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Helices, Hasimoto Surfaces and Bäcklund Transformations

Published online by Cambridge University Press:  20 November 2018

Thomas A. Ivey
Thomas A. Ivey*
Affiliation:
Department of Mathematics, College of Charleston, Charleston, SC 29464-0001, USA
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Abstract

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Travelling wave solutions to the vortex filament flow generated by elastica produce surfaces in ${{\mathbb{R}}^{3}}$ 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 Bäcklund transformation for constant torsion curves in ${{\mathbb{R}}^{3}}$, previously derived fromthe 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

53A0558F3752C4258A15surfacesfilament flowBäcklund transformations
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 43 , Issue 4 , 01 December 2000 , pp. 427 - 439
Copyright
Copyright © Canadian Mathematical Society 2000

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