Integrable Systems in n-dimensional Riemannian Geometry

Details Integrable Systems in n-dimensional Riemannian Geometry Integrable Systems in n-dimensional Riemannian Geometry

2003-01-20

arxiv.org/abs/math/0301212v1

Mathematics

Analysis of PDEs

Scientific paper

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 in (\orth{n+1}) with the vector modified Korteweg-De Vries equation (vmKDV) \[ \vk{t}= \vk{xxx}+\fr32 ||\vk{}||^2 \vk{x} \] as integrability condition. We indicate that other integrable vector evolution equations can be found by using a different Ansatz on the form of the Lax pair. We obtain these results by using the {\em natural} or {\em parallel} frame and we show how this can be gauged by a generalized Hasimoto transformation to the (usual) {\em Fren{\^e}t} frame. If one chooses the curvature to be zero, as is usual in the context of integrable systems, then one loses information unless one works in the natural frame.

Affiliated with

Also associated with

No associations

Rating

Integrable Systems in n-dimensional Riemannian Geometry does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Integrable Systems in n-dimensional Riemannian Geometry, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Integrable Systems in n-dimensional Riemannian Geometry will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-307053