Motions of Curves in the Projective Plane Inducing the Kaup-Kupershmidt Hierarchy

The equation of a motion of curves in the projective plane is deduced. Local flows are defined in terms of polynomial differential functions. A family of local flows inducing the Kaup-Kupershmidt hierarchy is constructed. The integration of the congruence curves is discussed. Local motions defined by the traveling wave cnoidal solutions of the fifth-order Kaup-Kupershmidt equation are described

Symplectic Applicability of Lagrangian Surfaces

We develop an approach to affine symplectic invariant geometry of Lagrangian surfaces by the method of moving frames. The fundamental invariants of elliptic Lagrangian immersions in affine symplectic four-space are derived together with their integrability equations. The invariant setup is applied to discuss the question of symplectic applicability for elliptic Lagrangian immersions. Explicit examples are considered

Quantization of the conformal arclength functional on space curves

By a conformal string in Euclidean space is meant a closed critical curve with non-constant conformal curvatures of the conformal arclength functional. We prove that (1) the set of conformal classes of conformal strings is in 1-1 correspondence with the rational points of the complex domain $\{q\in \mathbb{C} \,:\, 1/2 0,\,\, |q| < 1/\sqrt{2}\}$ and (2) any conformal class has a model conformal string, called symmetrical configuration, which is determined by three phenomenological invariants: the order of its symmetry group and its linking numbers with the two conformal circles representing the rotational axes of the symmetry group. This amounts to the quantization of closed trajectories of the contact dynamical system associated to the conformal arclength functional via Griffiths' formalism of the calculus of variations.Comment: 24 pages, 6 figures. v2: final version; minor changes in the exposition; references update

Tableaux over Lie algebras, integrable systems, and classical surface theory

Starting from suitable tableaux over finite dimensional Lie algebras, we provide a scheme for producing involutive linear Pfaffian systems related to various classes of submanifolds in homogeneous spaces which constitute integrable systems. These include isothermic surfaces, Willmore surfaces, and other classical soliton surfaces. Completely integrable equations such as the G/G_0-system of Terng and the curved flat system of Ferus-Pedit may be obtained as special cases of this construction. Some classes of surfaces in projective differential geometry whose Gauss-Codazzi equations are associated with tableaux over sl(4,R) are discussed.Comment: 16 pages, v3: final version; changes in the expositio

Deformation of Surfaces in Lie Sphere Geometry

The theory of surfaces in Euclidean space can be naturally formulated in the more general context of Legendre surfaces into the space of contact elements. We address the question of deformability of Legendre surfaces with respect to the symmetry group of Lie sphere contact transformations from the point of view of the deformation theory of submanifolds in homogeneous spaces. Necessary and sufficient conditions are provided for a Legendre surface to admit non-trivial deformations, and the corresponding existence problem is discusse