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

Abstract

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 ⋉ R n ) / G \mathcal {M} \cong (G\ltimes \mathbb {R}^n)/G where G ⊂ G L ( n , R ) 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 M \mathcal {M} so that the induced evolution on their differential invariants is Hamiltonian with respect to our main Hamiltonian bracket.