Bi-Hamiltonian flows and their realizations as curves in real semisimple homogeneous manifolds

Gloria Marí Beffa

Vol. 247 (2010), No. 1, 163–188

DOI: 10.2140/pjm.2010.247.163

Abstract

We describe a reduction process that allows us to define Hamiltonian structures on the manifold of differential invariants of parametrized curves for any homogeneous manifold of the form G∕H, with G semisimple. We also prove that equations that are Hamiltonian with respect to the first of these reduced brackets automatically have a geometric realization as an invariant flow of curves in G∕H. This result applies to some well-known completely integrable systems. We study in detail the Hamiltonian structures associated to the sphere SO(n + 1)∕SO(n).

Keywords

invariant curve evolutions, Poisson structures, differential invariants, moving frames, homogeneous spaces

Mathematical Subject Classification 2000

Primary: 37K10, 37K25, 37K30

Secondary: 53A55

Milestones

Received: 3 December 2008

Accepted: 18 February 2010

Published: 1 July 2010