Helices, Hasimoto Surfaces and Bäcklund Transformations

http://dx.doi.org/10.4153/CMB-2000-051-9
Canad. Math. Bull. 43(2000), 427-439

  Published:2000-12-01
 Printed: Dec 2000

Travelling wave solutions to the vortex filament flow generated by elastica produce surfaces in $\R^3$ 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 B\"acklund transformation for constant torsion curves in $\R^3$, previously derived from the well-known transformation for pseudospherical surfaces. A similar investigation for surfaces in $H^3$ or $S^3$ leads to a new transformation for constant torsion curves in those spaces that is also derived from pseudospherical surfaces.