AN ELEMENTARY GEOMETRIC CHARACTERIZATION OF THE INTEGRABLE MOTIONS OFA CURVE

We show that the following elementary geometric properties of the moti on of a curve select hierarchies of integrable dynamics: (i) the curve moves in an N-dimensional sphere of radius R; (ii) the motion is nons tretching; (iii) the dynamics does not depend explicitly on the radius of the sphere. For N = 2 we obtain the modified Korteweg-de Vries hie rarchy, for N = 3 the nonlinear Schrodinger hierarchy and for N > 3 we obtain integrable multicomponent generalizations of the above hierarc hies.