1. CMB 2000 (vol 43 pp. 427)
|Helices, Hasimoto Surfaces and BÃ¤cklund Transformations |
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.
Keywords:surfaces, filament flow, BÃ¤cklund transformations
Categories:53A05, 58F37, 52C42, 58A15