A Lie algebra structure on variation vector fields along curves in $2$-dimensional space forms

Abstract

A Lie algebra structure on variation vector fields along an immersed curve in a $2$-dimensional real space form is investigated. This Lie algebra particularized to plane curves is the cornerstone in order to define a Hamiltonian structure for plane curve motions. The Hamiltonian form and the integrability of the planar filament equation is finally discussed from this point of view.

arXiv comment14 pages

References in corpus (3)

• Motion of Curves on Two Dimensional Surfaces and Soliton Equations
• Bi-Hamiltonian operators, integrable flows of curves using moving frames, and geometric map equations
• Differential Geometry of the Vortex Filament Equation

Cited by in corpus (1)

• Integrability aspects of the vortex filament equation for pseudo-null curves