Geometric Poisson brackets on Grassmannians and conformal spheres

Abstract

We relate the geometric Poisson brackets on the 2-Grassmannian in R 4 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.

Quantization of the conformal arclength functional on space curves

“…Further developments of the subject as an instance of the conformal geometry of submanifolds can be found in [2] and the literature therein. The subject has also received much attention for its many fields of application, including the theory of integrable systems [5,8], the topology and Möbius energy of knots [1,11,18], and the geometric approach to shape analysis and medical imaging [34].…”

Conformal Geometry of Arc length Functional on Space Curves and Space time

“…The conformal geometry of space curves was mainly developed in the first half of the past century and later taken up starting from the early 1980's. This subject has got much attention for its many fields of application, including the theory of integrable systems [3], [12], [8], topology and M• bius energy of knots [2], [6], [9], and the geometric approach to shape analysis and medical imaging [13]. Suppose that ⊂ , ≥ 3, be a smooth curve parameterized by arclength s. The conformal arclength parameter of is defined by = , − , 2 1 4 =: , where , is the standard scalar product on and , stands for double and triple derivative of .…”

Closed trajectories of the conformal arclength functional

“…In more recent times, the topic has been considered in connection with the regularization of the Kepler problem [6,9,14], within the theory of integrable systems and in the topology of knots [3,1,2,5,8,10,11]. In a previous paper, published several years ago [12], I studied the variational problem defined by the conformal density of a space curve, improperly called conformal arc-element.…”