Isotropic curve flows on $R^{n+1, n}$

TL;DR

This paper studies isotropic curves in a specific vector space, establishing a natural parameter and moving frame, and links their differential invariants to integrable soliton hierarchies, providing a geometric understanding of these systems.

Contribution

It introduces a natural isotropic parameter and a moving frame for isotropic curves, connecting their invariants to KdV-type hierarchies and developing related soliton theory tools.

Findings

Differential invariants satisfy Drinfeld-Sokolov KdV hierarchies.

Constructed bi-Hamiltonian structures and conservation laws.

Developed Bäcklund transformations and permutability formulas.

Abstract

Let $R^{n+1, n}$ be the vector space $R^{2n+1}$ equipped with the bilinear form $(X,Y)=X^t C_n Y$ of index $n$, where $C_n= \sum_{i=1}^{2n+1} (-1)^{n+i-1} e_{i, 2n+2-i}$. A smooth $\gamma: R\to R^{n+1,n}$ is {\it isotropic} if $\gamma, \gamma_x, \ldots, \gamma_x^{(2n)}$ are linearly independent and the span of $\gamma, \ldots, \gamma_x^{(n-1)}$ is isotropic. Given an isotropic curve, we show that there is a unique up to translation parameter such that $(\gamma_x^{(n)}, \gamma_x^{(n)})=1$ (we call such parameter the isotropic parameter) and there also exists a natural moving frame. In this paper, we consider two sequences of curve flows on the space of isotropic curves parametrized by isotropic parameter. We show that differential invariants of these isotropic curves satisfy Drinfeld-Sokolov's KdV type soliton hierarchies associated to the affine Kac-Moody algebra $\hat B_n^{(1)}$ and $\hat A_{2n}^{(2)}$ Then we use techniques from soliton theory to construct bi-Hamiltonian structure, conservation laws, Backlund transformations and permutability formulas for these curve flows.