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

In this paper we describe a family of compatible Poisson structures defined on the space of coframes (or differential invariants) of curves in. at homogeneous spaces of the form M congruent to (G x R-n)/G where G subset of 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.