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We relate the geometric Poisson brackets on the 2-Grassmannian in ℝ4 and on the (2, 2) Möbius sphere. We show that, when written in terms of local moving frames, the geometric Poisson bracket on the Möbius sphere does not restrict to the space of differential invariants of Schwarzian type. But when the concept of conformal natural frame is transported from the conformal sphere into the Grassmannian, and the Poisson bracket is written in terms of the Grassmannian natural frame, it restricts and results in either a decoupled system or a complexly coupled system of Korteweg–de Vries (KdV) equations, depending on the character of the invariants. We also show that the bi-Hamiltonian Grassmannian geometric brackets are equivalent to the non-commutative KdV bi-Hamiltonian structure. Both integrable systems and Hamiltonian structure can be brought back to the conformal sphere. [ABSTRACT FROM PUBLISHER]

In this paper we introduce a new infinite-dimensional pencil of Hamiltonian structures. These Poisson tensors appear naturally as the ones governing the evolution of the curvatures of certain flows of curves in 3-dimensional Riemannian manifolds with constant curvature. The curves themselves are evolving following arclength-preserving geometric evolutions for which the variation of the curve is an invariant combination of the tangent, normal, and binormal vectors. Under very natural conditions, the evolution of the curvatures will be Hamiltonian and, in some instances, bi-Hamiltonian and completely integrable. [ABSTRACT FROM AUTHOR]