Projective-type differential invariants and geometric curve evolutions of KdV-type in flat homogeneous manifolds
A new concept of moving frame was introduced by Fels and Olver in 1999--2000. Instead of the classical concept of moving frame along a curve, they looked at the moving frame as an equivariant map. A fundamental problem with the traditional moving frame was that in non-affine geometries, the classical Serret-Frenet equation did not provide a complete set of generators of differential invariants for the curve. In the new setting, however, a complete set of generators can always be found among the entries of the matrix defining its Serret-Frenet equations. In this paper, the author describes moving frames and differential invariants for curves in two different parabolic manifolds \(G/H\), where \(G= O(p+1,q+1)\) and \(G= O(2m,2m)\). He shows that, in the first case, there are geometric flows inducing equations of the KdV type in the projective-type differential invariants when proper initial conditions are chosen. In the second case, he classifies differential invariants and shows that for some choices of moving frames, one can find geometric evolutions inducing a decoupled system of KdV equations on the projective-type differential invariants. The differences between this case and the Lagrangian Grassmannian case are also described in detail.
scientific article; zbMATH DE number 5303676