AN ELEMENTARY GEOMETRIC CHARACTERIZATION OF THE INTEGRABLE MOTIONS OF A CURVE

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