2010/06/30 by G. Mari Beffa, Beffa, G. Mari, M. Eastwood +1
Mathematics · Physics and Astronomy · #53A55 #Differential Geometry (math.DG) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Primary 37K10 #Secondary 37K25 #math.DG #msc:37K10 #msc:37K25 #msc:53A55 #nlin.SI
paper · pdf · doi:10.48550/arxiv.1006.5753
33 pages
arxiv created 2010/06/30 · arxiv updated 2010/07/01
In this paper we relate the geometric Poisson brackets on the Grassmannian of 2-planes in R4 and on the (2,2) Moebius sphere. We show that, when written in terms of local moving frames, the geometric Poisson bracket on the Moebius 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 into either a decoupled system or a complexly coupled system of KdV equations, depending on the character of the invariants. We also show that the biHamiltonian Grassmannian geometric brackets are equivalent to the non-commutative KdV biHamiltonian structure. Both integrable systems and Hamiltonian structure can be brought back to the conformal sphere.