Poisson brackets associated to invariant evolutions of Riemannian curves

G. Marí Beffa

Vol. 215 (2004), No. 2, 357–380

DOI: 10.2140/pjm.2004.215.357

Abstract

In this paper we show that Poisson brackets linked to geometric flows of curves on flat Riemannian manifolds are Poisson reductions of the Kac–Moody bracket of SO(n). The bracket is reduced to submanifolds defined by either the Riemannian or the natural curvatures of the curves. We show that these two cases are (formally) Poisson equivalent and we give explicit conditions on the coefficients of the geometric flow guaranteeing that the induced flow on the curvatures is Hamiltonian.

Milestones

Received: 2 December 2002

Revised: 27 October 2003

Published: 1 June 2004