Poisson brackets associated to the conformal geometry of curves

In this paper we present an invariant moving frame, in the group theoretical sense, along curves in the Möbius sphere. This moving frame will describe the relationship between all conformal differential invariants for curves that appear in the literature. Using this frame we first show that the Kac-Moody Poisson bracket on can be Poisson reduced to the space of conformal differential invariants of curves. The resulting bracket will be the conformal analogue of the Adler-Gel'fand-Dikii bracket. Secondly, a conformally invariant flow of curves induces naturally an evolution on the differential invariants of the flow. We give the conditions on the invariant flow ensuring that the induced evolution is Hamiltonian with respect to the reduced Poisson bracket. Because of a certain parallelism with the Euclidean case we study what we call Frenet and natural cases. We comment on the implications for completely integrable systems, and describe conformal analogues of the Hasimoto transformation.

     

Poisson geometry of differential invariants of curves in some nonsemisimple homogeneous spaces

In this paper we describe a family of compatible Poisson structures defined on the space of coframes (or differential invariants) of curves in flat homogeneous spaces of the form where is semisimple. This includes Euclidean, affine, special affine, Lorentz, and symplectic geometries. We also give conditions on geometric evolutions of curves in the manifold so that the induced evolution on their differential invariants is Hamiltonian with respect to our main Hamiltonian bracket.

     

Discrete Moving Frames and Discrete Integrable Systems

Group-based moving frames have a wide range of applications, from the classical equivalence problems in differential geometry to more modern applications such as computer vision. Here we describe what we call a discrete group-based moving frame, which is essentially a sequence of moving frames with overlapping domains. We demonstrate a small set of generators of the algebra of invariants, which we call the discrete Maurer–Cartan invariants, for which there are recursion formulas. We show that this offers significant computational advantages over a single moving frame for our study of discrete integrable systems. We demonstrate that the discrete analogues of some curvature flows lead naturally to Hamiltonian pairs, which generate integrable differential-difference systems. In particular, we show that in the centro-affine plane and the projective space, the Hamiltonian pairs obtained can be transformed into the known Hamiltonian pairs for the Toda and modified Volterra lattices, respectively, under Miura transformations. We also show that a specified invariant map of polygons in the centro-affine plane can be transformed to the integrable discretization of the Toda Lattice. Moreover, we describe in detail the case of discrete flows in the homogeneous 2-sphere and we obtain realizations of equations of Volterra type as evolutions of polygons on the sphere.     

On the Poisson geometry of the Adler-Gel’fand-Dikii brackets

In this paper, we consider a symplectic leaf that goes through a singular point of the Adler-Gel’fand-Dikii Poisson bracket associated to SL(n,R). We find a finite-dimensional transverse section2 at the singular point and we prove that one can induce a Poisson structure on2 (the transverse structure) that is linearizable and equivalent to the Lie-Poisson structure on sl(n,R)*. This problem is closely related to finding normal forms for nth order scalar differential operators with periodic coefficient. We partially generalize a well-known result for Hill’s operators to the higher order case.     

On Generalizations of the Pentagram Map: Discretizations of AGD Flows

In this paper we investigate discretizations of AGD flows whose projective realizations are defined by intersecting different types of subspace in $\mathbb{RP}^{m}$ . These maps are natural candidates to generalize the pentagram map, itself defined as the intersection of consecutive shortest diagonals of a convex polygon, and a completely integrable discretization of the Boussinesq equation. We conjecture that the r-AGD flow in m dimensions can be discretized using one (r?1)-dimensional subspace and r?1 different (m?1)-dimensional subspaces of $\mathbb{RP}^{m}$ .     

Integrable Systems in Three-Dimensional Riemannian Geometry

Summary. 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. Received May 31, 2001; accepted January 2, 2002 Online publication March 11, 2002 Communicated by A. Bloch Communicated by A. Bloch rid="     

Poisson structures for geometric curve flows in semi-simple homogeneous spaces

We apply the equivariant method of moving frames to investigate the existence of Poisson structures for geometric curve flows in semi-simple homogeneous spaces. We derive explicit compatibility conditions that ensure that a geometric flow induces a Hamiltonian evolution of the associated differential invariants. Our results are illustrated by several examples of geometric interest.     

Transverse sections for the Second Hamiltonian KdV Structure

In this paper we study regular and singular operators for the Second KdV Hamiltonian Structure. We prove that at some point of any regular symplectic leaf, one can move transversally to the leaf by simply adding constants to the coefficients of the operator. We also prove that given any leaf, one can move transversally to the leaf at some point by adding certain trigonometric polynomials to the coefficient of the operator. We discuss the implications of this result for the Poisson geometry of Adler-Gel’fand-Dikii manifolds and for the normal forms of scalar operators with periodic coefficients.     

Zur Frage der Koordinationsweise von Chromkomplexen aus dem Azofarbstoff 1-Amino-2-hydroxy-naphtalin-4-sulfonsäurepiperidid →→ 1-Phenyl-3-methyl-5-pyrazolon

On chroming the dyestuff 1-amino-2-hydroxy-naphthalene-4-sulfonic acid piperidide → 1-phenyl-3-methyl-5-pyrazolone, only one 1:2 chromium complex is formed. This fact as well as the absorption spectra and the great stability of the complex indicate that the complex must be coordinated in the Drew-Pfitzner arrangement, and that the sandwich arrangement must be excluded. Since the steric structure of the 1:2-complex is already preformed in the adequate 1:1-complex, our results disprove the conclusions presented by IDELSON et al. [1] [3].