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Geometric Hamiltonian Structures on Flat Semisimple Homogeneous Manifolds

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).