New integrable ( $$3+1$$ 3 + 1 )-dimensional systems and contact geometry

On Integrable Systems in 3-dimensional Riemannian Geometry

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 flow of curves in three dimensional Riemannian manifolds with constant curvature. The curves themselves are evolving following arc-length preserving geometric evolutions for which the variation of the curv...

Integrable Systems in n-dimensional Riemannian Geometry

In this paper we show that if one writes down the structure equations for the evolution of a curve embedded in an n-dimensional Riemannian manifold with constant curvature this leads to a symplectic, a Hamiltonian and an hereditary operator. This gives us a natural connection between finite dimensional geometry, infinite dimensional geometry and integrable systems. Moreover one finds a Lax pair...

Integrable systems in n-dimensional conformal geometry

3-dimensional Methods in Contact Geometry

Discrete Integrable Systems and Geometry