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The bi-Hamiltonian structure of the two known vector generalizations of themKdV hierarchy of soliton equations is derived in a geometrical fashion fromflows of non-stretching curves in Riemannian symmetric spaces G/SO(N). Thesespaces are exhausted by the Lie groups G=SO(N+1),SU(N). The derivation of thebi-Hamiltonian structure uses a parallel frame and connection along the curves,tied to a zero curvature Maurer-Cartan form on G, and this yields the vectormKdV recursion operators in a geometric O(N-1)-invariant form. The kernel ofthese recursion operators is shown to yield two hyperbolic vectorgeneralizations of the sine-Gordon equation. The corresponding geometric curveflows in the hierarchies are described in an explicit form, given by wave mapequations and mKdV analogs of Schrodinger map equations.

Hamiltonian Flows of Curves in G/SO(N) and Vector Soliton Equations of mKdV and Sine-Gordon Type