ON DARBOUX LINES

Abstract

  In this paper, we first prove that the only surface (other then a sphere) on which the two families of Darboux lines form a Tschebycheff net and the third family of Darboux lines is transversal to one of the two families of Darboux lines is a cylinder of revolution. We next show that the surface (other than a sphere or a developable surface) on which the Darboux lines correspond to those on its parallel surface is a surface of constant mean curvature. Moreover, if the two families of Darboux lines on a surface of constant mean curvature form a Tschebyscheff net, then a surface becomes a cylinder of revolution or a plane. Furthermore, we prove that surfaces (other than a sphere or a plane) whose Darboux lines are preserved under inversion are Dupin's cyclides or, in particular, a pipe surface of revolution. Finally, we show that Molure surface on which the two families of Darboux lines which are different from the lines of curvature form two semi-Tschebyscheff nets together with a family of lines of curvature are either a surface of revolution or a pipe surface.