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In this paper we describe a family of compatible Poisson structures defined on the space of coframes (or differential invariants) of curves in flat homogeneous spaces of the form M ≅ (G x R n )/G where G C GL(n,R) is semisimple. This includes Euclidean, affine, special affine, Lorentz, and symplectic geometries. We also give conditions on geometric evolutions of curves in the manifold M so that the induced evolution on their differential invariants is Hamiltonian with respect to our main Hamiltonian bracket.

Poisson geometry of differential invariants of curves in some nonsemisimple homogeneous spaces