Geometric Poisson brackets on Grassmannians and conformal spheres (Q2895827)

This paper studies the geometric Poisson brackets of flat Grassmannian of two dimensional planes in \(\mathbb R^4\). The existence of integrable systems and geometric flows have been extensively studied in the literature. In this work, the authors concentrate on moving frames for curves in Grassmannian manifolds. They prove that the geometric Poisson brackets on the 2-Grassmannian in \(\mathbb R^4\) are linked to the Möbius sphere. They also prove that the bi-Hamiltonian geometric structure on the complete Grassmannian is equal to the non-commutative KdV bi-Hamiltonian structure. They finally show that the non-commutative KdV equation has a Grassmannian geometric realization.