Integrable systems and invariant curve flows in symplectic Grassmannian space

In this paper, local geometry of curves in the symplectic Grassmannian homogeneous space Sp(4,R)/(Sp(2,R)×Sp(2,R)) and its connection with that of the pseudo-hyperbolic space H2,2 are studied. The group-based Serret-Frenet equations and the associated Maurer-Cartan differential invariants for the Grassmannian curves are obtained by using the equivariant moving frame method. The Grassmannian natural frame is also constructed by a gauge transformation from the Serret-Frenet frame, relating to the hyperbolic natural frame by the local Lie group isomorphism. Using the natural frames, invariant curve flows in the Grassmannian and the hyperbolic spaces are studied. It is shown that certain intrinsic curve flows induce the bi-Hamiltonian integrable matrix mKdV equation on the Maurer-Cartan differential invariants.