Projective-type differential invariants and geometric curve evolutions of KdV-type in flat homogeneous manifolds

In this paper we describe moving frames and differential invariants for curves in two different |1|-graded parabolic manifolds G/H, G = O(p + 1, q + 1) and G = O(2m, 2m), and we define differential invariants of projective-type. We then show that, in the first case, there are geometric flows in G/H inducing equations of KdV-type in the projective-type differential invariants when proper initial conditions are chosen. We also show that geometric Poisson brackets in the space of differential invariants of curves in G/H can be reduced to the submanifold of invariants of projective-type to become Hamiltonian structures of KdV-type. The study is based on the use Fels and Olver moving frames. In the second case we classify differential invariants and we show that for some choices of moving frames we can find geometric evolutions inducing a decoupled system of KdV equations on the projectivetype differential invariants, if proper initial values are chosen. We describe the differences between this case and the Lagrangian Grassmannian case in detail.