Helices, Hasimoto Surfaces and Bäcklund Transformations

Abstract

AbstractTravelling wave solutions to the vortex filament flow generated by elastica produce surfaces in ℝ3 that carry mutually orthogonal foliations by geodesics and by helices. These surfaces are classified in the special cases where the helices are all congruent or are all generated by a single screw motion. The first case yields a new characterization for the Bäcklund transformation for constant torsion curves in R3, previously derived fromthe well-known transformation for pseudospherical surfaces. A similar investigation for surfaces in H3 or S3 leads to a new transformation for constant torsion curves in those spaces that is also derived from pseudospherical surfaces.