A smooth curve gamma in Rn+1, n is isotropic if gamma, gamma(x),..., gamma((2 n))(x) are linearly independent and the span of gamma, gamma(x),..., gamma((n-1))(x) is isotropic. We construct two hierarchies of isotropic curve flows on R-n+1,R-n, whose differential invariants are solutions of Drinfeld-Sokolov's KdV type soliton hierarchies associated to the affine Kac-Moody algebra (B) over cap ((1))(n) and (A) over cap ((2))(2n). For example, the (B) over cap ((1))(1)-KdV is the KdV hierarchy and the (A) over cap ((2))(2)-KdV hierarchy is the Kupershmidt-Kaup (KK) hierarchy. Hence we our study gives geometric interpretations of the KdV and KK equations as the curvature flows of natural geometric curve flows on the light cone of R-2,R-1. Bi-Hamiltonian structures and conservation laws for isotropic curve flows on R-n+1,R-n are also given.
展开▼
