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The deep geometrical relationships holding among the NLS equation, the vortexfilament equation,the Heisenberg spin model, and the Schrodinger map equationare extended to the general setting of Hermitian symmetric spaces. New resultsare obtained by utilizing a generalized Hasimoto variable which arises fromapplying the general theory of parallel moving frames. The example of complexprojective space CP^N= SU(N+1)/U(N) is used to illustrate the method andresults.

Hasimoto variables, generalized vortex filament equations, Heisenberg models and Schrodinger maps arising from group-invariant NLS systems