Helices, Hasimoto Surfaces and Bäcklund Transformations

Ivey, Thomas A.

Canadian mathematical bulletin, Tome 43 (2000) no. 4, pp. 427-439

Résumé

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.

Détail

DOI : 10.4153/CMB-2000-051-9

Mots-clés : 53A05, 58F37, 52C42, 58A15, surfaces, filament flow, Bäcklund transformations

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