Helices, Hasimoto Surfaces and Bäcklund Transformations

Travelling wave solutions to the vortex filament flow generated by elastica produce surfaces in R 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 R, previously derived from the well-known transformation for pseudospherical surfaces. A similar investigation for surfaces in H or S leads to a new transformation for constant torsion curves in those spaces that is also derived from pseudospherical surfaces.