The symplectic structure of curves in three dimensional spaces of constant curvature and the equations of mathematical physics

TL;DR

This paper introduces a symplectic structure on the space of framed curves in constant curvature spaces and connects Hamiltonian systems derived from geometric invariants to equations in mathematical physics.

Contribution

It defines a symplectic form on the manifold of curves and links Hamiltonian systems from geometric invariants to physical equations.

Findings

Symplectic form established on the space of framed curves.

Hamiltonian systems related to geometric invariants.

Connections made to equations of mathematical physics.

Abstract

This paper defines a symplectic form on the infinite dimensional Fr\'echet manifold of framed curves of fixed length over a simply connected Riemannian manifold of constant curvature. The paper then considers Hamiltonian systems generated by the geometric invariants of the curves on the base manifold and relates them to the equations of mathematical physics.