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

Abstract

In this paper, by means of characteristic lines a sufficient condition for a surface of positive curvature in 3-dimensional euclidean space to be isometric with a rotation surface is given, i.e. it is shown that if the characteristic lines of such a surface constitute two families of geodesic parallels, then the surface is isometric with a rotation surface. Furthermore, it is proved that the lines of curvature on such surfaces are isothermal-conjugate or, equivalently, the characteristic lines and the lines of curvature form a hexagonal 4-web.