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

José del Amor; Ángel Giménez; Pascual Lucas

doi:10.1016/j.geomphys.2014.11.010

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arXiv · 1404.0524

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

José del Amor·Ángel Giménez·Pascual Lucas

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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.

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José del Amor, Ángel Giménez, Pascual Lucas. 2014-04-02. A Lie algebra structure on variation vector fields along curves in $2$-dimensional space forms. https://doi.org/10.1016/j.geomphys.2014.11.010

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