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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 surfaces, filament flow, Bäcklund transformations
MSC Classifications: 53A05, 58F37, 52C42, 58A15 show english descriptions Surfaces in Euclidean space
unknown classification 58F37
unknown classification 52C42
Exterior differential systems (Cartan theory)
53A05 - Surfaces in Euclidean space
58F37 - unknown classification 58F37
52C42 - unknown classification 52C42
58A15 - Exterior differential systems (Cartan theory)
 

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