Geometric Hamiltonian Structures on Flat Semisimple Homogeneous Manifolds

Abstract

Abstract. In this paper we describe Poisson structures defined on the space of Serret-Frenet equations of curves in a flat homogeneous space G/H where G is semisimple. These structures are defined via Poisson reduction from Poisson brackets on Lg * , the space of Loops in g * . We also give conditions on invariant geometric evolution of curves in G/H which guarantee that the evolution induced on the differential invariants is Hamiltonian with respect to the most relevant of the Poisson brackets. Along the way we prove that differential invariants of curves in semisimple flat homogeneous spaces have order equal to 2 or higher, and we also establish the relationship between classical moving frames (a curve in the frame bundle) and group theoretical moving frames (equivariant G-valued maps on the jet space).

Geometric Poisson brackets on Grassmannians and conformal spheres

“…For a given choice of normalization equations, let us denote the space of Maurer-Cartan matrices by K. Marí Beffa [14] showed that, locally around a generic curve u(x) ∈ G/H with G semisimple, the space K can be written as a quotient U/LH, where U ⊂ Lg * is open. The elements δH/δL, δF/δL ∈ g are the variational derivatives at L and ·, · is the invariant pairing of g with g * .…”

“…The following theorem can be found in [14]. Perhaps the most interesting part is the direct relation between invariant evolutions and evolutions that are Hamiltonian with respect to the reduction of (4.1).…”

“…where δH(K)/δL is calculated similarly to the variational derivative of F. In [14] Marí Beffa proved that this bracket is always a Poisson bracket. To show that the bracket can be further restricted to the submanifold κ 3 = κ 4 = 0, we shall check that, if f depends only on k 1 and k 2 (that is, f 3 = f 4 = 0), while h depends only on κ 3 and κ 4 (that is, h 2 = h 1 = 0), then {f, h}(k) = 0 along the submanifold κ 3 = κ 4 = 0.…”

“…Given a flat homogeneous space, one can define a Hamiltonian structure on the space of differential invariants (curvatures) of parametrized curves [14]. These structures are often linked to completely integrable partial differential equations and to their geometric realizations (invariant curve flows on the homogeneous space inducing the integrable system on its invariants).…”

On characteristic integrals of Toda field theories

“…In sequels to this work, Anderson and the author will apply the characteristic integrals found in this paper and the structure theory in [1] to realize the Toda field theory (1.3) as the quotient of two standard differential systems [20] depending on x and y separately. We will also apply the method and result of this paper to the setting of differential invariants for parabolic geometry [4] with the standard differential system, generalizing works done by Mari Beffa [17]. This paper is organized as follows.…”