Geometric Hamiltonian Structures on Flat Semisimple Homogeneous Manifolds

G. Mari Beffa
2008 Asian Journal of Mathematics  
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
more » ... 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).
doi:10.4310/ajm.2008.v12.n1.a1 fatcat:xnfhssxvpvaalfoybnx6hr4dmm