Surfaces of positive curvature whose characteristic lines constitute two families of geodesic parallels

Abstract

Upon a surface of positive curvature there is a unique conjugate system for which the angle between the directions at any point is the minimum angle between conjugate directions at the point; it is the only conjugate system whose directions are symmetric with respect to the directions of the lines of curvature. In this paper the authors prove: Theorem 1. Surfaces of positive curvature in \(E_ 3\) on which the five families of characteristic lines are geodesic parallels are isometric with rotation surfaces. Theorem 2. A necessary condition for the characteristic lines of a surface \(S\) of positive curvature to be geodesic parallels is that its lines of curvature form an isothermal-conjugate net or, equivalently, its characteristic lines and the lines of curvature form a hexagonal 4- web.