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# Helices, Hasimoto Surfaces and Bäcklund Transformations

Published:2000-12-01
Printed: Dec 2000
• Thomas A. Ivey
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## Abstract

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, Bäcklund transformations
 MSC Classifications: 53A05 - Surfaces in Euclidean space 58F37 - unknown classification 58F3752C42 - unknown classification 52C4258A15 - Exterior differential systems (Cartan theory)