  Moving frames of various kinds are used to derive bi-Hamiltonian operators
and associated hierarchies of multi-component soliton equations from
group-invariant flows of non-stretching curves in constant curvature manifolds
and Lie group manifolds. The hierarchy in the constant-curvature case consists
of a vector mKdV equation coming from a parallel frame, a vector potential mKdV
equation coming from a covariantly-constant frame, and higher order
counterparts generated by an underlying vector mKdV recursion operator. In the
Lie-group case the hierarchy comprises a group-invariant analog of the vector
NLS equation coming from a left-invariant frame, along with higher order
counterparts generated by a recursion operator that is like a square-root of
the mKdV one. The corresponding respective curve flows are found to be given by
geometric nonlinear PDEs, specifically mKdV and group-invariant analogs of
Schrodinger maps. In all cases the hierarchies also contain variants of vector
sine-Gordon equations arising from the kernel of the respective recursion
operators. The geometric PDEs that describe the corresponding curve flows are
shown to be wave maps. Full details of these results are presented for two main
cases: $S^2,S^3\simeq SU(2)$.
