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 flat homogeneous spaces of the form $\mathcal{M} \cong (G\ltimes\mathbb{R} ^n)/G$ where $G\subset {\mathrm{GL}}(n,\mathbb{R} )$ is semisimple. This includes Euclidean, affine, special affine, Lorentz, and symplectic geometries. We also give conditions on geometric evolutions of curves in the manifold $\mathcal{M}$ so that the induced evolution on their differential invariants is Hamiltonian with respect to our main Hamiltonian bracket.