

Solutions can be constructed iteratively under weaker regularity assumptions.

Abstract

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.